Modeling Wave Packet Dynamics and Exploring Applications: A Comprehensive Guide to the Nonlinear Schrödinger Equation

Abstract

The nonlinear Schrödinger (NLS) equation stands as a cornerstone model for exploring the intricate behavior of weakly nonlinear, quasi-monochromatic wave packets in dispersive media. Its reach extends across diverse physical domains, from surface gravity waves to the captivating realm of Bose–Einstein condensates. This article delves into the dual facets of the NLS equation: its capacity for modeling wave packet dynamics and its remarkable breadth of applications. We illuminate the derivation of the NLS equation through both heuristic and multiple-scale approaches, underscoring how distinct interpretations of physical variables and governing equations give rise to varied wave packet dynamics and tailored values for dispersive and nonlinear coefficients. To showcase its versatility, we present an overview of the NLS equation’s compelling applications in four research frontiers: nonlinear optics, surface gravity waves, superconductivity, and Bose–Einstein condensates. This exploration reveals the NLS equation as a powerful tool for unifying and understanding a vast spectrum of physical phenomena.

1. Introduction

From the graceful dance of light in a crystal to the turbulent flow of a river, a vast tapestry of natural and physical phenomena unfolds through the language of partial differential equations (PDEs). While linear PDEs, with their elegant principle of superposition, offer solutions in neat and tidy packages, the intricate dynamics of real-world systems often defy this simplicity. Enter nonlinear PDEs, powerful tools that capture the hidden complexities of phenomena as diverse as the growth of populations, the behavior of light, and the swirling galaxies beyond our planet. Spanning practically all areas of the natural sciences, the theory of PDEs, both linear and nonlinear, pulsates with activity, constantly evolving and pushing the boundaries of our understanding.

The legacy of these early insights continues to inspire physicists and mathematicians alike, as they delve deeper into the intricate tapestry woven by nonlinear PDEs, building upon the foundations laid by the study of wave propagation and the development of general nonlinear PDE theory. Today, researchers grapple with challenges like predicting the behavior of complex materials, deciphering the brain’s neural networks, and unlocking the secrets of the early universe, all fueled by the ever-expanding power of PDEs.

Born from the quest to understand the geometry of surfaces, the study of partial differential equations (PDEs) embarked on a remarkable journey. In its early days, it grappled with problems in mechanics, paving the way for the calculus of variations to bridge the gap between surface theory and physical phenomena. A pivotal turning point came with the study of wave propagation, triggering a surge in the development of general nonlinear PDE theory [1,2]. This fertile ground yielded iconic equations like the Korteweg–de Vries (KdV) equation, renowned for its elegant soliton solutions, and the nonlinear Schrödinger (NLS) equation, governing diverse systems from Bose–Einstein condensates to optical fibers. Today, the legacy of these early insights continues to inspire physicists and mathematicians alike, as they delve deeper into the intricate tapestry woven by nonlinear PDEs.

In the captivating realm of nonlinear waves, the cubic Schrödinger equation (CSE) stands out for its elegance and versatility. Defined in one space and one time dimension, the CSE models the dynamic evolution of weakly nonlinear wave packets in dispersive media. Its cubic nonlinearity, also known as the Gross–Pitaevskii equation in Bose–Einstein condensates, grants it the remarkable property of complete integrability. This means that solutions can be precisely analyzed using the inverse scattering transform, revealing a treasure trove of mathematical insights and connections to a broader wave theory [3,4,5]. Mastering the NLS equation, therefore, becomes fundamental not only for unraveling the complexities of nonlinear dispersive waves but also for unlocking valuable tools and knowledge applicable to diverse fields ranging from Bose–Einstein condensates to optical fibers.

While the seemingly innocuous removal of the nonlinear term in the NLS equation reveals the familiar face of the linear Schrödinger equation, this simplification belies a universe of hidden complexities. The Schrödinger equation, a cornerstone of quantum mechanics, governs the behavior of wavefunctions that define the state of a quantum system [6]. Its foundation lies in the Dirac–von Neumann axioms, where the mathematical machinery of quantum mechanics takes shape with operators in Hilbert space [7,8]. While delving deeper into the Schrödinger equation is readily available in numerous quantum mechanics textbooks [9,10,11], this article embarks on a different journey—into the heart of the NLS equation, where the dance between linearity and nonlinearity unlocks a vibrant tapestry of new phenomena and behaviors.

From the rhythmic dance of ocean waves to the enigmatic world of superfluids, the NLS equation stretches its tentacles across a breathtaking range of physical phenomena. It sculpts the mesmerizing patterns of surface gravity waves on oceans and lakes, guiding their majestic rise and fall. In the realm of solid-state physics, the NLS equation orchestrates the delicate dance of thermal pulses as they ripple through the crystalline lattice. Beyond these, it illuminates the intricate interactions in plasma physics and magnetohydrodynamics, unravels the mysteries of solitary wave propagation in semiconductors, and even sheds light on the complex dynamics of financial markets. While this article delves deeper into the captivating stories of nonlinear optics, water waves, superconductivity, and Bose–Einstein condensates, countless other exciting narratives await exploration under the umbrella of the NLS equation.

A wealth of research exists on the NLS equation and its diverse variations. The body of published literature offers abundant results, and the list is far from exhaustive. Notably, Sulem and Sulem offer a detailed exploration of this equation, focusing on the properties of soliton solutions and their applications in nonlinear physical models. Their monograph examines the cubic NLS equation with both attractive and repulsive nonlinearities, along with its generalization to other power-law forms. It also delves into the stability of soliton solutions, the phenomenon of wave collapse, and the theoretical and practical implications of NLS systems, both continuous and discrete [12].

In his encyclopedia article, Malomed provides an overview of the basic properties and solutions of NLS equations. He discusses their integrability through the inverse scattering transform (IST), the existence of soliton and continuous wave solutions, their stability, and their representation in Lagrangian and Hamiltonian forms. He also explores their applications in various physical systems, such as optical fibers, waveguides that lie in a plane, and plasma waves. Furthermore, Malomed examines the derivation of NLS equations as universal equations and provides insights into their generalized forms and integrability [13].

Ablowitz and Prinari provide an informative discussion of the NLS equation, including its historical background from Ginzburg–Landau equations for superconductivity in the 1950s and the focusing behavior of light beams in materials that respond nonlinearly to light in the 1960s [14]. Their follow-up article in an online encyclopedia explores the NLS equation’s rich history and diverse applications in physics. They delve into its derivation, universality, and integrability, analyzing both continuous and discrete NLS systems across various dimensions. The article showcases applications in fields like fluid dynamics, optics, plasma physics, and BECs, emphasizing its role in studying localized pulses and vector wave propagation. It details the mathematical framework, including the IST, making it a valuable resource for researchers and students in nonlinear physics [15].

A book by Linares and Ponce introduces the well-posedness theory for initial-value problems of nonlinear, dispersive partial differential equations, focusing on the KdV and NLS equations. It equips readers to grapple with the main topics covered by providing a concise treatment of background material. Each chapter concludes with an expert account of recent developments, open problems, and challenging exercises. The final chapter delves into local well-posedness for the NLS equation, propelling readers to the cutting edge of current research. This valuable resource is ideal for readers interested in the mathematics and applications of the NLS equation and related models [16].

Recently, Liu and Kengne thoroughly investigated various types of Schrödinger equations in nonlinear systems in their well-regarded book [17]. They discuss both the well-posedness and ill-posedness of boundary value problems (BVPs) for both linear Schrödinger and NLS equations, and further investigate dynamical properties of solitons in matter waves, relevant to problems in transmission networks and Bose–Einstein condensates (BECs). Beyond the focusing cubic NLS equation, research delves into various modifications and applications of this versatile equation. Studies have explored areas like compiling known solutions [18], investigations into their behavior when quantum numbers become large [19], understanding qualitative properties of fractional forms [20], investigating higher-order variations [21], and even exploring the NLS with point-concentrated nonlinearities [22]. While these exciting areas deserve further exploration, they fall outside the scope of this review.

Before concluding this introduction, it is worth mentioning that as a key player in diverse physical phenomena, the NLS equation is intricately linked to the AKNS hierarchy as a fundamental member. Named after its inventors Ablowitz, Kaup, Newell, and Segur, the AKNS hierarchy is a set of integrable PDEs that arise in the context of soliton theory and integrable systems [3]. It is characterized by its rich algebraic and geometric structure. The hierarchy has been extensively studied due to its significance in describing various physical phenomena, such as rogue waves and solitons. The NLS’s connection to the AKNS hierarchy not only sheds light on complex wave behaviors but also paves the way for a unified structure of rogue wave and multiple rogue wave solutions for all equations of the hierarchy, as explored in recent research [23,24,25].

This article embarks on an exciting exploration of the NLS equation, a versatile tool that unlocks the secrets of diverse wave phenomena. We begin by diving into its intricate relationship with light, where the NLS equation paints a vivid picture of its nonlinear dance within optical fibers (Section 2). From there, we journey deeper into the enchanting world of water waves, witnessing their elegant dance in shaping their mesmerizing patterns (Section 3). We derive both linear Schrödinger and NLS equations heuristically. By implementing the method of multiple-scale [26,27,28], it follows with the derivation of the temporal and spatial NLS equations. Next, we turn our gaze to the realm of superconductivity, where the NLS equation sheds light on the captivating behavior of exotic solitons (Section 4). Finally, we arrive at the fascinating frontier of BEC, where the NLS equation guides our understanding of these enigmatic superfluid states (Section 5). This journey culminates in a deeper appreciation of the NLS equation’s power and versatility, not just in explaining known phenomena, but also in opening doors to exciting scientific frontiers yet to be explored. Finally, Section 6 concludes our discussion and offers future implications of this review.

2. Nonlinear Optics

2.1. Deriving NLS Equation from Maxwell’s and Helmholtz’ Equations

Maxwell’s equations, also known as Maxwell–Heaviside equations, are a set of four fundamental equations that form the theoretical basis for classical electromagnetism. They concisely describe the behavior of electric and magnetic fields, as well as how they interact with each other and with charged particles. While Maxwell’s equations provide a comprehensive description of electromagnetism, Helmholtz’s equation offers a simpler framework applicable to specific cases and approximations. Helmholtz’s equation can be derived from Maxwell’s equations under specific assumptions, such as assuming monochromatic waves (single frequency) in a source-free region. This allows for exploring specific aspects of wave behavior in a simpler mathematical framework.

The derivation in this subsection follows the argument presented in [12,29,30]. See also [31,32,33]. In differential form and the International System of Units, Maxwell’s equations governing optical fields in fibers are written as follows:

$$\begin{array}{}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{(1)}& \hfill \hspace{1em}\nabla \times \mathbf{E}=& -{\partial }_t\mathbf{B},\hfill & (\mathrm{Faraday}\text{’}\mathrm{s}\mathrm{law}\mathrm{of}\mathrm{induction})\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{(2)}& \hfill \nabla \times \mathbf{H}=& \mathbf{J}+{\partial }_t\mathbf{D},\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hfill & (\text{Ampère’s}\mathrm{circuital}\mathrm{law}\mathrm{with}\mathrm{Maxwell}\text{’}\mathrm{s}\mathrm{addition})\hspace{1em}& \hfill \hspace{1em}\nabla ·\mathbf{D}=& \rho ,\hfill & (\mathrm{Gauss}\text{’}\mathrm{s}\mathrm{law}\mathrm{for}\mathrm{electricity})\hspace{1em}& \hspace{1em}\nabla ·\mathbf{B}=& 0.& (\mathrm{Gauss}\text{’}\mathrm{s}\mathrm{law}\mathrm{for}\mathrm{magnetism})\end{array}$$

Here, $\mathbf{E}$ and $\mathbf{H}$ denote electric and magnetic vector fields, respectively; $\mathbf{D}$ and $\mathbf{B}$ denote the corresponding electric and magnetic flux densities; $\rho$ is the charge density; and $\mathbf{J}$ is the corresponding current density of free charges, and both represent the sources for the electromagnetic field.

We also have the following constitutive relations for electric and magnetic flux densities:

$$\mathbf{D}={ϵ}_0\mathbf{E}+\mathbf{P},$$

(3)

$$\mathbf{B}={\mu }_0\mathbf{H}+\mathbf{M},$$

(4)

where ${ϵ}_0$ is the vacuum permittivity, ${\mu }_0$ is the vacuum permeability, and $\mathbf{P}$ and $\mathbf{M}$ are the induced electric and magnetic polarizations, respectively.

We will adopt the following common assumptions in nonlinear fiber optics: the absence of free charges ($\rho =0$ and $\mathbf{J}=\mathbf{0}$) and a nonmagnetic medium like fiber optics ($\mathbf{M}=\mathbf{0}$). By taking the curl of Faraday’s law of induction (1), using (2), (3), and (4), we can eliminate $\mathbf{B}$ and $\mathbf{D}$ to obtain an expression in $\mathbf{E}$ and $\mathbf{P}$:

$$\nabla \times \nabla \times \mathbf{E}+\frac{1}{c^2}{\partial }_t^2\mathbf{E}=-{\mu }_0{\partial }_t^2\mathbf{P},$$

(5)

where $1/c^2={\mu }_0{ϵ}_0$ is the speed of light in a vacuum.

We adopt a common relationship between the induced polarization $\mathbf{P}$ and the electric field $\mathbf{E}$, which is valid in the electric dipole approximation and under the assumption of local medium response. The induced polarization $\mathbf{P}$ is written as the combination of the linear and the nonlinear parts, ${\mathbf{P}}_L$ and ${\mathbf{P}}_{NL}$, respectively, that is, $\mathbf{P}(\mathbf{r},t)={\mathbf{P}}_L(\mathbf{r},t)+{\mathbf{P}}_{NL}(\mathbf{r},t)$, where

$$\begin{array}{cc}\hfill {\mathbf{P}}_L(\mathbf{r},t)& ={ϵ}_0{\int }_{-\infty }^t{\chi }^{\left(1\right)}(t-t_0)\mathbf{E}(\mathbf{r},t_0)\mathrm{d}{t}_0,\hfill \\ \hfill {\mathbf{P}}_{NL}(\mathbf{r},t)& ={ϵ}_0{\int }_{-\infty }^t{\int }_{-\infty }^t{\int }_{-\infty }^t{\chi }^{\left(3\right)}(t-t_1,t-t_2,t-t_3)\mathbf{E}(\mathbf{r},t_1)\mathbf{E}(\mathbf{r},t_2)\mathbf{E}(\mathbf{r},t_3)\mathrm{d}{t}_1\mathrm{d}{t}_2\mathrm{d}{t}_3,\hfill \end{array}$$

where ${\chi }^{\left(j\right)}$ is a tensor of rank $j+1$, the j-th order of susceptibility. Consider the case where ${\mathbf{P}}_{NL}=\mathbf{0}$. Let $\widehat{\mathbf{E}}(\mathbf{r},\omega )$ be the Fourier transform of $\mathbf{E}(\mathbf{r},t)$, defined as follows:

$$\widehat{\mathbf{E}}(\mathbf{r},\omega )={\int }_{-\infty }^{\infty }\mathbf{E}(\mathbf{r},t)e^{i\omega t}\mathrm{d}t,$$

and let ${\widehat{\chi }}^{\left(1\right)}\left(\omega \right)$ be the Fourier transform of the linear, first-order susceptibility of ${\chi }^{\left(1\right)}\left(t\right)$; then the curl Equation (5) in the frequency domain can be written as follows:

$$\nabla \times \nabla \times \widehat{\mathbf{E}}=ϵ\left(\omega \right)\frac{{\omega }^2}{c^2}\widehat{\mathbf{E}}(\mathbf{r},\omega ),$$

where $ϵ\left(\omega \right)=1+{\widehat{\chi }}^{\left(1\right)}\left(\omega \right)={\left(n+i\alpha c/\left(2\omega \right)\right)}^2$ is the frequency-dependent dielectric constant and its real and imaginary parts are related to the refractive index $n\left(\omega \right)$ and the absorption coefficient $\alpha \left(\omega \right)$.

Due to low optical loss in fibers within the wavelength region of interest, that is, Im$\left\{{\chi }^{\left(1\right)}\left(\omega \right)\right\}\ll 1$, and hence, $ϵ\left(\omega \right)\approx n^2\left(\omega \right)$. In addition, because usually $n\left(\omega \right)$ is independent of spatial coordinates, then $\nabla ·\mathbf{D}=ϵ\nabla ·\mathbf{E}=0$, and hence, $\nabla \times \nabla \times \mathbf{E}=\nabla (\nabla ·\mathbf{E})-{\nabla }^2\mathbf{E}=-{\nabla }^2\mathbf{E}$. Finally, we arrive at the Helmholtz equation:

$${\nabla }^2\widehat{\mathbf{E}}+n^2\left(\omega \right)\frac{{\omega }^2}{c^2}\widehat{\mathbf{E}}=0.$$

(6)

By including the nonlinear effect of the induced polarization, the Helmholtz Equation (6) can be written as follows:

$${\nabla }^2\widehat{\mathbf{E}}+ϵ\left(\omega \right)k_0^2\widehat{\mathbf{E}}=0,$$

(7)

where $k_0=\omega /c$ and the dielectric constant $ϵ\left(\omega \right)=1+{\widehat{\chi }}^{\left(1\right)}\left(\omega \right)+\frac{3}{4}\frac{d^4{\chi }^{\left(3\right)}}{d{x}^4}{\left|E(\mathbf{r},t)\right|}^2$. The Helmholtz Equation (7) can be solved using the method of separation of variables. We assume an ansatz in the following form:

$$\widehat{E}(\mathbf{r},\omega -{\omega }_0)=\widehat{A}(z,\omega -{\omega }_0)B(x,y)e^{i{\beta }_{0}z},$$

(8)

where $\widehat{A}(z,\omega )$ is a slowly varying function of z and ${\beta }_0$ is a wavenumber that needs to be determined. The Helmholtz Equation (7) leads to the following equations for $\widehat{A}(z,\omega )$ and for $B(x,y)$, where we have neglected ${\partial }_z^2\widehat{A}$ due to an assumption of a slowly varying function in z:

$$\begin{array}{cc}\hfill 2i{\beta }_0{\partial }_z\widehat{A}+{({\widehat{\beta }}^2-{\beta }_0^2)}^2\widehat{A}& =0,\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill {\nabla }^{2}B+\left[ϵ\left(\omega \right)k_0^2-{\widehat{\beta }}^2\right]B& =0.\hfill \end{array}$$

(10)

The wavenumber $\widehat{\beta }$ is determined by solving the eigenvalue Equation (10) using the first-order perturbation theory. We obtain

$$\widehat{\beta }\left(\omega \right)=\beta \left(\omega \right)+\Delta \beta ,$$

(11)

where

$$\Delta \beta =\frac{{\omega }^{2}n\left(\omega \right)}{c^2\beta \left(\omega \right)}\frac{{\displaystyle {\int }_{\infty }^{\infty }{\int }_{-\infty }^{\infty }\Delta n\left(\omega \right){\left|B(x,y)\right|}^2\mathrm{d}x\mathrm{d}y}}{{\displaystyle {\int }_{\infty }^{\infty }{\int }_{-\infty }^{\infty }{\left|B(x,y)\right|}^2\mathrm{d}x\mathrm{d}y}}.$$

Using (11) and approximating ${\widehat{\beta }}^2-{\beta }_0^2\approx 2{\beta }_0(\widehat{\beta }-{\beta }_0)$, the Fourier transform $\widehat{A}(z,\omega -{\omega }_0)$ satisfying (9) can be written as follows:

$${\partial }_z\widehat{A}=i\left[\beta \left(\omega \right)+\Delta \beta \left(\omega \right)-{\beta }_0\right]\widehat{A}.$$

Because an exact form of the propagation constant $\beta \left(\omega \right)$ is rarely known, it is beneficial to expand both $\beta \left(\omega \right)$ and $\Delta \beta \left(\omega \right)$ in a Taylor series about the carrier frequency ${\omega }_0$:

$$\begin{array}{cc}\hfill \beta \left(\omega \right)& =\beta \left({\omega }_0\right)+{\beta }^{\prime }\left({\omega }_0\right)(\omega -{\omega }_0)+\frac{1}{2}{\beta }^{″}\left({\omega }_0\right){(\omega -{\omega }_0)}^2+\cdots ,\hfill \\ \hfill \Delta \beta \left(\omega \right)& =\Delta \beta \left({\omega }_0\right)+\Delta {\beta }^{\prime }\left({\omega }_0\right)(\omega -{\omega }_0)+\frac{1}{2}\Delta {\beta }^{″}\left({\omega }_0\right){(\omega -{\omega }_0)}^2+\cdots .\hfill \end{array}$$

Replacing $\omega -{\omega }_0$ with the differential operator $i{\partial }_t$ and taking back the inverse Fourier transform of $\widehat{A}(z,\omega -{\omega }_0)$, we obtain the following equation for $A(z,t)$:

$$i{\partial }_{z}A+i{\beta }^{\prime }\left({\omega }_0\right){\partial }_{t}A-\frac{1}{2}{\beta }^{″}\left({\omega }_0\right){\partial }_t^{2}A+\Delta \beta \left({\omega }_0\right)A=0.$$

(12)

Using the transformation of a moving frame of reference $T=t-{\beta }^{\prime }\left({\omega }_0\right)z$ and considering that the last term of (12) contains the fiber loss and nonlinearity effects, we obtain the NLS equation:

$$i{\partial }_{z}A+\beta {\partial }_T^{2}A+\gamma {\left|A\right|}^{2}A=0.$$

(13)

The dispersive coefficient $\beta$ and the nonlinear coefficient $\gamma$ are given as follows, respectively:

$$\begin{array}{cc}\hfill \beta & =\beta \left({\omega }_0\right)=-\frac{1}{2}{\beta }^{″}\left({\omega }_0\right),\hfill \\ \hfill \gamma & =\gamma \left({\omega }_0\right)=-\frac{{\omega }_0}{c}{n}_2\left({\omega }_0\right)\frac{{\displaystyle {\int }_{\infty }^{\infty }{\int }_{-\infty }^{\infty }{\left|B(x,y)\right|}^4\mathrm{d}x\mathrm{d}y}}{{\displaystyle {\left({\int }_{\infty }^{\infty }{\int }_{-\infty }^{\infty }{\left|B(x,y)\right|}^2\mathrm{d}x\mathrm{d}y\right)}^2}}.\hfill \end{array}$$

For a single-mode fiber, the modal distribution $B(x,y)$ corresponds to the fundamental fiber mode, given by one of the following expressions:

$$\begin{array}{cc}\hfill B(x,y)& =\left\{\begin{array}{cc}{J}_0\left(p\sqrt{x^2+y^2}\right),\hfill & \sqrt{x^2+y^2}\le a,\hfill \\ \frac{\sqrt{a}}{\sqrt[4]{x^2+y^2}}{J}_0\left(pa\right)e^{-q(\sqrt{x^2+y^2}-a)},\hfill & \sqrt{x^2+y^2}\ge a,\hfill \end{array}\right.\hfill \\ \hfill \mathrm{or}B(x,y)& =e^{-\frac{(x^2+y^2)}{w^2}}.\hfill \end{array}$$

Here, $J_0$ denotes the Bessel function of the first kind of order zero, a is the radius of the fiber core, w is a width parameter, and the quantities $p=\sqrt{n_1^2{k}_0^2-{\beta }^2}$ and $q=\sqrt{{\beta }^2-n_c^2{k}_0^2}$.

2.2. Applications in Nonlinear Optics

In the context of nonlinear optics, an object of interest to study is the “soliton,” a solitary wave that maintains its shape during propagation. Depending on whether the light confinement occurs in time or space during wave propagation, solitons can be classified into two main types: temporal and spatial. On the one hand, the NLS Equation (13) governs the time-dependent pulse envelope propagation in optical fibers, which leads to temporal solitons. On the other hand, spatial solitons, continuous wave beams propagating inside a nonlinear optical medium with Kerr (or cubic) nonlinearity, are governed by the following (1 + 1)D NLS equation:

$$i{\partial }_{z}A+\beta {\partial }_X^{2}A+\gamma {\left|A\right|}^{2}A=0.$$

(14)

A classical example of the latter is a bell-shaped spatial wave packet with the self-induced lensing effect, a phenomenon of self-trapping in dielectric waveguide modes discovered by Chiao and his collaborators [34]. Furthermore, a stable spatial soliton was also observed experimentally with self-trapping laser beams propagating through homogeneous transparent dielectrics [35]. For extensive coverage of spatial solitons, please consult [36].

Next, we delve into the fascinating world of temporal solitons within optical fibers. These flexible and transparent strands, manufactured from highly purified silica glass or plastic, serve as light highways for communication. Their inner core is surrounded by a cladding—an outer layer of optical material with a lower refractive index, guiding the light within. For added protection from moisture and physical damage, a buffer and a jacket encase the fiber. Crucially, total internal reflection keeps the light confined within the core, transforming the fiber into a waveguide. Additionally, it is here that temporal solitons play a vital role, maintaining their shape as they pulse through the fiber, revolutionizing the transmission of information across vast distances.

An interaction between fiber dispersion and nonlinearity gives rise to fiber optic solitons. In this process of self-phase modulation, an ultrashort light pulse causes a varying refractive index due to the Kerr effect, balancing the dispersion with a nonlinearity effect to create an optical soliton. The existence of temporal optical solitons in optical fibers was predicted theoretically in 1973 [37,38] and confirmed experimentally in 1980 [39]. These solitons are optical pulses that maintain their shape during propagation and belong to a group of exact solutions of the NLS equation.

Nonlinear fiber optics continues to blossom, thanks to the development of erbium-doped fiber amplifiers [40,41]. Since the turn of the 21st century, new types of fiber optic amplifiers exploiting nonlinear effects, including stimulated Raman scattering and four-wave mixing, have been developed [42,43]. These advancements paved the way for diverse soliton types like dispersion-managed and dissipative solitons [30,44,45,46,47]. For further exploration of theoretical and experimental challenges in optical solitons, readers can consult a dedicated volume edited by Porsezian and collaborators [48].

Furthermore, Lembrikov edited a book discussing the significance of nonlinear optics in modern physics. It goes beyond highlighting various nonlinear optical phenomena, such as self-focusing, soliton formation, and four-wave mixing, by delving into the mathematical analysis of these phenomena. Additionally, the book offers novel results in areas like soliton formation and propagation in optical fibers, exploring the unique complexities of nonlinear optical phenomena in micro- and nanostructures [49].

Recently, there has been significant interest in nematicons, known as spatial optical solitons in liquid crystals, thanks to their potential applications in optical information processing. To enhance the understanding of light localization in nematicons, Altawallbeh et al. present a new generalized, time-conformable NLS equation for nematicons. They use a specific analytical approach called the generalized Riccati simple equation method, along with a new type of nonlinearity (namely, the nonlinear quadruple power law), to derive bright, dark, and singular soliton solutions [50].

Equally interesting is the higher-order NLS equation and its applications in nonlinear optics. For example, Az-Zo’bi et al. examine the generalized higher-order cubic-quintic-septic NLS equation, which includes an eight-order dispersion term, employing two techniques, that is, the generalized Riccati simplest equation method (GRSEM) and the modified simple equation method (MSEM). They find a variety of optical soliton solutions, including bright, dark, and singular types, subject to certain conditions [51]. Furthermore, Az-Zo’bi et al. also investigate the conformable generalized Kudryashov equation, describing pulse propagation with power nonlinearity, a higher-order equation modeling diverse wave propagation phenomena in nonlinear media. Using the generalized Riccati equation mapping method, the study obtains various types of optical solitons, including kinks, bisymmetric, periodic, singular, bright, and dark solitons [52].

As briefly mentioned in the introduction, models involving fractional derivatives are active areas of research in nonlinear optics. One important example is the Fokas–Lenells (FL) equation, which describes the propagation of ultrashort optical pulses passing through single-mode optical fibers. Its relevance lies in capturing nonlinear effects beyond the standard cubic NLS model, making it valuable for understanding the dynamics of these pulses within optical fiber systems. Similar to the relationship between the Camassa–Holm and KdV equations, the FL equation is completely integrable and has been extensively studied through analytical and theoretical methods to understand its properties and behavior [53,54,55].

The FL equation has been comprehensively investigated, with studies focusing on finding various types of solutions, exact solutions, and exploring dispersive optical solitons. One example is Al-Askar’s work, employing the modified mapping method to derive novel elliptic, hyperbolic, rational, and trigonometric solutions for the equation with truncated M- and $\beta$-derivatives [56]. Additionally, Muhammad et al. delve into the complexities of the FL equation, utilizing an extended algebraic technique to find new wave solutions with nonlinear properties and analyze their propagation and stability. Notably, they introduce a disturbance to the equation and observe that the system exhibits chaotic dynamics and a strong dependence on initial conditions [57].

Another example is the Chen–Lee–Liu model, a variant of the NLS equation, which Nurul Islam et al. investigate in the context of optical fiber communication systems. The paper utilizes the increasingly popular fractional $\beta$-derivative, capable of modeling phenomena beyond the scope of integer-order derivatives. Using a sophisticated analytical technique called the generalized $\left(G^{\prime }/G\right)$ expansion approach, the authors present various soliton solutions [58].

Beyond the well-known bright, dark, and gray solitons, “optical rogue waves” have attracted significant attention in nonlinear optics over the past decade. Supporting this surge of interest, numerical simulations based on probabilistic supercontinuum in highly nonlinear microstructured optical fibers and a generalized NLS model were conducted [59,60,61]. A comprehensive overview of the research dynamics and state-of-the-art in optical rogue wave science can be found in [62]. For insights into the ongoing debate on whether rogue wave science is moving towards a unifying concept, please consult the papers published by various authors in [63].

3. Hydrodynamics

3.1. Heuristic Derivation for the NLS Equation

In simple terms, a heuristic is a mental shortcut or technique used to solve problems, make decisions, and learn in situations where strict rules or complete information is unavailable. In the context of scientific research and problem-solving, a “heuristic derivation” refers to a process of deriving an equation or model using intuitive reasoning, approximations, and practical considerations. This contrasts with “rigorous derivation”, which relies on exact mathematical principles and proofs. Using intuitive reasoning and approximation adapted from the monographs written by Debnath and Dingemans, the following heuristic derivation demonstrates how we arrive at both the linear Schrödinger and NLS equations [64,65].

For a linear dispersive wave equation, we can express its general solution in terms of a Fourier transform representation. Suppose we have the following linear equation governing the evolution of the surface elevation $\eta (x,t)$:

$${\partial }_t\eta +i\Omega (-i{\partial }_x)\eta =0,$$

(15)

then the general solution of the governing Equation (15) can be expressed by a Fourier representation. It is given by

$$\eta (x,t)=\frac{1}{2\pi }{\int }_{-\infty }^{\infty }F\left(\zeta \right)e^{i(kx-\omega t)}\mathrm{d}\zeta .$$

(16)

In this expression (16), we can replace the variable $\zeta$ either with the wavenumber k or with the frequency $\omega$, and both are related by the linear dispersion relationship $\omega =\Omega \left(k\right)$ or $k=K\left(\omega \right)$, where $K={\Omega }^{-1}$. The spectrum function $F\left(\zeta \right)$ will be determined from a given initial or boundary condition. For an initial value problem (IVP), $F\left(k\right)$ is the Fourier transform of the initial condition $\eta (x,0)$. Correspondingly, the Fourier transform of the initial signal $\eta (0,t)$ is given by $F\left(\omega \right)$ in the case of BVP. Both quantities are given as follows, respectively:

$$\begin{array}{cc}\hfill F\left(k\right)& ={\int }_{-\infty }^{\infty }\eta (x,0)e^{-ikx}\mathrm{d}x,\hfill \\ \hfill F\left(\omega \right)& ={\int }_{-\infty }^{\infty }\eta (0,t)e^{i\omega t}\mathrm{d}t.\hfill \end{array}$$

We adopt an assumption of a slowly modulated wave as it propagates in a dispersive medium, and hence, $F\left(k\right)$ and $F\left(\omega \right)$ are narrow-banded spectra around $k_0$ and ${\omega }_0$, respectively. The linear dispersion relationship can be expressed in its Taylor expansion series about the basic state wavenumber $k_0$ and frequency ${\omega }_0$, written as follows:

$$\begin{array}{cc}\hfill \Omega \left(k\right)& ={\omega }_0+{\Omega }^{\prime }\left(k_0\right)(k-k_0)+\frac{1}{2!}{\Omega }^{″}\left(k_0\right){(k-k_0)}^2+\cdots ,\hfill \\ \hfill K\left(\omega \right)& =k_0+K^{\prime }\left({\omega }_0\right)(\omega -{\omega }_0)+\frac{1}{2!}{K}^{″}\left({\omega }_0\right){(\omega -{\omega }_0)}^2+\cdots ,\hfill \end{array}$$

where primes denote the derivatives with respect to the associated variable. Therefore, we can rewrite the surface elevation (16) as $\eta (x,t)=A(\xi ,\tau )e^{i(k_{0}x-{\omega }_{0}t)}$, where $A(\xi ,\tau )$ is the corresponding complex-valued amplitude of the wave packet $\eta (x,t)$, written in two different versions as follows:

$$\begin{array}{cc}\hfill A({\xi }_1,{\tau }_1)& =\frac{1}{2\pi }{\int }_{-\infty }^{\infty }F(k_0+\kappa )e^{i({\xi }_1-{\Omega }_{res}\left(k_0\right){\tau }_1/{\kappa }^2)}\mathrm{d}\kappa ,\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill A({\xi }_2,{\tau }_2)& =\frac{1}{2\pi }{\int }_{-\infty }^{\infty }F({\omega }_0+\nu )e^{-i({\tau }_2-K_{res}\left({\omega }_0\right){\xi }_2/{\nu }^2)}\mathrm{d}\nu .\hfill \end{array}$$

(18)

Here, $\kappa =k-k_0=\mathcal{O}\left(ϵ\right)$, ${\xi }_1=\kappa (x-{\Omega }^{\prime }\left(k_0\right)t)$, ${\tau }_1={\kappa }^{2}t$, $\nu =\omega -{\omega }_0=\mathcal{O}\left(ϵ\right)$, ${\xi }_2={\nu }^{2}x$, $\tau =\nu (t-K^{\prime }\left({\omega }_0\right)x)$, and $0<ϵ\ll 1$ is a small positive parameter. The residual terms appearing in the exponential term read as follows:

$$\begin{array}{cc}\hfill {\Omega }_{res}\left(k_0\right)& =\Omega \left(k\right)-[{\omega }_0+{\Omega }^{\prime }\left(k_0\right)\kappa ]={\kappa }^2\left(\frac{1}{2!}{\Omega }^{″}\left(k_0\right)+\frac{1}{3!}{\Omega }^{‴}\left(k_0\right)\kappa +\cdots \right),\hfill \\ \hfill K_{res}\left({\omega }_0\right)& =K\left(\omega \right)-[k_0+K^{\prime }\left({\omega }_0\right)\nu ]={\nu }^2\left(\frac{1}{2!}{K}^{″}\left({\omega }_0\right)+\frac{1}{3!}{K}^{‴}\left({\omega }_0\right)\nu +\cdots \right).\hfill \end{array}$$

From the complex amplitude representations (17) and (18), it follows that $\kappa$ (respectively $\nu$) is associated with the differential operator $i{\partial }_{\xi }$ (respectively $-i{\partial }_{\tau }$), and thus, ${\kappa }^2=-{\partial }_{\xi }^2$ (respectively ${\nu }^2=-{\partial }_{\tau }^2$). Thus, the complex-valued amplitude A satisfies

$$\begin{array}{cc}\hfill {\partial }_{\tau }A+i{\Omega }_{res}\left(i{\partial }_{\xi }\right)A& =0,\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill {\partial }_{\xi }A+i{K}_{res}(-i{\partial }_{\tau })A& =0.\hfill \end{array}$$

(20)

For a narrow-banded spectrum, the evolution Equations (19) and (20) reduce to approximate equations called the temporal and the spatial “linear Schrödinger” equations by approximating ${\Omega }_{res}$ and $K_{res}$ in their lowest-order terms, respectively. They are given as follows:

$$\begin{array}{cc}\hfill i{\partial }_{\tau }A+{\beta }_1{\partial }_{\xi }^{2}A& =0,\hfill \end{array}$$

(21)

$$\begin{array}{cc}\hfill i{\partial }_{\xi }A+{\beta }_2{\partial }_{\tau }^{2}A& =0.\hfill \end{array}$$

(22)

In (21) and (22), ${\beta }_1$ and ${\beta }_2$ are the dispersion coefficients. They are given by

$${\beta }_1=\frac{1}{2}{\Omega }^{″}\left(k_0\right),\mathrm{and}{\beta }_2=-\frac{1}{2}{K}^{″}\left({\omega }_0\right)=\frac{1}{2}\frac{{\Omega }^{″}\left(k_0\right)}{{\left[{\Omega }^{\prime }\left(k_0\right)\right]}^3}.$$

Before proceeding to derive the NLS equation heuristically, we briefly discuss the local existence and uniqueness for the IVP of the temporal linear Schrödinger Equation (21). Dispersion is a characteristic of this equation, meaning that the group velocity, denoted by ${\Omega }^{\prime }\left(k\right)$, depends on the wavenumber k. Consequently, waves with different frequencies travel at different speeds. As a result, a traveling and localized wave packet will eventually disperse.

Definition 1

(Fundamental solution). The function

$$\Psi (\xi ,\tau ):=\frac{1}{\sqrt{4\pi i{\beta }_1\tau }}{e}^{\frac{{i\left|\xi \right|}^2}{4{\beta }_1\tau }},\xi \in \mathbb{R},\tau \ne 0.$$

(23)

is called the fundamental solution of the linear Schrödinger Equation (21).

Theorem 1.

The corresponding initial value problem linear Schrödinger Equation (21) with an initial condition $A(\xi ,0)$ admits an exact solution using the fundamental solution (23), given explicitly as follows:

$$A(\xi ,\tau )=\frac{1}{\sqrt{4\pi i{\beta }_1\tau }}{\int }_{-\infty }^{\infty }{e}^{-\frac{{(\xi -\zeta )}^2}{4i{\beta }_1\tau }}A(\zeta ,0)\mathrm{d}\zeta ,\xi \in \mathbb{R},\tau >0.$$

The proof of this theorem can be found in [66,67] using the Fourier transform method, and we have adopted the following convention for the Fourier transform definition in this article.

Definition 2

(Fourier transform). Let f be a defined function on $\xi \in \mathbb{R}$; then the Fourier transform of f and its inverse Fourier transform are given as follows:

$$\begin{array}{cc}\hfill \widehat{f}\left(\kappa \right)=\mathcal{F}\left\{f\left(\xi \right)\right\}& =\frac{1}{\sqrt{2\pi }}{\int }_{-\infty }^{\infty }f\left(\xi \right)e^{-i\kappa \xi }\mathrm{d}\xi \hfill \\ \hfill f\left(\xi \right)={\mathcal{F}}^{-1}\left\{\widehat{f}\left(\kappa \right)\right\}& =\frac{1}{\sqrt{2\pi }}{\int }_{-\infty }^{\infty }\widehat{f}\left(\kappa \right)e^{i\kappa \xi }\mathrm{d}\kappa .\hfill \end{array}$$

Proof. 

Let $\widehat{A}(\kappa ,\tau )=\mathcal{F}\left\{A(\xi ,\tau )\right\}$ be the Fourier transform of the complex-valued amplitude $A(\xi ,\tau )$. Taking the Fourier transform of the linear Schrödinger Equation (21) and its initial condition yields the following, respectively:

$$i{\partial }_{\tau }\widehat{A}-{\beta }_1{\kappa }^2\widehat{A}=0,\mathrm{and}\widehat{A}(\kappa ,0).$$

(24)

The solution to this IVP (24) is given by

$$\widehat{A}(\kappa ,\tau )=\widehat{A}(\kappa ,0)e^{-i{\beta }_1{\kappa }^2\tau }.$$

(25)

Taking the inverse of the Fourier transform (25), we obtain the solution of the linear Schrödinger Equation (21) in the convolution form

$$A(\xi ,\tau )=\frac{1}{\sqrt{2\pi }}A(\xi ,0)\ast {\mathcal{F}}^{-1}\left\{e^{-i{\beta }_1{\kappa }^2\tau }\right\}.$$

Using the fact that the Fourier transform of a Gaussian shape function is also a Gaussian shape profile, that is,

$$\mathcal{F}\left\{e^{-\frac{p}{2}{\xi }^2}\right\}=\frac{1}{\sqrt{p}}{e}^{-\frac{{\kappa }^2}{2p}},\mathrm{and}{\mathcal{F}}^{-1}\left\{e^{-i{\beta }_1{\kappa }^2\tau }\right\}=\frac{1}{\sqrt{2i{\beta }_1\tau }}{e}^{-\frac{{\xi }^2}{4i{\beta }_1\tau }},$$

with $p=1/\left(2i{\beta }_1\tau \right)$, we then obtain the desired expression

$$A(\xi ,\tau )=\frac{1}{\sqrt{4\pi i{\beta }_1\tau }}{\int }_{-\infty }^{\infty }{e}^{-\frac{{(\xi -\zeta )}^2}{4i{\beta }_1\tau }}A(\zeta ,0)\mathrm{d}\zeta ,\xi \in \mathbb{R},\tau >0.$$

This completes the proof. □

This solution does make sense for all positive time $t>0$ provided that the initial condition $A(\xi ,0)\in L^1\left(\mathbb{R}\right)$. From this expression, we obtain the following basic $L^{\infty }$-estimate [68,69,70]:

$${∥A(\xi ,\tau )∥}_{L^{\infty }\left(\mathbb{R}\right)}=\underset{\xi \in \mathbb{R}}{sup}\left|A(\xi ,\tau )\right|\le \frac{1}{\sqrt{4\pi {\beta }_1\tau }}{\int }_{-\infty }^{\infty }\left|A(\xi ,0)\right|\mathrm{d}\xi =\frac{1}{\sqrt{4\pi {\beta }_1\tau }}{∥A(\xi ,0)∥}_{L^1\left(\mathbb{R}\right)}.$$

This indicates that any solution with spatially localized initial conditions will decay uniformly toward zero with a rate of $\sqrt{\tau }$ [71]. Furthermore, using Plancherel’s theorem, if the initial condition $A(\xi ,0)\in L^1\left(\mathbb{R}\right)\cap L^2\left(\mathbb{R}\right)$, then the $L^2$-norm is preserved:

$${∥A(\xi ,\tau )∥}_{L^2\left(\mathbb{R}\right)}={∥A(\xi ,0)∥}_{L^2\left(\mathbb{R}\right)},\forall t>0.$$

In the following, we will derive the NLS equation from the nonlinear dispersion relationship that includes a slowly varying real-valued amplitude $a(x,t)\sim \mathcal{O}\left(ϵ\right)$:

$$\omega =\Omega (k,a^2)\mathrm{or}k=K(\omega ,a^2).$$

(26)

We perform a Taylor series expansion of (26) about the basic state wavenumber $k=k_0$ (or the basic state frequency $\omega ={\omega }_0$) and the zero amplitude ${\left|a\right|}^2=0$:

$$\begin{array}{cc}\hfill \omega & =\Omega (k_0,0)+\frac{\partial \Omega }{\partial k}(k_0,0)(k-k_0)+\frac{{\partial }^2\Omega }{{\partial \left|a\right|}^2}(k_0,0){\left|a\right|}^2+\frac{1}{2}\frac{{\partial }^2\Omega }{\partial k^2}(k_0,0){(k-k_0)}^2\hfill \\ \hfill & +\frac{{\partial }^2\Omega }{{\partial k\partial \left|a\right|}^2}(k_0,0)(k-k_0){\left|a\right|}^2+\frac{1}{2}\frac{{\partial }^2\Omega }{{\partial \left|a\right|}^4}(k_0,0){\left|a\right|}^4+\cdots .\hfill \end{array}$$

(27)

Rewriting and considering only the essential terms in (27), we obtain

$$\nu -{\Omega }^{\prime }\left(k_0\right)\kappa -\frac{1}{2}{\Omega }^{″}\left(k_0\right){\kappa }^2-{ϵ}^2\frac{{\partial }^2\Omega }{{\partial \left|A\right|}^2}(k_0,0){\left|A\right|}^2-\cdots =0.$$

(28)

Associating $\nu$ and $\kappa$ in (28) with differential operators $i{\partial }_t$ and $-i{\partial }_x$, respectively, and letting them operate on the complex-valued amplitude A, it yields the following nonlinear evolution equation:

$$i({\partial }_t+{\Omega }^{\prime }\left(k_0\right){\partial }_x)A+\frac{1}{2}{\Omega }^{″}\left(k_0\right){\partial }_x^{2}A-{ϵ}^2\frac{{\partial }^2\Omega }{{\partial \left|a\right|}^2}(k_0,0){\left|A\right|}^{2}A=0.$$

(29)

Introducing the slowly moving coordinate $\xi =ϵ(x-{\Omega }^{\prime }\left(k_0\right)t)$ and slower time variable $\tau ={ϵ}^{2}t$ to (29), we obtain the temporal NLS equation:

$$i{\partial }_{\tau }A+{\beta }_1{\partial }_{\xi }^{2}A+{\gamma }_1{\left|A\right|}^{2}A=0,$$

(30)

where the dispersion and the nonlinear coefficients are respectively given by

$${\beta }_1=\frac{1}{2}{\Omega }^{″}\left(k_0\right),\mathrm{and}{\gamma }_1=-\frac{{\partial }^2\Omega }{{\partial \left|A\right|}^2}(k_0,0).$$

For the temporal NLS Equation (30), the following result on the local existence and uniqueness solution for an initial value problem in Sobolev spaces is known in the literature. A detailed proof of the following theorem can be found in [71,72].

Theorem 2.

Let $m\ge 1$; let $H^m\left(\mathbb{R}\right)$ be the Sobolev space, the space of m times weakly differentiable functions $u:\mathbb{R}\to \mathbb{R}$ with derivatives in $L^2$-space for $j\in \{0,1,\cdots ,m\}$. Let the space $H^m$ be equipped with the norm

$${\parallel u\parallel }_{H^m}=\underset{j\in \{0,1,\cdots ,m\}}{max}{\parallel {\partial }_{\xi }^{j}u\parallel }_{L^2}.$$

Let $A_0\in H^m\left(\mathbb{R}\right)$ be a complex-valued function. Then there exists a time ${\tau }_0={\tau }_0\left(\parallel A_0{\parallel }_{H^m}\right)>0$ and a unique solution $A\in \mathbb{C}\left(\left[0,{\tau }_0\right],H^m\right)$ of the temporal NLS Equation (30) with the initial condition $A_0$.

Readers who are interested in the well-posedness of the Cauchy problem and the long-time behavior of the corresponding global solutions to the NLS equation may consult [68,69,73,74,75].

Similarly, by writing out the Taylor series expansion in two variables for $k=K(\omega ,a^2)$ and retaining only the essential terms, we acquire

$$k=K({\omega }_0,0)+\frac{\partial K}{\partial \omega }({\omega }_0,0)(\omega -{\omega }_0)+\frac{1}{2}\frac{{\partial }^{2}K}{\partial {\omega }^2}({\omega }_0,0){(\omega -{\omega }_0)}^2+\frac{{\partial }^2\Omega }{{\partial \left|a\right|}^2}({\omega }_0,0){\left|a\right|}^2+\cdots .$$

(31)

Rearranging the terms to the left-hand side of (31) yields

$$\kappa -K^{\prime }\left({\omega }_0\right)\nu -\frac{1}{2}{K}^{″}\left({\omega }_0\right){\nu }^2-\frac{{\partial }^2\Omega }{{\partial \left|a\right|}^2}({\omega }_0,0){\left|a\right|}^2-\cdots =0.$$

(32)

By associating the parameters $\kappa$ and $\nu$ in (32) with the differential operators $-i{\partial }_x$ and $i{\partial }_t$, respectively, and let them act on the complex-valued amplitude A, we obtain another, slightly different, nonlinear evolution equation:

$$i\left({\partial }_x+K^{\prime }\left({\omega }_0\right){\partial }_t\right)A-\frac{1}{2}{K}^{″}\left({\omega }_0\right){\partial }_t^{2}A+ϵ\frac{{\partial }^2\Omega }{{\partial \left|A\right|}^2}({\omega }_0,0){\left|A\right|}^{2}A=0.$$

(33)

Introducing the slowly moving variables $\xi ={ϵ}^{2}x$ and $\tau =ϵ(t-K^{\prime }\left({\omega }_0\right)x)$ to (33), we obtain the spatial NLS equation:

$$i{\partial }_{\xi }A+{\beta }_2{\partial }_{\tau }^{2}A+{\gamma }_2{\left|A\right|}^{2}A=0,$$

where the dispersion and the nonlinear coefficients are given as follows, respectively:

$${\beta }_2=-\frac{1}{2}{K}^{″}\left({\omega }_0\right)\mathrm{and}{\gamma }_2=\frac{{\partial }^2\Omega }{{\partial \left|A\right|}^2}({\omega }_0,0).$$

3.2. Derivation of the Temporal NLS Equation

The following derivation, as well as the derivation for the spatial NLS equation in Section 3.3, follows the argument presented in [76,77]. Consider a KdV type of equation with an exact dispersion relationship property, given as follows:

$${\partial }_t\eta +i\Omega (-i{\partial }_x)\eta +c\eta {\partial }_x\eta =0,$$

(34)

where $c\in \mathbb{R}$ is a nonlinear coefficient of the KdV Equation (34). We will observe that it also contributes to the nonlinear coefficient of the NLS equation. The function $\Omega$ acts as both a differential operator and a dispersion relationship.

We seek a solution for the surface wave elevation $\eta (x,t)$ in the form of a wave packet, or a wave group. This wave packet consists of a superposition of the first-order harmonic wave, the second-order double harmonic wave, and the second-order nonharmonic long wave, explicitly given as follows:

$$\eta (x,t)=ϵA(\xi ,\tau )e^{i\theta }+{ϵ}^2[B(\xi ,\tau )e^{2i\theta }+C(\xi ,\tau )]+\mathrm{c}.\mathrm{c}.,$$

(35)

where $0<ϵ\ll 1$ is a small positive parameter as used commonly in perturbation theory. The term in the exponent is $\theta (x,t)=k_{0}x-{\omega }_{0}t$, where $k_0$ and ${\omega }_0$ are related by the linear dispersion relation. The functions $A(\xi ,\tau )$, $B(\xi ,\tau )$, and $C(\xi ,\tau )$ are complex-valued wave packet envelopes, and they are allowed to vary slowly in the slower-moving frame of reference, where the spatial and temporal variables are given by $\xi =ϵ(x-{\Omega }^{\prime }\left(k_0\right)t)$ and $\tau ={ϵ}^{2}t$, respectively. As usual, c.c. denotes the complex conjugation of the preceding terms.

Substituting the ansatz (35) into the KdV Equation (34) yields the residue $R(x,t)$ that can be expressed in the following form:

$$R(x,t)=\sum _{n,m}{ϵ}^n{R}_{nm}{e}^{im\theta }+\mathrm{c}.\mathrm{c}.,$$

(36)

where $n\ge 1$, $m\ge 0$, and the coefficients $R_{nm}$ in (36) contain expressions in $A(\xi ,\tau )$, $B(\xi ,\tau )$, $C(\xi ,\tau )$, and their respective partial derivatives. All coefficients of $R(x,t)$ must vanish in order to satisfy the KdV Equation (34). We obtain the vanishing of the first-order residue coefficients $R_{1m}=0$, $m\ge 0$ when $k_0$ and ${\omega }_0$ are related by the linear dispersion relationship. The second-order residue coefficients are given by

$$\begin{array}{cc}\hfill R_{20}& =0;R_{21}=0;R_{2m}=0,m\ge 3;\hfill \\ \hfill R_{22}& =i\left\{\left[\Omega \left(2k_0\right)-2{\omega }_0\right]B+c{k}_0{A}^2\right\}.\hfill \end{array}$$

The vanishing of $R_{22}$ leads to an expression for $B(\xi ,\tau )$ as a function of $A(\xi ,\tau )$:

$$B(\xi ,\tau )=\frac{c{k}_0{A}^2(\xi ,\tau )}{2{\omega }_0-\Omega \left(2k_0\right)}.$$

The third-order residue coefficients read as follows:

$$\begin{array}{cc}\hfill R_{30}& =\left[{\Omega }^{\prime }\left(0\right)-{\Omega }^{\prime }\left(k_0\right)\right]{\partial }_{\xi }C+\frac{1}{2}c{\partial }_{\xi }{\left|A\right|}^2;\hfill \\ \hfill R_{31}& ={\partial }_{\tau }A-\frac{1}{2}i{\Omega }^{″}\left(k_0\right){\partial }_{\xi }^{2}A+ic{k}_0\left(A^{*}B+AC+A{C}^{*}\right).\hfill \end{array}$$

Requiring $R_{30}$ to vanish yields an expression for $C(\xi ,\tau )$ as a function of $A(\xi ,\tau )$ and a $\tau$-dependent constant of integration ${\alpha }_T$, given as follows:

$$C(\xi ,\tau )=\frac{1}{2}\frac{{c\left|A(\xi ,\tau )\right|}^2}{{\Omega }^{\prime }\left(k_0\right)-{\Omega }^{\prime }\left(0\right)}+{\alpha }_T\left(\tau \right).$$

To prevent the occurrence of a resonance phenomenon, $R_{31}$ has to vanish as well, and this leads to an evolution equation for $A(\xi ,\tau )$:

$${\partial }_{\tau }A+i{\beta }_1{\partial }_{\xi }^{2}A+i{\gamma }_1{\left|A\right|}^{2}A+2i{k}_{0}c\mathrm{R}\mathrm{e}\left[{\alpha }_T\left(\tau \right)\right]A=0.$$

(37)

In a similar assumption of unidirectional wave propagation, applying the “gauge transformation” by multiplying the evolution Equation (37) by

$$\exp \left\{2i{k}_{0}c\int \mathrm{R}\mathrm{e}\left[{\alpha }_T\left(\tau \right)\right]d\tau \right\},$$

where exp denotes the exponential function, we obtain the temporal NLS equation for $A(\xi ,\tau )$ [78]:

$${\partial }_{\tau }A+i{\beta }_1{\partial }_{\xi }^{2}A+i{\gamma }_1{\left|A\right|}^{2}A=0.$$

(38)

The dispersion and nonlinear coefficients of the NLS Equation (38) are given by ${\beta }_1$ and ${\gamma }_1$, respectively, where they are explicitly given as follows:

$$\begin{array}{cc}\hfill {\beta }_1& =\beta \left(k_0\right)=-\frac{1}{2}{\Omega }^{″}\left(k_0\right),\mathrm{and}\hfill \\ \hfill {\gamma }_1& =\gamma \left(k_0,c\right)=k_0{c}^2\left(\frac{1}{{\Omega }^{\prime }\left(k_0\right)-{\Omega }^{\prime }\left(0\right)}+\frac{k_0}{2{\omega }_0-\Omega \left(2k_0\right)}\right).\hfill \end{array}$$

In surface gravity waves, both the temporal NLS Equation (38) and the spatial NLS equation that will be derived in Section 3.3 are valid for intermediate water wave models.

In the following, we will derive the corresponding “energy equation” and “nonlinear dispersion relationship” for the temporal NLS equation. We express the complex-valued amplitude $A(\xi ,\tau )$ in the physical, faster-moving variables as $A_1(x,t)=ϵA(\xi ,\tau )$ using the relationship above $\xi =ϵ(x-{\Omega }^{\prime }\left(k_0\right)t)$ and $\tau ={ϵ}^{2}t$. Consequently, a temporal NLS equation in the physical variables x and t is now given by

$${\partial }_t{A}_1+{\Omega }^{\prime }\left(k_0\right){\partial }_x{A}_1+i{\beta }_1{\partial }_x^2{A}_1+i{\gamma }_1{\left|A_1\right|}^2{A}_1=0.$$

(39)

We then apply the Madelung transformation by writing the complex-valued amplitude $A_1(x,t)$ in its polar form $A_1(x,t)=a(x,t)e^{i\varphi (x,t)}$, where the amplitude $a(x,t)$ and the phase $\varphi (x,t)$ are both real-valued functions [79]. Upon substitution to (39), removing the complex exponential factor $e^{i\varphi (x,t)}$, and separating the real and the imaginary parts, it yields the following coupled phase-amplitude equations:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\displaystyle {\partial }_{t}a+{\Omega }^{\prime }\left(k_0\right){\partial }_{x}a-{\beta }_1\left(a{\partial }_x^2\varphi +2{\partial }_{x}a{\partial }_x\varphi \right)}\hfill & =0,\hfill \\ {\displaystyle {\partial }_t\varphi +{\Omega }^{\prime }\left(k_0\right){\partial }_x\varphi +{\beta }_1\left(\frac{{\partial }_x^{2}a}{a}-{\left({\partial }_x\varphi \right)}^2\right)+{\gamma }_1{a}^2}\hfill & =0.\hfill \end{array}\right.\end{array}$$

(40)

We also express the wavenumber $k(x,t)$ and the frequency $\omega (x,t)$ in the following form, respectively:

$$\begin{array}{cc}\hfill k(x,t)& =k_0+\kappa =k_0+{\partial }_x\varphi ,\mathrm{and}\hfill \\ \hfill \omega (x,t)& ={\omega }_0+\nu ={\omega }_0-{\partial }_t\varphi .\hfill \end{array}$$

(41)

The local wavenumber $\kappa ={\partial }_x\varphi$ and the local frequency $\nu =-{\partial }_t\varphi$ in (41) act as modulational quantities. Using these quantities, the phase-amplitude equations can be written in a more compact form. The amplitude and the phase equations are the first and the second expressions in (40), respectively. Expressing ${\partial }_x\varphi$ in terms of the wavenumber $k(x,t)$ and multiplying the amplitude equation with $a(x,t)$, we obtain

$$\frac{1}{2}{\partial }_t\left(a^2\right)+\frac{1}{2}{\partial }_x\left\{\left[{\Omega }^{\prime }\left(k_0\right)+{\Omega }^{″}\left(k_0\right)(k-k_0)\right]a^2\right\}=0.$$

(42)

By recognizing the terms inside the square brackets of the second term in (42) as a linear approximation for ${\Omega }^{\prime }\left(k\right)$, we arrive at the equation for the conservation of energy of the temporal NLS equation. It is given as follows:

$${\partial }_t\left(a^2\right)+{\partial }_x\left[{\Omega }^{\prime }\left(k\right)a^2\right]=0.$$

Similarly, by expressing the local wavenumber ${\partial }_x\varphi =k-k_0$ and the local frequency $-{\partial }_t\varphi =\omega -{\omega }_0$, we obtain the following expression:

$$\omega -\left[{\omega }_0+{\Omega }^{\prime }\left(k_0\right)(k-k_0)+\frac{1}{2}{\Omega }^{″}\left(k_0\right){(k-k_0)}^2\right]={\beta }_1\frac{{\partial }_x^{2}a}{a}+{\gamma }_1{a}^2.$$

(43)

By considering the terms inside the square brackets in (43) as a quadratic approximation for $\Omega \left(k\right)$, the phase equation leads to the following nonlinear dispersion relationship for the temporal NLS equation:

$$\omega -\Omega \left(k\right)={\beta }_1\frac{{\partial }_x^{2}a}{a}+{\gamma }_1{a}^2.$$

(44)

3.3. Derivation of the Spatial NLS Equation

A similar technique using the method of multiple-scale can be applied to the KdV equation with an exact dispersion relationship (34) to obtain the spatial NLS equation. The difference occurs in the choice of the slow-moving spatial and temporal variables, respectively chosen as $\xi ={ϵ}^{2}x$ and $\tau =ϵ(t-x/{\Omega }^{\prime }\left(k_0\right))$. Following an equivalent procedure as in Section 3.2, we obtain identical first-order and second-order residue coefficients:

$$\begin{array}{cc}\hfill R_{1m}& =0,m\ge 0,\hfill \\ \hfill R_{20}& =0;R_{21}=0;R_{2m}=0,m\ge 3,\hfill \\ \hfill R_{22}& =i\left(\left[\Omega \left(2k_0\right)-2{\omega }_0\right]B+c{k}_0{A}^2\right).\hfill \end{array}$$

The third-order residue coefficients are given as follows:

$$\begin{array}{cc}\hfill R_{30}& =\left(1-\frac{{\Omega }^{\prime }\left(0\right)}{{\Omega }^{\prime }\left(k_0\right)}\right){\partial }_{\tau }C-\frac{1}{2}\frac{c}{{\Omega }^{\prime }\left(k_0\right)}{\partial }_{\tau }{\left|A\right|}^2,\hfill \\ \hfill R_{31}& ={\Omega }^{\prime }\left(k_0\right){\partial }_{\xi }A-\frac{1}{2}i\frac{{\Omega }^{″}\left(k_0\right)}{{\left[{\Omega }^{\prime }\left(k_0\right)\right]}^2}{\partial }_{\tau }^{2}A+ic{k}_0(A^{*}B+AC+A{C}^{*}),\hfill \\ \hfill R_{32}& =-\frac{1}{{\Omega }^{\prime }\left(k_0\right)}\left({\Omega }^{\prime }\left(2k_0\right){\partial }_{\tau }B+\frac{1}{2}c{\partial }_{\tau }{A}^2\right),\hfill \\ \hfill R_{33}& =3i{k}_{0}cAB.\hfill \end{array}$$

Requiring $R_{30}$ to vanish leads to an expression for $C(\xi ,\tau )$ as a function of $A(\xi ,\tau )$ and a $\xi$-dependent constant of integration ${\alpha }_S\left(\xi \right)$ for all $\theta (x,t)$:

$$C(\xi ,\tau )=\frac{1}{2}\frac{{c\left|A(\xi ,\tau )\right|}^2}{{\Omega }^{\prime }\left(k_0\right)-{\Omega }^{\prime }\left(0\right)}+{\alpha }_S\left(\xi \right).$$

In order to prevent the occurrence of the resonance phenomenon, the residue $R_{31}$ has to vanish, which leads to a dynamic evolution equation for $A(\xi ,\tau )$:

$${\partial }_{\xi }A+i{\beta }_2{\partial }_{\tau }^{2}A+i{\gamma }_2{\left|A\right|}^{2}A+\frac{2i{k}_{0}c}{{\Omega }^{\prime }\left(k_0\right)}\mathrm{R}\mathrm{e}\left[{\alpha }_S\left(\xi \right)\right]A=0.$$

(45)

We can remove the term containing Re$\left[{\alpha }_S\left(\xi \right)\right]$ in (45) by applying the gauge transformation [78]. Multiply the evolution Equation (45) by

$$\exp \left\{\frac{2i{k}_{0}c}{{\Omega }^{\prime }\left(k_0\right)}\int \mathrm{R}\mathrm{e}\left[{\alpha }_S\left(\xi \right)\right]d\xi \right\};$$

then the new complex amplitude $\tilde{A}$, defined as follows:

$$\tilde{A}(\xi ,\tau )=\exp \left\{\frac{2i{k}_{0}c}{{\Omega }^{\prime }\left(k_0\right)}\int \mathrm{R}\mathrm{e}\left[{\alpha }_S\left(\xi \right)\right]d\xi \right\}A(\xi ,\tau ),$$

satisfies the spatial NLS equation, which, after dropping the tilde, can be written in the following form:

$${\partial }_{\xi }A+i{\beta }_2{\partial }_{\tau }^{2}A+i{\gamma }_2{\left|A\right|}^{2}A=0.$$

(46)

The dispersion coefficient ${\beta }_2$ and the nonlinear coefficient ${\gamma }_2$ of (46) are given as follows, respectively:

$$\begin{array}{cc}\hfill {\beta }_2& =\beta \left(k_0\right)=-\frac{1}{2}\frac{{\Omega }^{″}\left(k_0\right)}{{\left[{\Omega }^{\prime }\left(k_0\right)\right]}^3},\hfill \\ \hfill {\gamma }_2& =\gamma (k_0,c)=\frac{k_0{c}^2}{{\Omega }^{\prime }\left(k_0\right)}\left(\frac{1}{{\Omega }^{\prime }\left(k_0\right)-{\Omega }^{\prime }\left(0\right)}+\frac{k_0}{2{\omega }_0-\Omega \left(2k_0\right)}\right).\hfill \end{array}$$

We apply a similar approach as in the previous section to derive the corresponding energy equation and nonlinear dispersion relationship for the spatial NLS equation. We write $A_2(x,t)=ϵA(\xi ,\tau )$, where $\xi ={ϵ}^{2}x$ and $\tau =ϵ(t-x/{\Omega }^{\prime }\left(k_0\right))$. The spatial NLS equation in the physical variables is expressed as follows:

$${\partial }_x{A}_2+\frac{1}{{\Omega }^{\prime }\left(k_0\right)}{\partial }_t{A}_2+i{\beta }_2{\partial }_t^2{A}_2+i{\gamma }_2{\left|A_2\right|}^2{A}_2=0.$$

(47)

Apply the Madelung transformation by writing $A_2(x,t)$ in its polar form $A_2(x,t)=a(x,t)e^{i\varphi (x,t)}$, where $a(x,t)$ and $\varphi (x,t)$ are real-valued quantities [79]. After substituting this to the spatial NLS Equation (47), removing the factor $e^{i\varphi (x,t)}$, and collecting the real and the imaginary parts, we obtain the following coupled phase-amplitude equations in the original physical variables:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\displaystyle {\partial }_{x}a+\frac{{\partial }_{t}a}{{\Omega }^{\prime }\left(k_0\right)}-{\beta }_2\left(a{\partial }_t^2\varphi +2{\partial }_{t}a{\partial }_t\varphi \right),}\hfill & =0\hfill \\ {\displaystyle {\partial }_x\varphi +\frac{{\partial }_t\varphi }{{\Omega }^{\prime }\left(k_0\right)}+{\beta }_2\left(\frac{{\partial }_t^{2}a}{a}-{\left({\partial }_t\varphi \right)}^2\right)+{\gamma }_2{a}^2}\hfill & =0.\hfill \end{array}\right.\end{array}$$

(48)

Using the previous definition for wavenumber and frequency expressed in terms of modulated local wavenumber and local frequency, respectively, we can write these phase-amplitude equations in a more compact form. We also adopt the linear dispersion relationship $\omega =\Omega \left(k\right)$ or $k=K\left(\omega \right)$, where $K={\Omega }^{-1}$. From $k=K[\Omega (k\left)\right]$, we can derive the relationship between its derivatives, up to the second order given explicitly as follows:

$$K^{\prime }\left({\omega }_0\right)=\frac{1}{{\Omega }^{\prime }\left(k_0\right)}\mathrm{a}\mathrm{n}\mathrm{d}{K}^{″}\left({\omega }_0\right)=-\frac{{\Omega }^{″}\left(k_0\right)}{{\left[{\Omega }^{\prime }\left(k_0\right)\right]}^3}.$$

Expressing the local frequency $-{\partial }_t\varphi =\omega -{\omega }_0$ and multiplying the amplitude Equation (48) with $a(x,t)$, we obtain the following expression:

$$\frac{1}{2}{\partial }_x\left(a^2\right)+\frac{1}{2}{\partial }_t\left\{\left[K^{\prime }\left({\omega }_0\right)+K^{″}\left({\omega }_0\right)(\omega -{\omega }_0)\right]a^2\right\}=0.$$

(49)

Recognizing the terms in the square brackets of (49) as a linear approximation for $K^{\prime }\left(\omega \right)$, we can write the amplitude Equation (49) as the energy equation:

$${\partial }_x\left(a^2\right)+{\partial }_t\left[K^{\prime }\left(\omega \right)a^2\right]=0.$$

Furthermore, by expressing the local wavenumber ${\partial }_x\varphi =k-k_0$ and the local frequency $-{\partial }_t\varphi =\omega -{\omega }_0$, we can write the phase Equation (48) as follows:

$$\left[k_0+K^{\prime }\left({\omega }_0\right)(\omega -{\omega }_0)+\frac{1}{2}{K}^{″}\left({\omega }_0\right){(\omega -{\omega }_0)}^2\right]-k={\beta }_2\frac{{\partial }_t^{2}a}{a}+{\gamma }_2{a}^2.$$

(50)

By considering the terms inside the square brackets of (50) as a quadratic approximation for $K\left(\omega \right)$, the phase Equation (50) leads to the nonlinear dispersion relationship:

$$K\left(\omega \right)-k={\beta }_2\frac{{\partial }_t^{2}a}{a}+{\gamma }_2{a}^2.$$

(51)

The nonlinear dispersion relationship (44) or (51) describes the relationship between the wavenumber k and the frequency $\omega$ in the dispersion plane $(k,\omega )$. Since, generally, the right-hand side of (44) or (51) does not vanish, any combination of $(k,\omega )$ does not always satisfy the linear dispersion relationship.

Introduced by Chu and Mei [80,81] in their discussion of phase singularity and wavefront dislocation in surface gravity wave propagation, the ratio $\frac{{\partial }_x^{2}a}{a}$ in (44) or $\frac{{\partial }_t^{2}a}{a}$ in (51) is known as the Chu–Mei quotient [82]. This unbounded quotient is responsible for the occurrence of these phenomena. While the quotient has appeared earlier within the context of modulated waves in nonlinear media [83,84], it was Chu and Mei who first introduced it when deriving Whitham’s modulation equations for slowly varying Stokes waves [80,81]. Several authors subsequently call it the “Fornberg–Whitham term,” referring to Fornberg and Whitham’s paper [85,86].

3.4. Applications in Surface Gravity Waves

The temporal NLS equation derived in Section 3.2 models absolute dynamics, whereas the spatial NLS equation derived in Section 3.3 models convective dynamics [12]. When equipped with a specific initial condition, the temporal NLS equation formulates an initial value problem, governing the evolution of a wave packet from a given profile at a particular time to any future time. Similarly, the spatial NLS equation paired with a boundary condition acts as boundary value problem, also known as a signaling problem. This model specifically applies to wave signal generation in a hydrodynamic laboratory wave tank. By inputting an initial wave signal to a wavemaker and allowing it to propagate along the tank, we can measure the “experimental” wave signal at various downstream points and compare it with the “theoretical” wave signal predicted by the spatial NLS equation.

The spatial NLS equation with solitons on a nonvanishing background has been used as a model for deterministic freak wave generation in intermediate water depth at MARIN’s high-speed wave basin in Wageningen, the Netherlands [77,87,88]. The experiments confirm the occurrence of a phase singularity and its related phenomenon of wavefront dislocation at the location where the wave signals reach maximum amplitude, as predicted theoretically [82]. While the model does not quantitatively predict the signal evolution and spectrum in accurate detail, it exhibits an extraordinary qualitative agreement. Our results qualitatively mirror this behavior, consistent with other testings where the corresponding wave spectra demonstrate frequency downshift as the wave signals propagate along the tank [89,90]. The authors also introduced the term “Wessel curves” to represent the evolution of the real and imaginary parts of the complex-valued amplitude $A(\xi ,\tau )$ in the absence of the oscillating component generated by the continuous-wave or plane-wave solution.

Modulational instability, a well-known phenomenon in both fluid dynamics and nonlinear optics, manifests as the Benjamin–Feir instability in hydrodynamics. It refers to the amplification of sidebands in the optical spectrum, ultimately breaking up a periodic waveform into a train of pulses. Hence, it is also called sideband instability. This instability exhibits a rich structure with an infinite number of sideband pairs at non-equally spaced frequencies, arising from various nonlinearities (e.g., Kerr nonlinearity) in optics, typically due to anomalous chromatic dispersion [29,91].

The history of modulational instability can be traced back to the 1960s, a period when the Soviet Union housed a thriving scientific community making significant contributions across various fields of science and technology. In 1965, Soviet scientists Piliptetskii and Rustamov observed spatial modulation instability in high-power lasers using organic solvents [92]. The following year, Bespalov and Talanov published the mathematical theory behind this instability [93]. Meanwhile, in the United Kingdom, Benjamin and Feir at the University of Cambridge successfully predicted and experimentally confirmed the instability of Stokes wave trains for periodic surface gravity waves in deep water in 1967 [94,95,96]. Interestingly, modulational instability has been suggested as a potential mechanism for rogue wave generation [97,98].

Through linear perturbation analysis, the plane-wave solution of the NLS equation is shown to be modulationally unstable. Its corresponding nonlinear solution, the “Akhme-diev–Eleonskii–Kulagin breather” (also known as “solitons on a nonvanishing background”, “solitons on a nonzero background”, or “solitons on a constant background”) [99,100,101,102], reinforces periodic wave trains in the spectral domain. This reinforcement leads to the generation of spectral sidebands and the subsequent breakup of the waveform into a train of pulses. The breather’s analytical expression for ${\beta }_1=1$ and ${\gamma }_1=2$ in the temporal NLS Equation (38) can be written as follows:

$$A_{\mathrm{AEK}}(\xi ,\tau )=e^{2i\tau }\left(\frac{{\nu }^{3}cosh\left[\sigma (\tau -{\tau }_0)\right]+i\nu \sigma sinh\left[\sigma (\tau -{\tau }_0)\right]}{2\nu cosh\left[\sigma (\tau -{\tau }_0)\right]-\sigma cos\left[\nu (\xi -{\xi }_0)\right]}-1\right).$$

(52)

The family of solitons on a nonvanishing background is defined for the parameter values $0<\nu <2$, and $\sigma =\nu \sqrt{4-{\nu }^2}$. The soliton is a holomorphic function for $\sqrt{3}<\nu <2$, and it reaches local maxima and local minima at $(\xi ,\tau )=({\xi }_0+2n\pi /\nu ,{\tau }_0)$ and $(\xi ,\tau )=({\xi }_0+[2n+1]\pi /\nu ,{\tau }_0)$, respectively, for $n\in \mathbb{Z}$.

Other exact soliton solutions of the NLS equation known as “breather solutions” have been proposed to model hydrodynamic freak wave formation [103]. One notable family of such solitons on a nonvanishing background is the “Kuznetsov–Ma breather” [104], derived by Kuznetsov in 1977 and later by Kawata and Inoue in 1978, as well as by Ma in 1979 [105,106,107].

Delving into the history of the Kuznetsov breather reveals a fascinating journey. Evgenii A. Kuznetsov’s original Russian paper was published as a preprint in 1976 by the Institute of Automation and Electrometry of the Siberian Branch of the USSR (now Russian) Academy of Sciences in Novosibirsk. This preprint was later translated into English and appeared in the Proceedings of the 13th International Conference on Phenomena in ionized Gases and Plasma, held in East Berlin, German Democratic Republic, on 12–17 September 1977. During this period (1976–1977), Kuznetsov frequently interacted with Japanese physicist Tutomu Kawata (his given name is sometimes also spelled as Tsutomu, 川田 勉), who was a postdoctoral researcher under the mentorship of Professor Vladimir E. Zakharov at the Landau Institute for Theoretical Physics in Chernogolovka, near Moscow. Kuznetsov even shared his preprint on the breather soliton solution with Kawata. While Yan-Chow Ma from the Massachusetts Institute of Technology never personally met Kuznetsov, he was certainly aware of this early work, referencing another paper co-authored by Kuznetsov and Alexander V. Mikhailov on the stability of stationary waves using the KdV equation is his own research [108].

For the temporal NLS Equation (38) with specific dispersive and nonlinear coefficients corresponding to ${\beta }_1=1$ and ${\gamma }_1=2$, the Kuznetsov–Ma breather takes the following explicit form:

$$A_{\mathrm{KM}}(\xi ,\tau )=e^{2i\tau }\left(\frac{{\mu }^{3}cos\left[\rho (\tau -{\tau }_0)\right]+i\mu \rho sin\left[\rho (\tau -{\tau }_0)\right]}{2\mu cos\left[\rho (\tau -{\tau }_0)\right]-\rho cosh\left[\mu (\xi -{\xi }_0)\right]}+1\right),$$

(53)

where $\rho =\mu \sqrt{4+{\mu }^2}$, $\mu \in \mathbb{R}$. Another exact solution is known as the “Peregrine breather” or the “rational soliton” solution [109]. It is given as follows:

$$A_{\mathrm{PR}}(\xi ,\tau )=e^{2i\tau }\left(\frac{4(1+4i(\tau -{\tau }_0)}{1+16{(\tau -{\tau }_0)}^2+4{(\xi -{\xi }_0)}^2}-1\right).$$

(54)

This Peregrine breather solution can be obtained as a limiting case for the Akhmediev–Eleonskii–Kulagin breather when the parameters $\nu$ and $\mu$ approach zero [110,111]. In a recent breakthrough for nonlinear waves, Karjanto analytically derived the Fourier spectrum of various breather solutions, including the fundamental bright soliton, providing valuable insights into their behaviors in the spectral domain [112,113]. Moreover, it has been successfully realized experimentally as a rogue wave in a water wave tank [114]. All members of this family of soliton on a constant background have also been discussed in the context of spectral curves, loci of zeros of the characteristic polynomial of the Lax matrix of a system in the AKNS hierarchy. All of them are associated with singular algebraic curves of the arithmetic genus $g_a=2$ and of the topological genus $g=0$ [24,115].

There are other classes of solutions to the NLS equation, and some of them also share some characteristics with freak waves [100]. For example, using the Krichever construction, a mathematical method that linearizes the KdV hierarchy on the Jacobian of a curve, especially when applied to spectral curves, Smirnov discussed two approaches for finding elliptic solutions of the NLS equation [116,117]. Smirnov studied the properties and characteristics of a periodic two-phase “rogue waves” solution of the NLS equation, offering insights into these wave behaviors [118,119]. Together with collaborators, Smirnov developed periodic two-phase solutions of the NLS equation using algebro-geometric methods [120]. Smirnov and Matveev demonstrated that the scaling and Galilean invariance properties of NLS solutions can be extended with appropriate modifications to the entire AKNS hierarchy, offering insights into the system’s continuous symmetries [121]. Leveraging the success of finding two-phase freak wave solutions, Smirnov and collaborators extended into three-phase solutions of the NLS equation using finite-gap algebro-geometric methods [122].

The NLS equation has also been used to model the formation of oceanic rogue waves, those massive and unexpected waves arising from nonlinear energy transfer in open ocean deep waters. This phenomenon has been extensively studied, both deterministically and stochastically, as documented in the published literature [98,123,124,125,126,127,128,129,130]. Similar phenomena have been proposed, predicted, observed, and studied in other fields beyond hydrodynamics, where the NLS equation serves as a mathematical model. These include optical rogue waves [61], atmospheric rogue waves [131], matter rogue waves in Bose–Einstein condensates [132], and even financial rogue waves [133,134].

4. Superconductivity

4.1. Deriving the NLS Equation from a Nonlinear Klein–Gordon Equation

A nonlinear Klein–Gordon equation is a specific type of the sine-Gordon equation. Both are nonlinear second-order PDEs resembling the wave equation. The key difference with the wave equation is that the sine-Gordon equation involves the sine of the field value, while a Klein–Gordon equation includes a Taylor series expansion of the sine term beyond the linear term. Hence, the lowest order of a nonlinear Klein–Gordon equation has a cubic power of nonlinearity.

Imagine a series of closely spaced pendulums hanging vertically due to gravity. Each pendulum is allowed to twist around a separate horizontal torsion wire. Let $u(x,t)$ represent the twist angle of the pendulum at position x and time t. The motion of each pendulum can then be modeled by a sine-Gordon equation:

$${\partial }_t^{2}u-a{\partial }^{2}xu+bsinu=0,a,b\ge 0.$$

In this model, the term $-bsinu$ acts as an external force due to gravitational acceleration, whereas the term $a{\partial }_x^{2}u$ models the force arising from the twist’s effect. If we wiggle one end of the pendulum chain with a small amplitude with the frequency $\omega$, then the term $sinu$ can be approximated by its truncated Maclaurin series about $u=0$. By keeping only the first two terms, this yields a nonlinear Klein–Gordon equation with a cubic nonlinearity.

The following derivation of the NLS equation from a nonlinear Klein–Gordon equation follows the argument in [12,71,72,135,136,137]. Consider a cubic nonlinearity Klein–Gordon equation describing a model for a wave packet $u(x,t)$ that moves at a constant group velocity c, presented as the following initial value problem (IVP):

$$\begin{array}{cc}\hfill {\partial }_t^{2}u-a{\partial }_x^{2}u+bu-\lambda u^3& =0,a,b\ge 0,\lambda \in \mathbb{R},\hfill \end{array}$$

(55)

$$\begin{array}{cc}\hfill u(x,0)=u_0,{\partial }_{t}u(x,0)& =u_1.\hfill \end{array}$$

(56)

An ansatz for $u(x,t)$ is expressed as a perturbation series, where again $0<ϵ\ll 1$ is a small parameter

$$u(x,t)={\widehat{u}}_0+ϵ{\widehat{u}}_1+{ϵ}^2{\widehat{u}}_2+\cdots ,$$

(57)

with $u_n(x,X,t,{\tau }_1,{\tau }_2)$, where $X=ϵx$, ${\tau }_1=ϵt$, and ${\tau }_2={ϵ}^{2}t$ act as slower variables. Substituting (57) to the IVP (55) and (56) yields a series progressing in the order of $ϵ$. The vanishing of the lowest-order term simply reduces to a linear Klein–Gordon IVP:

$$\begin{array}{cc}\hfill {\partial }_t^2{\widehat{u}}_0-a{\partial }_x^2{\widehat{u}}_0+b{\widehat{u}}_0& =0,\hfill \\ \hfill {\widehat{u}}_0(x,X,0,0,0;ϵ)& =u_0/ϵ,{\partial }_t{\widehat{u}}_0(x,X,0,0,0;ϵ)=u_1/ϵ.\hfill \end{array}$$

We seek the solution in the form

$${\widehat{u}}_0(x,X,t,{\tau }_1,{\tau }_2)=A(X,{\tau }_1,{\tau }_2)e^{i(kx-\omega t)}+\mathrm{c}.\mathrm{c}.,A\in \mathbb{C},$$

where c.c. denotes the complex conjugation of the preceding term and $(k,\omega )$ satisfies the dispersion relationship $\omega =\Omega \left(k\right)=\sqrt{a{k}^2+b}$. Requiring the first-order terms to vanish yields the following IVP:

$$\begin{array}{cc}\hfill {\partial }_t^2{\widehat{u}}_1-a{\partial }_x^2{\widehat{u}}_1+b{\widehat{u}}_1& =-2{\partial }_{{\tau }_1}{\partial }_t{\widehat{u}}_0+2a{\partial }_X{\partial }_x{\widehat{u}}_0\hfill \\ \hfill & =2i\left(\omega {\partial }_{{\tau }_1}A+ak{\partial }_{X}A\right)e^{i(kx-\omega t)}+\mathrm{c}.\mathrm{c}.,\hfill \\ \hfill {\widehat{u}}_1(x,X,0,0,0;ϵ)& =0,\hfill \\ \hfill {\partial }_t{\widehat{u}}_1(x,X,0,0,0;ϵ)& =-{\partial }_{{\tau }_1}{\widehat{u}}_0(x,X,0,0,0;ϵ).\hfill \end{array}$$

(58)

Because the right-hand side of (58) represents secular terms that would lead to unbounded growth in ${\widehat{u}}_1$ over a long period of time, we need to eliminate them by taking $\omega {\partial }_{{\tau }_1}A+ak{\partial }_{X}A=0$. This condition is equivalent to the group velocity ${\Omega }^{\prime }\left(k\right)=ak/\Omega \left(k\right)$. The solution of the linear Klein–Gordon equation for ${\widehat{u}}_1$ is similar to the one for ${\widehat{u}}_0$:

$${\widehat{u}}_1(x,X,t,{\tau }_1,{\tau }_2)=B(X,{\tau }_1,{\tau }_2)e^{i(kx-\omega t)}+\mathrm{c}.\mathrm{c}.,B\in \mathbb{C}.$$

Collecting the second-order term and requiring it to vanish gives the following IVP:

$$\begin{array}{cc}\hfill {\partial }_t^2{\widehat{u}}_2-\alpha {\partial }_x^2{\widehat{u}}_2+\beta {\widehat{u}}_2& =-2{\partial }_{{\tau }_2}{\partial }_t{\widehat{u}}_0-{\partial }_{{\tau }_1}^2{\widehat{u}}_0+a{\partial }_X^2{\widehat{u}}_0+\lambda {\widehat{u}}_0^3+2{\partial }_{{\tau }_1}{\partial }_t{\widehat{u}}_1-2a{\partial }_X{\partial }_x{\widehat{u}}_1\hfill \\ \hfill & =\left\{2i\Omega \left(k\right){\partial }_{{\tau }_2}A+(a-{\Omega }^{\prime }{\left(k\right)}^2){\partial }_{\xi }^{2}A+3\lambda {\left|A\right|}^{2}A+2i\left(\omega {\partial }_{{\tau }_1}B+ak{\partial }_{X}B\right)\right\}{e}^{i(kx-\omega t)}\hfill \\ \hfill & +\lambda A^3{e}^{3i(kx-\omega t)}+\mathrm{c}.\mathrm{c}.,\hfill \\ \hfill {\widehat{u}}_2(x,X,0,0,0;ϵ)& =0,\hfill \\ \hfill {\partial }_t{\widehat{u}}_2(x,X,0,0,0;ϵ)& =-{\partial }_{{\tau }_1}{\widehat{u}}_0(x,X,0,0,0;ϵ)-{\partial }_{{\tau }_2}{\widehat{u}}_0(x,X,0,0,0;ϵ).\hfill \end{array}$$

Here, we have used $\xi =ϵ\left(x-{\Omega }^{\prime }\left(k\right)t\right)$. Similar to the previous step, we would like to remove secular terms by requiring ${\widehat{u}}_2$ to be bounded, and B must satisfy $\omega {\partial }_{{\tau }_1}B+ak{\partial }_{X}B=0$. It follows that the quantity $A(\xi ,{\tau }_2)$ satisfies the temporal NLS equation:

$$i{\partial }_{\tau }A+\beta {\partial }_{\xi }^{2}A+\gamma {\left|A\right|}^{2}A=0,$$

where we have dropped the subscript 2 from the variable ${\tau }_2$. The dispersive and the nonlinear coefficients are given as follows, respectively:

$$\beta =\frac{1}{2}{\Omega }^{″}\left(k\right),\mathrm{and}\gamma =\frac{3}{2}\frac{\lambda }{\Omega \left(k\right)}.$$

(59)

The additional term $\lambda A^3{e}^{3i(kx-\omega t)}+\mathrm{c}.\mathrm{c}.$ is not a resonant term and, thus, is not problematic as generally $\Omega \left(3k\right)\ne 3\Omega \left(k\right)$.

4.2. Applications of Sine-Gordon Model

Both the nonlinear Klein–Gordon and sine-Gordon equations hold significant importance in various fields, from physics to engineering. Understanding their solutions and behaviors has attracted researchers across disciplines, leading to the development of diverse analytical and numerical methods to tackle these complex equations. Two notable analytical approaches are the Zakharov–Shabat method and the sine–cosine and tanh methods. The elegant Zakharov–Shabat technique decomposes the equations into coupled linear systems, often resulting in exact solutions in the form of scattering data [138,139]. The sine–cosine and tanh methods utilize trigonometric and hyperbolic functions to construct exact solutions, offering valuable insights into specific solution types [140].

Among numerical approaches, the finite difference method discretizes the equations into solvable algebraic equations. References [141,142,143,144,145] showcase its implementation for various problems. The thin plate splines-radial basis functions method offers flexibility in handling complex geometries and boundary conditions, highlighting its strengths in particular situations [146]. It is important to note that choosing the most suitable method depends on several factors, including the desired level of accuracy, computational efficiency, and specific problem characteristics. Researchers often combine or adapt these methods to achieve their research goals.

The allure of nonlinear evolution equations like the sine-Gordon and Klein–Gordon extends far beyond theoretical exercises. These equations permeate various physical domains, showcasing their remarkable versatility and ability to model diverse phenomena [147,148].

Take the sine-Gordon equation, for instance. Its historical journey began in the realm of differential geometry, gracefully representing surfaces with negative curvature—a testament to its inherent mathematical beauty [149]. Moving beyond academia, this equation finds practical applications in understanding the dynamics of crystal dislocations with their characteristic sinusoidal patterns [150]. If that was not enough, the sine-Gordon equation even ventures into the subatomic realm, playing a role in describing the interactions and behavior of fundamental particles like mesons and baryons [151]. Its applications do not stop there—it can even be found influencing the dynamics of weakly unstable baroclinic shear flows, vital for understanding atmospheric and oceanic phenomena [152].

This expansive range of applications highlights the true power of nonlinear evolution equations. They are not merely abstract mathematical constructs; they serve as potent tools for unveiling the hidden connections and underlying principles governing diverse physical systems across vast scales—from the microscopic dance of elementary particles to the majestic movements of large-scale fluid flows. This versatility paves the way for further exploration and potentially groundbreaking discoveries in various scientific fields.

Another important application lies in superconductivity, specifically the Josephson effect across a Josephson junction. A Josephson junction is a quantum mechanical device consisting of two superconducting electrodes separated by a thin, non-superconducting barrier. This barrier permits superconducting electrons to undergo quantum tunneling, leading to a fascinating macroscopic quantum phenomenon known as the Josephson effect. Unlike ordinary currents, this effect gives rise to a supercurrent, which flows continuously without applied voltage, showcasing the manifestation of quantum mechanical principles at a macroscopic level [153,154,155]. British physicist Brian David Josephson investigated the mathematical relationship between supercurrent through the junction and the voltage across the weak link [156,157].

Josephson junctions have wide-ranging applications, extending from electronic circuits and ultrafast computers to sensitive magnetometers and voltmeters. Additionally, they are crucial components in superconducting quantum interference devices (SQUIDs), known for their exceptional magnetic field sensitivity [158,159,160,161]. Moreover, the Josephson effect extends beyond SQUIDs, finding applications in superfluid helium quantum interference devices (SHeQUIDs) and various areas of science and engineering, particularly in precision metrology, where its highly accurate frequency-to-voltage conversion proves invaluable [162,163,164]. Reviews on Josephson junctions and superconducting soliton oscillators within the sine-Gordon model are given in [165,166].

The derivation of the NLS equation from the sine-Gordon system for a small-amplitude limit was considered by Kaup [167], who discovered a phase-locked breather with an applied alternating current field by means of perturbation theory. The nonlinear breather dynamics of an alternating current parametric force in the presence of loss in a sine-Gordon system have been analyzed in [168]. In the case of a small-amplitude limit where the system can be described by an effective NLS equation, a correct threshold value for the driving force amplitude was obtained when the breather frequency equaled one.

Inspired by the recent progress in quantum graph theory and its applications [169,170,171], researchers have investigated interactions of traveling localized wave solutions with a vertex in a star graph from a tricrystal Josephson junction [172]. Other applications of the sine-Gordon and nonlinear Klein–Gordon models include a mechanical model with springs, wires, and bearings and Bloch wall dynamics in magnetic crystals [173]. For a summary of contemporary developments of the sine-Gordon model and its wide range of applications, please consult [174].

5. Bose–Einstein Condensation

5.1. Deriving NLS Equation from Bose–Einstein Condensed State

A Bose–Einstein condensate (BEC) is a fascinating state of matter that occurs when a gas of bosons is cooled to extremely low temperatures, near absolute zero (−273.15 °C). At these frigid temperatures, a significant portion of the bosons occupy the lowest energy state possible, leading to a collective quantum phenomenon with unique properties. This phenomenon was predicted around a century ago by Bose and Einstein [175,176].

In a BEC, a large number of bosons are in the same quantum state, exhibiting a phenomenon called wavefunction coherence. This means that their individual wavefunctions overlap and behave as a single, macroscopic wavefunction. This coherence leads to unique properties like superfluid behavior (frictionless flow) and laserlike coherence in light emission. While quantum mechanics typically operates at the microscopic level, BECs bridge the gap, exhibiting quantum phenomena on a macroscopic scale. This allows scientists to observe and study these effects in a lab setting, providing valuable insights into quantum mechanics.

There are several mathematical models used to describe the behavior of BECs, depending on the level of complexity and specific phenomenon being studied. The Gross–Pitaevskii (GP) equation is the most common and widely used model for BECs, especially for systems near absolute zero and with weak interactions between bosons. It is a nonlinear mean-field equation describing the wavefunction of the condensate. The model was independently derived by Gross and Pitaevskii in the 1960s [177,178], and it is similar to the NLS equation. The derivation presented in this section builds upon the arguments presented in [179,180,181]. For a more rigorous treatment of the model, see [182,183].

Consider a system of a weakly interacting Bose gas where its Hamiltonian can be written in terms of the field operator $\widehat{\Psi }$:

$$\widehat{H}=\int \left(\frac{\hslash }{2m}\nabla {\widehat{\Psi }}^{†}\nabla \widehat{\Psi }\right)\mathrm{d}\mathbf{r}+\frac{1}{2}\int {\widehat{\Psi }}^{†}{\widehat{\Psi }}^{†\prime }V({\mathbf{r}}^{\prime }-\mathbf{r})\widehat{\Psi }{\widehat{\Psi }}^{\prime }\mathrm{d}{\mathbf{r}}^{\prime }\mathrm{d}\mathbf{r}.$$

(60)

Here, ℏ is the Planck constant, m is the particle mass, ${\widehat{\Psi }}^{†}\left(\mathbf{r}\right)$ and $\widehat{\Psi }\left(\mathbf{r}\right)$ are the field operators creating and annihilating a particle at the point $\mathbf{r}$, and $V\left(\mathbf{r}\right)$ is the two-body potential. The field operators satisfy the following commutation relationship:

$$\left[\widehat{\Psi }\left(\mathbf{r}\right),{\widehat{\Psi }}^{†}\left(\mathbf{r}\right)\right]=\delta (\mathbf{r}-{\mathbf{r}}^{\prime })\mathrm{and}\left[\widehat{\Psi }\left(\mathbf{r}\right),\widehat{\Psi }\left({\mathbf{r}}^{\prime }\right)\right]=0.$$

(61)

In the Heisenberg representation, the field operator $\widehat{\Psi }(\mathbf{r},t)$ satisfies the following condition:

$$\begin{array}{cc}\hfill i\hslash {\partial }_t\widehat{\Psi }(\mathbf{r},t)& =\left[\widehat{\Psi }(\mathbf{r},t),\widehat{H}\right],\hfill \\ \hfill & =\left[-\frac{{\hslash }^2{\nabla }^2}{2m}+V_{\mathrm{ext}}(\mathbf{r},t)+\int {\widehat{\Psi }}^{†}({\mathbf{r}}^{\prime },t)V({\mathbf{r}}^{\prime }-\mathbf{r})\widehat{\Psi }({\mathbf{r}}^{\prime },t)\mathrm{d}{\mathbf{r}}^{\prime }\right]\widehat{\Psi }(\mathbf{r},t),\hfill \end{array}$$

where $V_{\mathrm{ext}}$ is an external potential, and we have utilized the Hamiltonian (60) and the commutation relationship (61).

By applying an effective potential $V_{\mathrm{eff}}$ under which the Born approximation holds [184], we can replace the field operator $\widehat{\Psi }(\mathbf{r},t)$ with a classical field or the condensation wavefunction ${\Psi }_0(\mathbf{r},t)$ at very low temperatures and up to the lowest-order approximation. Assuming a slowly varying function ${\Psi }_0(\mathbf{r},t)$, with variations on distances comparable to the interatomic force range, ${\mathbf{r}}^{\prime }$ can be replaced with $\mathbf{r}$. We arrive at the GP equation:

$$i\hslash {\partial }_t{\Psi }_0(\mathbf{r},t)=\left(-\frac{{\hslash }^2{\nabla }^2}{2m}+V_{\mathrm{ext}}(\mathbf{r},t)+g{\left|{\Psi }_0(\mathbf{r},t)\right|}^2\right){\Psi }_0(\mathbf{r},t),$$

where ${\displaystyle g=\int V_{\mathrm{eff}}\left(\mathbf{r}\right)\mathrm{d}\mathbf{r}}$.

This GP equation governs the ground state of a quantum system of identical bosons, serving as a fundamental model for understanding BECs. It describes the characteristics of BECs through a wavefunction, encoding the likelihood of finding a boson at a specific point. Rather than tracking individual bosons, the GP equation captures their collective behavior through a single macroscopic wavefunction, effectively describing the behavior of the entire condensate.

Notably, the presence of an external potential $V_{\mathrm{ext}}$ enables us to simulate diverse scenarios where the external world interacts with the condensate. For example, $V_{\mathrm{ext}}$ can model the effects of magnetic traps, laser beams, or even other BECs on the condensate’s dynamics and behavior.

5.2. Applications in BEC

For decades, the theoretical prediction of BECs existed solely in the realm of quantum mechanics. First proposed in the 1920s by Satyendra Nath Bose and Albert Einstein, these intriguing states of matter promised unique properties at incredibly low temperatures and high densities. However, the technology and techniques necessary to observe them remained out of reach for many years.

However, the tide turned in the 1990s. Scientists finally achieved the experimental realization of BECs, marking a historic achievement in physics. Utilizing sophisticated techniques like magnetic trapping and laser cooling, researchers successfully brought vapors of rubidium ⁸⁷Rb, lithium ⁷Li, and sodium ²³Na atoms into the coveted BEC state [185,186,187]. This monumental discovery opened the door to a new era of exploration in quantum physics.

The initial success with ⁸⁷Rb, ⁷Li, and ²³Na ignited further exploration, pushing the boundaries of BEC research with diverse atomic species. Physicists turned their attention to dilute hydrogen gas [188,189], metastable helium ⁴He^* [190,191], and even elements like potassium ⁴¹K, cesium ¹³³Cs, and another rubidium isotope, rubidium ⁸⁵Rb [192,193,194]. Each breakthrough brought new insights and enriched the understanding of BEC behavior in different atomic systems.

A true landmark in BEC research came in 2001 when a team at the Joint Institute for Laboratory Astrophysics in Colorado made a groundbreaking discovery. They were the first to directly measure the collective excitations of a trapped, dilute Bose-condensed gas [195]. This remarkable feat provided crucial experimental evidence for the theoretical predictions of collective modes in BECs, solidifying their understanding and paving the way for further advanced studies.

For modeling purposes, the dynamics of a dilute trapped BEC can be effectively described using the mean-field theory, within which the GP equation can also be derived through the self-consistent Hartree–Fock–Bogoliubov (HFB) approximation [196,197,198]. The approximation is a powerful tool used in quantum mechanics to describe many-body systems, particularly those involving fermions. It offers an accurate and computationally efficient approach to studying ground states and low-energy excitations in these systems.

The Hartree–Fock–Bogoliubov (HFB) approximation treats interacting fermions as moving in an effective potential generated by the average interaction field of all other particles. It accounts for correlations and pairing effects, crucial for superconductors and nuclei with unpaired nucleons. Notably, it provides a variational framework, minimizing the system’s energy within a specific class of trial wavefunctions. The approximation comprises two main components: the Hartree–Fock (HF) part captures the average interaction, similar to the Hartree–Fock method for bosons, while the Bogoliubov part incorporates pairing correlations and allows for the creation and annihilation of particle-hole pairs, essential for describing superconductivity and pairing phenomena.

Most theoretical approaches to solving the GP equation rely on the Thomas–Fermi approximation, where the nonlinear atomic interactions significantly exceed the kinetic energy pressure, leading to its neglect [199,200,201,202,203]. The Thomas–Fermi approximation is a simplified yet powerful tool used in atomic physics and condensed matter physics to model systems of interacting fermions or bosons, particularly their ground state properties. In the context of BECs, it is often employed to solve the GP equation, which describes the condensate’s wavefunction and density distribution.

Although neglecting key aspects like kinetic energy and quantum fluctuations, the Thomas–Fermi approximation provides an initial guess for the wavefunction and density distribution of a BEC, allowing for quick and computationally efficient calculations of its properties like size, shape, and energy. The approximation hinges on three key assumptions: dominant interaction, constant local density, and mean-field approach. It assumes that the repulsive interactions between atoms are much stronger than the kinetic energy associated with their movement. This leads to a highly localized state where atoms minimize their potential energy due to interactions. Within this approximation, the density of atoms (or bosons) is considered constant within small local regions and varies smoothly across the system. It then replaces the complex interactions between individual particles with an average “mean field” felt by all.

Recognizing the limitation of the Thomas–Fermi approximation, Griffin’s work shed light on the necessity of incorporating additional factors [204]. First, he emphasized the importance of considering the depletion of the condensate, where a fraction of atoms escape the condensed state due to thermal excitations. This phenomenon, neglected in the Thomas–Fermi approach, significantly impacts the overall density profile and dynamics of the condensate. Second, Griffin highlighted the role of anomalous Bose correlations, arising from quantum statistics governing identical bosons. These correlations lead to unique interactions beyond simple pairwise potentials, influencing condensate properties and collective excitations.

Moreover, Griffin employed the Hohenberg–Martin classification scheme to critically analyze the self-consistent HFB approximation within the BEC context. This framework categorizes approximations based on their ability to conserve fundamental quantities and preserve a “gapless” single-particle spectrum, reflecting the presence of low-energy excitations. Notably, Griffin demonstrated that the Popov approximation to the full HFB theory falls short in maintaining a gapless spectrum at all temperatures, highlighting its limitations.

Alternative approaches have emerged to address these shortcomings. Notably, variational methods, as proposed in [205,206], offer tools for obtaining analytical solutions that capture the depletion of the condensate and incorporate beyond-mean-field correlations. These refined approaches provide a more comprehensive understanding of BEC dynamics, particularly for dilute ultracold atom clouds, where thermal effects and quantum fluctuations become significant. By delving deeper into the limitations of the Thomas–Fermi approximation and exploring alternative methods like those proposed by Griffin and others, researchers continue to refine our understanding of BECs and their remarkable behaviors.

The GP equation for BEC has been extensively explored and solved numerically, including direct numerical integration for both time-independent [207] and time-dependent GP equations [208]. Additional numerical techniques include the semi-implicit Crank–Nicholson scheme [209], an eigenfunction basis expansion method [210], an explicit finite-difference scheme [211], and the time-splitting spectral method [212], among others.

The GP equation finds another application for a 1D cloud of bosons [213]. This 1D bosonic gas with a roughly uniform potential is created by condensing atoms in an elongated trap. The phenomena occurring at the trap’s center, where the density is approximately uniform and the condensate behaves like a fluid, are particularly interesting. For repulsive interactions, the GP equation solution takes the form of a hyperbolic tangent profile, while for attractive interactions, it becomes a hyperbolic secant profile.

An extensive review of the mean-field theory applied to BEC is provided by Dalfovo and collaborators [181]. Additionally, consult reviews in [214,215] for deeper dives. Interested in a powerful interplay between theoretical and experimental contributions to BEC? Look no further than [216].

6. Conclusions

This review article has provided a brief glimpse of the remarkable tapestry woven by the NLS equation and its far-reaching applications in diverse fields. Section 2 outlined the derivation from Maxwell’s and Helmholtz’s equations of classical electromagnetism. Section 3 delved into the heuristic derivation of the NLS equation, its fundamental solutions, and the temporal and spatial NLS equations. It also discusses the local existence and uniqueness of solutions for the NLS equation, providing a detailed overview of the mathematical and physical aspects of NLS systems. Section 4 followed by deriving the NLS equation from a cubic Klein–Gordon equation, and Section 5 explained the derivation of the GP equation, where the NLS equation is often known, from the BEC state.

The intricate relationship between physical variables and governing equations can lead to varied forms for the NLS equation’s dispersive and nonlinear coefficients, resulting in the behavior of wave packets and specific values for these coefficients. Notably, NLS equation coefficients derived from Maxwell’s equations differ from those obtained from a nonlinear Klein–Gordon equation or the KdV equation with exact dispersion. Interestingly, despite depending on the same wave properties (fundamental frequency or its corresponding wavenumber), the dispersive coefficient typically scales with the second derivative of the linear dispersion relationship. In contrast, the nonlinear coefficient varies depending on the specific system it describes.

The journey through the review article culminates in a deeper appreciation of the NLS equation’s power and versatility, not only in explaining known phenomena but also in its ability to describe exciting scientific frontiers yet to be explored. We have witnessed its diverse talents, including modeling the vibrant pulses of light in nonlinear optics, the elegance of water waves, the enigmatic dance of solitons in superconductors, and the quantum mysteries of Bose–Einstein condensates. Thanks to its integrability, the NLS equation admits multiple families of exact analytical solutions, which serve as models for understanding a freak wave phenomenon, not only in light and water but also in specific environments like matter waves. Abundant research on this topic, both theoretical and experimental, exists in published literature. Despite its limitations, the NLS equation’s ability to bridge seemingly disparate worlds is truly extraordinary. This universal language unites diverse physical phenomena under a single elegant framework.

A natural extension of this study is to explore in greater depth the mathematical and physical aspects of other NLS systems, such as the discrete NLS equation, the vector NLS equation, higher spatial dimension NLS models, and the fractional NLS equation. Gaining a deeper understanding of their properties, solutions, and applications could reveal new insights and connections to broader theories of waves and solitons. The model’s versatility could provoke interest in unexplored connections and bring together researchers from various disciplines. We hope this exploration serves as a valuable resource for researchers, scientists, and engineers interested in nonlinear wave theory, inspiring a deeper desire to explore further and reveal the mysteries still held within the intricate language of the NLS equation.