# Nonlinear Dynamics of Wave Packets in Tunnel-Coupled Harmonic-Oscillator Traps

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## Abstract

**:**

## 1. Introduction

## 2. The Symmetric System

#### 2.1. The Coupled Equations

#### 2.2. The Transition from Regular to Chaotic Dynamics

#### 2.3. Spontaneous Symmetry Breaking (SSB) between the Coupled Components

## 3. The Half-Trapped System

#### 3.1. The Linearized System: Analytical and Numerical Results

#### 3.1.1. Emission of Radiation in the Untrapped Component

#### 3.1.2. The Shift of the GS and DM Existence Thresholds at Small Values of Coupling Constant $\lambda $

#### 3.1.3. The Analysis for Large Values of $\lambda $

#### 3.1.4. Exact Solutions for One- and Two-Dimensional Bound States in the Continuum (BIC) in the Linear System

#### 3.2. The Nonlinear Half-Trapped System

#### 3.2.1. The Thomas–Fermi Approximation (TFA)

#### 3.2.2. Existence Boundaries for Nonlinear States

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**A typical example of a regular Josephson dynamical regime, initiated by the DM (dipole mode) input launched in the u component (as given by Equation (5) with $z=0$ and ${A}_{0}=1$). The solution is produced by simulations of Equation (1) with $\lambda =\sigma =\mathsf{\Omega}=1$, $g=0$. Plots in the top row display the evolution of components $u\left(x,z\right)$ and $v\left(x,z\right)$. Left bottom: The evolution of the peak intensities of both components, ${U}_{max}^{2}\left(z\right)\equiv \underset{x}{\mathrm{max}}\left\{{\left|u(x,z)\right|}^{2}\right\}$ and ${V}_{max}^{2}\left(z\right)\equiv \underset{x}{\mathrm{max}}\left\{\left[{\left|v(x,z)\right|}^{2}\right]\right\}$. Right bottom: The power spectrum of oscillations of the two components, defined as per Equation (7). The spectra are virtually identical for both components.

**Figure 3.**Heatmaps of values of sharpness (8) of the central spectral peak quantifying proximity of the system’s dynamics to the regular regime. The maps are plotted in the plane of the linear-coupling strength, $\lambda $, and intensity of the input, ${A}_{0}^{2}$, which is launched in one component. (

**Top left**): The GS input, given by Equation (4) at $z=0$, in the case of the self-attraction ($\sigma =+1$). (

**Top right**): The same, but in the case of self-repulsion ($\sigma =-1$). (

**Bottom left**): The same as in the top left panel, but produced by the DM input, given by Equation (5) at $z=0$. (

**Bottom right**): The same as in the bottom left panel, but in the case of self-repulsion ($\sigma =-1$). In all cases, $g=0$ is set in Equation (1) (no XPM interaction between the components). Black curves cutting the left panels in their lower areas designate the onset of SSB (spontaneous symmetry breaking), signalized by appearance of $\theta \ne 0$, see Equation (9).

**Figure 4.**Plots of the SSB (spontaneous symmetry breaking) in the dynamical states initiated by the GS (ground state) and DM (dipole mode) inputs: the asymmetry parameter, defined as per Equation (9), is plotted versus the intensity of the input, ${A}_{0}^{2}$, for three different fixed values of the linear-coupling constant, $\lambda $, as indicated in the figure.

**Figure 5.**Simulations of the evolution of the linearized half-trapped system (10), displayed in the untrapped component by plotting Re$\left(v\left(x,z\right)\right)$. (

**Left**): Emission of radiation generated by the GS (ground state) populating the trapped component, $u\left(x,z\right)$ (see Equation (14)), with $\omega =0.25$ in Equation (10). (

**Right**): The same, but for the radiation generated by the DM (dipole mode) in the trapped component (see Equation (15)), with $\omega =0$. In both cases, the linear-coupling constant is $\lambda =0.05$.

**Figure 6.**(

**Left**): The evolution of fields u and v in the half-trapped system, as produced by simulations of Equation (10) with $\omega =0$ and coupling constant $\lambda =1$, initiated by the DM input in the u core, taken as per Equation (14) with ${A}_{0}=0.1$. Here and in Figure 7, spurious left-right asymmetry of the radiation field in the v component is an illusion produced by plotting. (

**Right**): The evolution of the peak intensities of both components, $\underset{x}{\mathrm{max}}\left\{{\left|u(x,t)\right|}^{2}\right\}$ and $\underset{x}{\mathrm{max}}\left\{{\left|v\left(x,t\right)\right|}^{2}\right\}$.

**Figure 7.**The same as in Figure 6, but for a large amplitude of the DM input in the u component, ${A}_{0}=8$.

**Figure 8.**The analytically predicted and numerically found threshold values of the mismatch parameter, $\omega $, above which the GS and DM solutions (the left and right panels, respectively) are produced, for the half-trapped system, by the linearized version of Equations (12) and (13), vs. coupling constant $\lambda $. The analytical results are produced by Equation (27). For the GS, they are shown (in the inset) only for relatively small values of $\lambda $, as in this case the analytical approximation is inaccurate at larger $\lambda $.

**Figure 9.**A bound state of the DM (dipole mode) type in the half-trapped system, found as a numerical solution of Equations (12) and (13) with $\omega =1.4$, $\lambda =0.225$, and $\sigma =0$ (the linearized version). The eigenvalue corresponding to this solution is $\mu \approx -0.036$.

**Figure 10.**(

**The top row**): left and right panels display the analytically predicted eigenvalues given by Equation (31) with $n=0$ and $n=1$, for the ground state (GS) and dipole mode (DM), respectively, of the half-trapped system, and their counterparts produced by the numerical solution of linearized coupled Equations (12) and (13), as functions of the linear-coupling constant, $\lambda $, and mismatch parameter, $\omega $. (

**The bottom row**): cross sections of the respective top panels along the diagonal connecting points $\left(\lambda ,\omega \right)=\left(0.10\right)$ and $\left(10,0\right)$. The results shown in these plots are relevant for relatively large values of $\lambda $.

**Figure 11.**The evolution initiated by the exceptional (BIC) exact solution (32) of the system of linearized Equations (12) and (13), as produced by simulations of the full nonlinear half-trapped system (10), with the attractive SPM, $\sigma =1$ in the top row, and repulsive $\sigma =-1$ in the bottom one. Other parameters are $g=0$, $\lambda =7/2$ and $\phantom{\rule{3.33333pt}{0ex}}\omega =1/2$, which are related as per Equation (32).

**Figure 12.**A typical example of the GS solution predicted by TFA (Thomas–Fermi approximation) for the half-trapped system, as per Equations (36)–(39), for $\sigma =-1$, $g=0$, $\lambda =8$, $\mu =-4$, and its comparison to the numerically found counterpart. The respective value of parameter m (see Equation (38)), which should be small for the applicability of TFA, is $m=0.125$.

**Figure 13.**The heatmap of threshold values of the mismatch parameter, ${\left({\omega}_{\mathrm{GS}}\right)}_{\mathrm{thr}}$, in the half-trapped system, based on Equations (12) and (13). For given values of the nonlinearity and linear-coupling coefficients, $\sigma $ and $\lambda $, the stable GS (ground state), subject to the normalization condition $P=1$ (see Equation (16)), exist above the threshold, i.e., at $\omega \ge {\left({\omega}_{\mathrm{GS}}\right)}_{\mathrm{thr}}$. Positive and negative values of $\sigma $ correspond to the attractive and repulsive sign of the self-interaction, respectively. The nontrivial region is one confined by red lines, in which the result is ${\left({\omega}_{\mathrm{GS}}\right)}_{\mathrm{thr}}<1/2$, i.e., the nonlinearity and linear coupling help to maintain self-trapped GSs in the parameter area where the decoupled system, with $\lambda =0$, cannot create such states.

**Figure 14.**The same as in Figure 13, but for DMs (dipole modes), obtained as numerical solutions to Equations (12) and (13) with $g=0$ and $g=1$ (the left and right panels, respectively). In this case, the nontrivial region, located between the red boundaries, is one with ${\left({\omega}_{\mathrm{DM}}\right)}_{\mathrm{thr}}<3/2$.

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**MDPI and ACS Style**

Hacker, N.; Malomed, B.A.
Nonlinear Dynamics of Wave Packets in Tunnel-Coupled Harmonic-Oscillator Traps. *Symmetry* **2021**, *13*, 372.
https://doi.org/10.3390/sym13030372

**AMA Style**

Hacker N, Malomed BA.
Nonlinear Dynamics of Wave Packets in Tunnel-Coupled Harmonic-Oscillator Traps. *Symmetry*. 2021; 13(3):372.
https://doi.org/10.3390/sym13030372

**Chicago/Turabian Style**

Hacker, Nir, and Boris A. Malomed.
2021. "Nonlinear Dynamics of Wave Packets in Tunnel-Coupled Harmonic-Oscillator Traps" *Symmetry* 13, no. 3: 372.
https://doi.org/10.3390/sym13030372