# Efficient Computation of the Nonlinear Schrödinger Equation with Time-Dependent Coefficients

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theory

#### 2.1. Explicit Runge–Kutta Pairs

#### 2.2. Phase-Lag and Stability

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

## 3. Construction and Analysis

- Constant coefficients, since no dominant frequency exists for different values of x, as observed in Figure 1.
- Maximised phase-lag and amplification-error orders m and r of Definition 1, for improved behaviour when solving Equation (1) with periodic/oscillatory solutions.
- Maximised real stability interval, based on Definition 4.
- Coefficients with similar orders of magnitude, to minimise the round-off error.
- Low-order method with similar stability characteristics to the high-order one, to improve the local error estimation for extreme step sizes.

## 4. A Modified Step Size Control Algorithm

## 5. Numerical Experiments

#### 5.1. Modified Step Size Control

#### 5.2. Results

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Notation

Nonlinear Schrödinger equation | |

$\psi (x,t)$ | mean-field matter wave function/complex electric field envelope |

t | time/propagation distance |

x | longitudinal coordinate/transverse coordinate |

$a\left(t\right)$ | dispersion |

$b\left(t\right)$ | nonlinearity |

$E\left(\psi \right)$ | global norm |

Runge–Kutta methods | |

$y\left(x\right)$ | theoretical solution |

${y}_{n}\left(x\right)$ | numerical solution |

h | step size |

$q,p$ | low and high algebraic orders |

${y}_{m}^{n}$, ${\widehat{y}}_{m}^{n}$ | low-order and high-order approximations of $\psi ({x}_{m},{t}_{n})$ |

${a}_{ij},{b}_{i},{\widehat{b}}_{i},{c}_{i}$ | Runge–Kutta coefficients |

$\mathbf{A},\mathbf{b},\widehat{\mathbf{b}},\mathbf{c}$ | Runge–Kutta coefficient matrices |

$EST$ | local error estimation |

$TOL$ | tolerance |

Phase-lag and stability | |

$\omega $ | frequency |

$R\left(v\right)$ | stability polynomial |

${I}_{I}$ | imaginary stability interval |

${I}_{R}$ | real stability interval |

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**Figure 1.**Real part $\Re \left(\psi (x,t)\right)$, showing different frequencies for various values of x.

**Figure 2.**Stability regions of both high-order and low-order methods of the new Runge–Kutta (RK) pair of Table 1.

**Figure 4.**Maximum absolute global norm error versus the function evaluations for the pairs PL8-AE9 (Table 1) (■), Papageorgiou et al. [29] (●) and Fehlberg [37] (▲). For each of the three pairs, Var II (solid line —) denotes the modified step size control algorithm described with Equation (10), while Var I (dashed line – –) denotes the original method in Equation (7).

**Figure 5.**Maximum absolute global norm error versus the function evaluations for all compared methods in Table 2.

**Figure 6.**The maximum absolute error of the solution $\underset{m}{max}}\left(\left|{y}_{m}^{n}-\psi ({x}_{m},{t}_{n})\right|\right)$ versus t.

**Figure 7.**The maximum absolute error of the square of the solution $\underset{m}{max}}\left(\left|{\left({y}_{m}^{n}\right)}^{2}-{\left(\psi ({x}_{m},{t}_{n})\right)}^{2}\right|\right)$ versus t.

**Figure 8.**The maximum absolute error of the solution $\underset{n}{max}}\left(\left|{y}_{m}^{n}-\psi ({x}_{m},{t}_{n})\right|\right)$ versus x.

0 | ||||||||

$\frac{1}{15}$ | $\frac{1}{15}$ | |||||||

$\frac{1}{5}$ | $-\frac{1}{10}$ | $\frac{3}{10}$ | ||||||

$\frac{1}{3}$ | $\frac{62-5\phantom{\rule{0.166667em}{0ex}}\sqrt{65}}{126}$ | $\frac{-55+5\phantom{\rule{0.166667em}{0ex}}\sqrt{65}}{84}$ | $\frac{125-5\phantom{\rule{0.166667em}{0ex}}\sqrt{65}}{252}$ | |||||

$\frac{2}{5}$ | $\frac{249-15\phantom{\rule{0.166667em}{0ex}}\sqrt{65}}{350}$ | $\frac{-141+9\phantom{\rule{0.166667em}{0ex}}\sqrt{65}}{140}$ | $\frac{89-3\phantom{\rule{0.166667em}{0ex}}\sqrt{65}}{140}$ | $\frac{3}{50}$ | ||||

$\frac{3}{5}$ | $\frac{192-7\phantom{\rule{0.166667em}{0ex}}\sqrt{65}}{350}$ | $\frac{-687+9\phantom{\rule{0.166667em}{0ex}}\sqrt{65}}{700}$ | $\frac{1019+37\phantom{\rule{0.166667em}{0ex}}\sqrt{65}}{700}$ | $-\frac{324+18\phantom{\rule{0.166667em}{0ex}}\sqrt{65}}{175}$ | $\frac{50+2\phantom{\rule{0.166667em}{0ex}}\sqrt{65}}{35}$ | |||

$\frac{4}{5}$ | $\frac{-2047+90\phantom{\rule{0.166667em}{0ex}}\sqrt{65}}{1750}$ | $\frac{591-27\phantom{\rule{0.166667em}{0ex}}\sqrt{65}}{350}$ | $\frac{285+18\phantom{\rule{0.166667em}{0ex}}\sqrt{65}}{700}$ | $-\frac{1071}{1000}$ | $\frac{21}{50}$ | $\frac{21}{40}$ | ||

1 | $\frac{396-15\phantom{\rule{0.166667em}{0ex}}\sqrt{65}}{119}$ | $\frac{-1020+45\phantom{\rule{0.166667em}{0ex}}\sqrt{65}}{238}$ | $\frac{225-30\phantom{\rule{0.166667em}{0ex}}\sqrt{65}}{476}$ | $-\frac{3261}{952}$ | $\frac{225}{34}$ | $-\frac{375}{136}$ | $\frac{125}{119}$ | |

b | $\frac{5}{96}$ | 0 | $\frac{125}{288}$ | $-\frac{81}{112}$ | $\frac{125}{144}$ | 0 | $\frac{625}{2016}$ | $\frac{17}{288}$ |

$\widehat{b}$ | $\frac{5}{96}$ | 0 | $\frac{383}{960}$ | $-\frac{333}{640}$ | $\frac{947}{1440}$ | $\frac{101}{1920}$ | $\frac{3}{10}$ | $\frac{17}{288}$ |

**Table 2.**Properties of the compared methods. H: High order, L: Low order (- means no step size control), S: Stages, PL: Phase-Lag order, AE: Amplification-Error order, IR: Real Stability Interval, II: Imaginary Stability Interval.

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**MDPI and ACS Style**

Kosti, A.A.; Colreavy-Donnelly, S.; Caraffini, F.; Anastassi, Z.A.
Efficient Computation of the Nonlinear Schrödinger Equation with Time-Dependent Coefficients. *Mathematics* **2020**, *8*, 374.
https://doi.org/10.3390/math8030374

**AMA Style**

Kosti AA, Colreavy-Donnelly S, Caraffini F, Anastassi ZA.
Efficient Computation of the Nonlinear Schrödinger Equation with Time-Dependent Coefficients. *Mathematics*. 2020; 8(3):374.
https://doi.org/10.3390/math8030374

**Chicago/Turabian Style**

Kosti, Athinoula A., Simon Colreavy-Donnelly, Fabio Caraffini, and Zacharias A. Anastassi.
2020. "Efficient Computation of the Nonlinear Schrödinger Equation with Time-Dependent Coefficients" *Mathematics* 8, no. 3: 374.
https://doi.org/10.3390/math8030374