Numerical computation for rogue waves in the coupled nonlinear Schrödinger equations with the coherent coupling effect

ABSTRACT

We report some efficient and accurate numerical experiments to simulate the vector rogue waves of the coupled nonlinear Schrödinger equations with the coherent coupling coefficients, which describe the propagation of orthogonally polarized optical pulses in an isotropic medium. We develop the time-splitting finite difference/spectral methods and exponential wave integrator spectral methods to compute the vector rogue-wave solutions. Because of the slowly decaying ratios at far field of the rogue waves with nonzero backgrounds, the truncation errors of numerical solutions are analysed numerically. A small computation interval of errors is considered to efficiently simulate rogue waves and compare different numerical methods. We find that the homogeneous Neumann boundary condition is better than periodic boundary condition, and the latter is often used by researchers to simulate rogue waves. The time-splitting cosine spectral method has the highest accuracy among the above methods for simulating the rogue waves. The stability of rogue waves is numerically analysed quantitatively and qualitatively. The vector rogue-wave solutions are always unstable under small perturbations, while such rogue wave dynamics may have a certain degree of stability that it has potential to be observed in experiments.

KEYWORDS:

• Coupled nonlinear Schrödinger equations
• vector rogue waves
• time-splitting method
• stability of rogue waves

2010 AMS SUBJECT CLASSIFICATIONS:

• 35C08
• 35C11
• 65M15
• 65M70

Acknowledgments

The authors thank fruitful discussions with Prof. Weizhu Bao and thank the Editors and Reviewers for their valuable comments.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work has been supported by the National Natural Science Foundation of China under Grant (Nos. 11801140,11947230), and by the China Postdoctoral Science Foundation (No. 2019M660430).