Methods and Applications of Analysis

Volume 29 (2022)

Number 1

Special issue dedicated to Professor Ling Hsiao on the occasion of her 80th birthday, Part IV

Guest editors: Qiangchang Ju (Institute of Applied Physics and Computational Mathematics, Beijing), Hailiang Li (Capital Normal University, Beijing), Tao Luo (City University of Hong Kong), and Zhouping Xin (Chinese University of Hong Kong)

Long-time simulations of rogue wave solutions in the nonlinear Schrödinger equation

Pages: 149 – 160

DOI: https://dx.doi.org/10.4310/MAA.2022.v29.n1.a5

Authors

Chenxi Zheng (HEDPS and LTCS, College of Engineering, Peking University, Beijing, China)

Shaoqiang Tang (HEDPS and LTCS, College of Engineering, Peking University, Beijing, China)

Abstract

Although several short-time simulations have been reported nicely reproducing rogue wave solutions in the nonlinear Schrödinger equation, rogue wave solutions are linearly unstable as shown by theoretical studies. In the present work, we perform long-time simulations for two kinds of rogue wave solutions, namely, the Akhmediev breather and Peregrine soliton. Numerical evidences demonstrate that spurious oscillations that emerge in the central domain in both simulations arise from round-off error and evolve under the mechanism of modulational instability. For the periodic approximation of the Peregrine soliton, the modulational instability also gives rise to additional oscillations on the boundary. We obtain a fitting formula to forecast the time when the boundary oscillations occur. Our simulation results show that a clean and faithful long-time reproduction of rogue wave solutions would be difficult because of the modulational instability.

Keywords

nonlinear Schrödinger equation, modulational instability, numerical simulation, rogue wave

2010 Mathematics Subject Classification

65N12, 76B15

Received 23 May 2020

Accepted 16 September 2020

Published 10 June 2022