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Nonlinear Dynamics

, Volume 97, Issue 4, pp 2443–2452 | Cite as

Rogue waves generation through multiphase solutions degeneration for the derivative nonlinear Schrödinger equation

  • Shuwei XuEmail author
  • Jingsong He
  • Dumitru Mihalache
Original paper
  • 124 Downloads

Abstract

The generation of rogue waves from the development of modulational instability of the plane waves due to small perturbations or interactions of solitons is usually modeled by some breather solutions, which are considered to as solitons on a finite background. The instability of small-amplitude perturbations is usually chaotic and may contain many frequencies in their spectra. Thus, a more general description of rogue wave generation can be achieved via the consideration of multiphase solutions and their interactions. The general N-order phase solutions of the derivative nonlinear Schrödinger (DNLS) equation are constructed from the trivial seed (zero solution) by using the determinant representation of the N-fold Darboux transformation. By adjusting the relative phases of the multiphase solutions in the interacting area, namely taking the limit \(\lambda _k \rightarrow \lambda _\mathrm{c}\) for some of \(\lambda _k\) (not for all of them), where \(\lambda _\mathrm{c}\) is a critical eigenvalue associated with the synchronization of each phase in the multiphase nonlinear waves of the DNLS equation, it is possible to obtain different types of quasi-rational solutions from the multiphase solutions degeneration mechanism: quasi-rational traveling wave solutions, rogue waves, and quasi-rogue waves with periodic conditions. Hence, this multiphase solution degeneration represents a new mechanism of rogue wave generation.

Keywords

Derivative nonlinear Schrödinger equation Multiphase solutions interactions Multiphase solutions degeneration Quasi-rational solutions Rogue waves Darboux transformation 

Notes

Acknowledgements

This work is supported by the National Natural Science Foundation of China under Grant Nos. 11601187 and 11671219 and Natural Science Foundation of Zhejiang Province under Grant No. LZ19A010001.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

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Authors and Affiliations

  1. 1.College of Mathematics Physics and Information EngineeringJiaxing UniversityJiaxingPeople’s Republic of China
  2. 2.Institute for Advanced StudyShenzhen UniversityShenzhenPeople’s Republic of China
  3. 3.Horia Hulubei National Institute for Physics and Nuclear EngineeringMagureleRomania

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