High-order nonlinear Schrödinger equation for the envelope of slowly modulated gravity waves on the surface of finite-depth fluid and its quasi-soliton solutions

Abstract

We consider the high-order nonlinear Schrödinger equation derived earlier by Sedletsky [Ukr. J. Phys. 48(1), 82 (2003)] for the first-harmonic envelope of slowly modulated gravity waves on the surface of finite-depth irrotational, inviscid, and incompressible fluid with flat bottom. This equation takes into account the third-order dispersion and cubic nonlinear dispersive terms. We rewrite this equation in dimensionless form featuring only one dimensionless parameter kh, where k is the carrier wavenumber and h is the undisturbed fluid depth. We show that one-soliton solutions of the classical nonlinear Schrödinger equation are transformed into quasi-soliton solutions with slowly varying amplitude when the high-order terms are taken into consideration. These quasi-soliton solutions represent the secondary modulations of gravity waves.

Keywords

Bibliographic record

BibTeX Citation

@article{Gandzha2014highordernonlinear,
  author = {Gandzha, I. and Sedletsky, Yu. and Dutykh, D.},
  title = {High-order nonlinear Schrödinger equation for the envelope of slowly modulated gravity waves on the surface of finite-depth fluid and its quasi-soliton solutions},
  journal = {Ukr. J. Phys.},
  year = {2014},
  volume = {59},
  number = {12},
  pages = {1201--1215},
  doi = {10.15407/ujpe59.12.1201},
  abstract = {We consider the high-order nonlinear Schrödinger equation derived earlier by Sedletsky [Ukr. J. Phys. 48(1), 82 (2003)] for the first-harmonic envelope of slowly modulated gravity waves on the surface of finite-depth irrotational, inviscid, and incompressible fluid with flat bottom. This equation takes into account the third-order dispersion and cubic nonlinear dispersive terms. We rewrite this equation in dimensionless form featuring only one dimensionless parameter kh, where k is the carrier wavenumber and h is the undisturbed fluid depth. We show that one-soliton solutions of the classical nonlinear Schrödinger equation are transformed into quasi-soliton solutions with slowly varying amplitude when the high-order terms are taken into consideration. These quasi-soliton solutions represent the secondary modulations of gravity waves.},
  keywords = {multiple-scale expansions, quasi-soliton, finite-depth fluid, third-order dispersion, nonlinear Schrödinger equation, gravity waves, fluid dynamics, wave propagation, modulation instability, slow modulations, cubic nonlinearity, finite depth, wave envelope, Nonlinear Schrödinger equation, quasi-soliton solutions, surface waves}
}