## 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.

Original language | British English |
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Pages (from-to) | 1201-1215 |

Number of pages | 15 |

Journal | Ukrainian Journal of Physics |

Volume | 59 |

Issue number | 12 |

DOIs | |

State | Published - 2014 |

## Keywords

- Finite depth
- Gravity waves
- Multiple-scale expansions
- Nonlinear Schr
- Odinger equation
- Quasi-soliton
- Slow modulations
- Wave envelope