• Українська
  • English

< | Next issue >

Ukr. J. Phys. 2015, Vol. 59, N 12, p.1201-1215
doi:10.15407/ujpe59.12.1201    Paper

Gandzha I.S.1, Sedletsky Yu.V.1, Dutykh D.S.2

1 Institute of Physics, Nat. Acad. of Sci. of Ukraine
(46, Prosp. Nauky, Kyiv 03028, Ukraine; e-mail: gandzha@iop.kiev.ua, sedlets@iop.kiev.ua)
2 Université de Savoie Mont Blanc
(CNRS–LAMA UMR 5127, Campus Universitaire, 73376 Le Bourget-du-Lac, France; e-mail: Denys.Dutykh@univ-savoie.fr)

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

Section: Nonlinear processes
Original Author's Text: English

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.

Key words: nonlinear Schrödinger equation, gravity waves, finite depth, slow modulations, wave envelope, quasi-soliton, multiple-scale expansions.