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

Ivan S. Gandzha (1) , Yury Sedletsky (1) , Denys Dutykh (2, 3, 4)

1 Department of Theoretical Physics
2 LAMA - Laboratoire de Mathématiques
3 USMB [Université de Savoie] [Université de Chambéry] - Université Savoie Mont Blanc
4 INSMI-CNRS - Institut National des Sciences Mathématiques et de leurs Interactions - CNRS Mathématiques

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.

Mots clés: finite depth; Nonlinear Schrödinger equation; multiple-scale expansions; wave envelope; gravity waves; quasi-soliton; slow modulations

Domaines: Mécanique des fluides [physics.class-ph]; Dynamique des Fluides [physics.flu-dyn]; Physique Atmosphérique et Océanique [physics.ao-ph]; Physique Numérique [physics.comp-ph]; Analyse numérique [math.NA]; Equations aux dérivées partielles [math.AP]; Formation de Structures et Solitons [nlin.PS]; Systèmes Solubles et Intégrables [nlin.SI]

Ivan S. Gandzha, Yury Sedletsky, Denys Dutykh. 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. Ukrainian Journal of Physics, 2014, 59 (12), pp.1201-1215. ⟨10.15407/ujpe59.12.1201⟩. ⟨hal-01084747v3⟩