Deep-water internal solitary waves near critical density ratio

Abstract

Bifurcations of solitary waves propagating along the interface between two ideal fluids are considered. The study is based on a Hamiltonian approach. It concentrates on values of the density ratio close to a critical one, where the supercritical bifurcation changes to the subcritical one. As the solitary wave velocity approaches the minimum phase velocity of linear interfacial waves (the bifurcation point), the solitary wave solutions transform into envelope solitons. In order to describe their behavior and bifurcations, a generalized nonlinear Schrödinger equation describing the behavior of solitons and their bifurcations is derived. In comparison with the classical NLS equation this equation takes into account three additional nonlinear terms: the so-called Lifshitz term responsible for pulse steepening, a nonlocal term analogous to that first found by Dysthe for gravity waves and the six-wave interaction term. We study both analytically and numerically two solitary wave families of this equation for values of the density ratio ρ that are both above and below the critical density ratio ρcr. For ρ > ρcr, the soliton solution can be found explicitly at the bifurcation point. The maximum amplitude of such a soliton is proportional to √ ρ − ρcr, and at large distances the soliton amplitude decays algebraically. A stability analysis shows that solitons below the critical ratio are stable in the Lyapunov sense in the wide range of soliton parameters. Above the critical density ratio solitons are shown to be unstable with respect to finite perturbations.

A Boussinesq system for two-way propagation of interfacial waves

“…It does not matter for the asymptotic expansions to be performed later. 1 There is some arbitrariness in this choice since there are two fluid depths in the problem. We could have also chosen the depth of the top layer as reference depth.…”

Long-wave transverse instability of interfacial gravity–capillary solitary waves in a two-layer potential flow in deep water

“…This definition for the total mechanical energy of interfacial gravity-capillary solitary waves was also adopted in [29,30], except for the way of nondimensionalization. Then, it is shown that…”

Intermittency in generalized NLS equation with focusing six-wave interactions

“…theory expansion beyond the four-wave interactions term |Ψ| 2 Ψ. One of these terms is the six-wave interactions α|Ψ| 4 Ψ, that appears in many physical models including Bose-Einstein condensation [19,20], surface water waves [21,22], pattern formation in the framework of the Ginzburg -Landau equation [23], dissipative solitons in lasers [24], optical fibers [25], and so on. Here α is the six-wave coupling coefficient.…”