Dispersive Nonlinear Waves in Two‐Layer Flows with Free Surface Part II. Large Amplitude Solitary Waves Embedded into the Continuous Spectrum

Abstract

In this paper we study the dispersive model derived in Part I, for the description of long wave propagation in two-layer flows with free surface. As in the case of the full water-wave problem, this model reproduces the resonance between short waves and long waves. The resulting wave is a generalized solitary wave, characterized by ripples in the far field in addition to the solitary pulse. In this work we focus on particular members of this family resulting from vanishing ripples. These are called embedded solitary waves and they correspond to true homoclinic orbits. Two wave regimes, characterized by elevation or depression of the interface between the layers, are presented. A critical depth ratio separates these two regimes. It is shown how this relates to a change of the global properties for the potential of the Hamiltonian system derived for traveling waves. In oceanic conditions, solitary waves are presented and their broadening is observed as the wave speed increases. We have observed that, for such waves to exist, their speed cannot exceed a certain limit value depending on the density ratio and thickness of each fluid. Finally, other sets of parameters were considered for which multihumped solitons exist, showing the richness and complexity of the Hamiltonian system considered here.

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

“…The existence of isolated solitary waves can be explained in terms of a homoclinic orbit that passes through the origin when the equations for steady solitary-wave solutions are expressed in an equivalent spatial dynamical system. For instance, isolated interfacial solitary waves were computed under the rotational-flow assumption in [31]. See [32] for general statements for the bifurcation of such isolated solitary waves.…”

Inhibiting Shear Instability Induced by Large Amplitude Internal Solitary Waves in Two‐Layer Flows with a Free Surface

“…One example is the generalized solitary waves that can only exist for the free-surface case (see [6,7]). In this case, internal solitary waves with multi-humped profiles have been observed by Barros and Gavrilyuk [8]. Also, on the hyperbolicity of the two-layer shallow water equations, distinct features between the two configurations can be found [9].…”

“…This same structure holds also for its traveling-wave solutions, which revealed in [8] as a valuable tool to characterize these solutions. In their paper, Barros and Gavrilyuk [8] have shown quite a rich diversity of solitary-wave solutions that is absent in the rigid-lid case. In particular, the multi-humped solitary-wave solutions exhibited there show the richness and complexity of the Hamiltonian system with two degrees of freedom describing traveling-wave solutions.…”

“…For this equation, we seek relations involving these parameters for which Equation (4) has only real roots. Notice that the system is always linearly stable for F = 0 since the dispersion relation (2) reduces to a biquadratic form with four distinct real solutions (see [8]). Several attempts have been made in the past to obtain conditions in terms of the literal coefficients of a polynomial, concerning a special root distribution (see [19] and references therein).…”

“…The system is endowed with a Hamiltonian structure and the momentum equations can be found as the Euler-Lagrange equations for an approximate Lagrangian to the Euler equations for two-layer flows with the free surface [11]. This same structure holds also for its traveling-wave solutions, which revealed in [8] as a valuable tool to characterize these solutions. In their paper, Barros and Gavrilyuk [8] have shown quite a rich diversity of solitary-wave solutions that is absent in the rigid-lid case.…”