We study both supercritical and subcritical bifurcations of internal solitary waves propagating along the interface between two ideal fluids. We derive a generalized nonlinear Schrödinger equation that describes solitons near the critical density ratio corresponding to the transition from a subcritical to a supercritical bifurcation. This equation takes into account gradient terms associated with the four-wave interactions, the so-called Lifshitz term and a nonlocal term analogous to that first found by Dysthe for pure gravity waves, as well as the term representing six-wave nonlinear interactions. Within this model we find two branches of solitons and analyze their Lyapunov stability. 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.
- Hamiltonian description
- Lyapunov stability
- Sobolev type inequalities
- Supercritical and subcritical bifurcations