Orbital Stability of Internal Waves

This paper studies the nonlinear stability of capillary-gravity waves propagating along interface dividing two immiscible fluid layers finite depth. The motion in both regions is governed by incompressible and irrotational Euler equations, with density each being constant but distinct. A diverse collection small-amplitude solitary wave solutions for this system have been constructed several authors case strong surface tension (as measured Bond number) slightly subcritical Froude number. We prove that all these are (conditionally) orbitally stable natural energy space. Moreover, trivial solution shown to be conditionally when numbers lie a certain unbounded parameter region. For near critical regime, we one can infer conditional orbital or instability traveling full from considerations dispersive PDE model equation. These results obtained reformulating problem as an infinite-dimensional Hamiltonian system, then applying version Grillakis–Shatah–Strauss method recently introduced Varholm et al. (Commun Pure Appl Math 73:2634–2684, 2020). key part analysis consists computing spectrum linearized augmented at shear flow wave. this, generalize idea used Mielke (R Soc Lond Philos Trans Ser Phys Eng Sci 360:2337–2358, 2002) treat water beneath vacuum.