We establish nonlinear stability and asymptotic behavior of traveling periodic waves of viscous conservation laws under localized perturbations or nonlocalized perturbations asymptotic to constant shifts in phase, showing that long-time behavior is governed by an associated secondorder formal Whitham modulation system. A key point is to identify the way in which initial perturbations translate ...

Full dispersive models of water waves, such as the Whitham equation and full dispersion Kadomtsev–Petviashvili (KP) equation, are interesting from both physical mathematical points view. This paper studies analogous KP nonlinear elastic waves propagating in a nonlocal medium. In particular we consider anti-plane shear which assumed to be small-amplitude long waves. We propose two different exte...

a modified f-expansion method to find the exact traveling wave solutions of  two-component nonlinear partial differential equations (nlpdes) is discussed. we use this method to construct many new solutions to the nonlinear whitham-broer-kaup system (1+1)-dimensional. the solutions obtained include jacobi elliptic periodic wave solutions which exactly degenerate to the soliton solutions, triangu...

In this paper, the two-component Fornberg–Whitham system is studied. We first investigate local well-posedness in classical Sobolev space and establish a blow-up criterion by local-in-time priori estimate, then, we give some sufficient conditions on initial data to lead wave breaking. Finally, discuss analytically existence of periodic traveling waves bifurcation theorem.

We analyze the spectral stability of small-amplitude, periodic, traveling-wave solutions a Boussinesq–Whitham system. These are shown numerically to exhibit high-frequency instabilities when subject bounded perturbations on real line. use formal perturbation method estimate asymptotic behavior these in small-amplitude regime. compare results with direct numerical computations.