Dynamics of generalized abcd Boussinesq solitary waves under a slowly variable bottom

Abstract

The Boussinesq $abcd$ system is a 4-parameter set of equations posed in $\mathbb R_t\times\mathbb R_x$, originally derived by Bona, Chen and Saut as first-order 2-wave approximations of the incompressible and irrotational, two-dimensional water wave equations in the shallow water wave regime, in the spirit of the original Boussinesq derivation. Among the various particular regimes, each determined by the values of the parameters $(a, b, c, d)$ appearing in the equations, the \emph{generic} regime is characterized by the conditions $b, d > 0$ and $a, c < 0$. If additionally $b=d$, the $abcd$ system is Hamiltonian. In this paper, we investigate the existence of generalized solitary waves and the corresponding collision problem in the physically relevant \emph{variable bottom regime}, introduced by M.\ Chen. More precisely, the bottom is represented by a smooth space-time dependent function $h=\varepsilon h_0(\varepsilon t,\varepsilon x)$, where $\varepsilon$ is a small parameter and $h_0$ is a fixed smooth profile. This formulation allows for a detailed description of weak long-range interactions and the evolution of the solitary wave without its destruction. We establish this result by constructing a new approximate solution that captures the interaction between the solitary wave and the slowly varying bottom.