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

TL;DR

This work investigates the weak interaction between stable solitary waves of the one-dimensional abcd Boussinesq system and a slowly varying bottom. By developing a background-adapted energy/virial framework and a modulated approximation, the authors construct a generalized solitary wave in the uneven medium and prove that, after a finite interaction window, the solitary wave persists with an error of order $\varepsilon^{1/2}$. The approach hinges on detailed energy and momentum estimates, a coercive linearization around the soliton, and a carefully crafted linear-correction analysis that captures both spatial and temporal bottom variations. The results provide a rigorous global dynamics description for solitary waves in non-homogeneous dispersive media and offer a blueprint for extending to other shallow-water models with variable bottom effects.

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.

Figures (1)

• Figure 1: Left: The momenta $P(\boldsymbol{Q}^-_\omega)$ (in red) and $P(\boldsymbol{Q}^+_\omega)$ (in blue), as functions of $\omega\in (0,1)$. Right: Parametric curves $(P(\boldsymbol{Q}^-_\omega), E(\boldsymbol{Q}^-_\omega))$ (in red) and $(P(\boldsymbol{Q}^+_\omega), E(\boldsymbol{Q}^+_\omega))$ (in blue), plotted in solid lines for $\omega \in (0, 1)$, and in dotted lines for $\omega \in (-1,0)$.

Theorems & Definitions (43)

• Lemma 1.1
• proof
• Theorem 1.2
• Definition 2.1: Canonical variable
• Lemma 2.2: Equivalence of local $L^2$ and $H^1$ norms, KMPP
• Lemma 2.3: ElDika2005-2KMPP
• Lemma 2.4
• proof
• Lemma 2.5
• proof