2025/11/26 by André de Laire, Olivier Goubet, de Laire, André +7
Earth and Planetary Sciences · Mathematics · Physics and Astronomy · #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Navier-Stokes equation solutions #Nonlinear Waves and Solitons #Ocean Waves and Remote Sensing
paper · pdf · doi:10.48550/arxiv.2511.21632
openalex publication_date 2025/11/26 · openalex created_date 2025/11/28 · openalex updated_date 2026/07/28
The Boussinesq abcd system is a 4-parameter set of equations posed in \mathbb Rt×\mathbb Rx, 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 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 variable bottom regime, introduced by M. Chen. More precisely, the bottom is represented by a smooth space-time dependent function h=ε h0(ε t,ε x), where ε is a small parameter and h0 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.