The scattering problem for the $abcd$ Boussinesq system in the energy space

TL;DR

This paper studies the decay and scattering of small solutions to the energy space for the $abcd$ Boussinesq system, revealing conditions under which solutions decay locally in the energy space despite long-range nonlinearities.

Contribution

It introduces a novel virial functional approach to prove decay and scattering for small solutions in the energy space without requiring parity or additional decay assumptions.

Findings

Small solutions decay locally in the energy space within the light cone.

Constructed virial functionals enable control of solutions over time.

Identified parameter regimes where solutions exhibit decay despite long-range 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 many particular regimes, depending each of them in terms of the value of the parameters $(a,b,c,d)$ present in the equations, the "generic" regime is characterized by the setting $b,d>0$ and $a,c<0$. The system is hamiltonian if also $b=d$. The equations in this regime are globally well-posed in the energy space $H^1\times H^1$, provided one works with small solutions. In this paper, we investigate decay and the scattering problem in this regime, which is characterized as having (quadratic) long-range nonlinearities, very weak linear decay $O(t^{-1/3})$ because of the one dimensional setting, and existence of non scattering solutions (solitary waves). We prove, among other results, that for a sufficiently dispersive $abcd$ systems (characterized only in terms of parameters $a, b$ and $c$), all small solutions must decay to zero, locally strongly in the energy space, in proper subset of the light cone $|x|\leq |t|$. We prove this result by constructing three suitable virial functionals in the spirit of works by Kowalczyk, Martel and the second author, and more recently by the last three authors (valid for the simpler scalar "good Boussinesq" model), leading to global in time decay and control of all local $H^1\times H^1$ terms. No parity nor extra decay assumptions are needed to prove decay, only small solutions in the energy space.