The scattering problem for Hamiltonian ABCD Boussinesq systems in the energy space

Abstract

The Boussinesq abcd system is a 4-parameter set of equations posed in Rt×Rx, originally derived by Bona, Chen and Saut [11,12] 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 [17]. 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. If additionally b=d, the abcd system is Hamiltonian. The equations in this regime are globally well-posed in the energy space H1×H1, provided one works with small solutions [12]. 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 dispersiveabcd 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|≤|t|. We prove this result by constructing three suitable virial functionals in the spirit of works [27,28], and more precisely [42] (valid for the simpler scalar “good Boussinesq” model), leading to global in time decay and control of all local H1×H1 terms. No parity nor extra decay assumptions are needed to prove decay, only small solutions in the energy space.