Asymptotic dynamics for the small data weakly dispersive one-dimensional Hamiltonian ABCD system

TL;DR

This paper studies the asymptotic decay of small solutions to the weakly dispersive 1D Hamiltonian ABCD system, extending decay results to broader parameter regimes and larger spatial regions using improved virial estimates.

Contribution

It advances decay analysis for the Hamiltonian ABCD system by enlarging parameter regimes, covering the entire light cone, and analyzing exterior regions with new virial estimates.

Findings

Decay established for broader parameter regimes in weakly dispersive case.

Decay shown across the entire light cone, ruling out nonzero speed solitary waves.

Decay of solutions in exterior regions, excluding super-luminical solitary waves.

Abstract

Consider the Hamiltonian $abcd$ system in one dimension, with data posed in the energy space $H^1\times H^1$. This model, introduced by Bona, Chen and Saut, is a well-known physical generalization of the classical Boussinesq equations. The Hamiltonian case corresponds to the regime where $a,c<0$ and $b=d>0$. Under this regime, small solutions in the energy space are globally defined. A first proof of decay for this $2\times 2$ system was given by the two authors and Poblete and Pozo, in a strongly dispersive regime, i.e. under essentially the conditions \[ b=d > \frac29, \quad a,c<-\frac1{18}. \] Additionally, decay was obtained inside a proper subset of the light cone $(-|t|,|t|)$. In this paper, we improve the last result in three directions. First, we enlarge the set of parameters $(a,b,c,d)$ for which decay to zero is the only available option, considering now the so-called weakly dispersive regime $a,c\sim 0$: we prove decay if now \[ b=d > \frac3{16}, \quad a,c<-\frac1{48}. \] This result is sharp in the case where $a=c$, since for $a,c$ bigger, some $abcd$ linear waves of nonzero frequency do have zero group velocity. Second, we sharply enlarge the interval of decay to consider the whole light cone, that is to say, any interval of the form $|x|\sim |v|t$, for any $|v|<1$. This result rules out, among other things, the existence of nonzero speed solitary waves in the regime where decay is present. Finally, we prove decay to zero of small $abcd$ solutions in exterior regions $|x|\gg |t|$, also discarding super-luminical small solitary waves. These three results are obtained by performing new improved virial estimates for which better decay properties are deduced.