Asymptotic stability of solutions to the good Boussinesq equation in dispersive wave region

Abstract

This work studies the asymptotic stability of solutions to the good Boussinesq equation in dispersive wave region when the reflection coefficients associated with the initial data belong to weighted Sobolev space. The Dbar-steepest descent method is applied to the Riemann-Hilbert problem and the long-time asymptotic expansion of the solution are obtained up to an optimal error of order $\mathcal{O}(t^{-3/4})$. Compared with previous results, we extend the initial data from the rapidly decaying Schwartz space to a weighted Sobolev space, and prove the asymptotic stability of the solution in dispersive wave region.