On the Cauchy problem for the damped Boussinesq equation. (English) Zbl 0844.35095

Summary: A classic solution to the Cauchy problem for the damped Boussinesq equation \[ u_{tt}- 2Bu_{txx}=- \alpha u_{xxxx}+ u_{xx}- \beta(u^2)_{xx},\quad x\in \mathbb{R}^1,\quad t> 0, \] \(\alpha\), \(B= \text{const}> 0\), \(\beta= \text{const}\in \mathbb{R}^1\), with small initial data is constructed by means of the application of both the spectral and perturbation theories. Large time asymptotics of this solution are obtained. Its main term accounts for two solitons traveling in opposite directions. Each of them is governed by the Burgers equation with a transfer.


35Q35 PDEs in connection with fluid mechanics
35B40 Asymptotic behavior of solutions to PDEs
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems