Existence of traveling-wave solutions to Boussinesq systems

Abstract

In this manuscript, the existence of traveling-wave solutions to Boussinesq systems \begin{equation*} \left\{ \begin{matrix} \eta_t +u_x+(\eta u)_x +au_{xxx}-b \eta_{xxt}=0, &\\ u_t + \eta_x +uu_x +c\eta_{xxx} - d u_{xxt}=0,&\\ \end{matrix} \right. \end{equation*} is established. We prove that all the systems with $ a<0,$ $ c<0$ and $ b=d$ exhibit traveling-wave solutions with small propagation speeds. The result complements our earlier work [6] on a restricted family of the systems where both existence and stability of traveling-wave solutions were established in the presence of large surface tension, namely when $a+b+c+d<0$.