Existence of traveling-wave solutions to Boussinesq systems

Abstract

In this manuscript, the existence of traveling-wave solutions to Boussinesq systems \eta_t + u_x + (\eta u)_x + a u_{xxx}-b \eta_{xxt} = 0, u_t +\eta_x + uu_x + c\eta_{xxx}- d u_{xxt} = 0, 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.