Existence of traveling-wave solutions to Boussinesq systems. (English) Zbl 1249.35286

Chen, Min; Nguyen, Nghiem V.; Sun, Shu-Ming

Differ. Integral Equ. 24, No. 9-10, 895-908 (2011).

Summary: In this manuscript the existence of traveling-wave solutions to Boussinesq systems \(\eta _t+u_x+(\eta u)_x+au_{xxx}-b\eta _{xxt}=0\), \(u_t+\eta _x+uu_x+c\eta _{xxx}-du_{xxt}=0\) 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 [Discrete Contin. Dyn. Syst. 26, No. 4, 1153–1184 (2010; Zbl 1198.35191)] 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 \(a+b+c+d<0\).

MSC:

35Q53	KdV equations (Korteweg-de Vries equations)
35C07	Traveling wave solutions
35G50	Systems of nonlinear higher-order PDEs

Keywords:

Boussinesq system; traveling wave; existence