A note on the existence of traveling-wave solutions to a Boussinesq system

Abstract

We obtain a one-parameter family $$(u_{\mu }(x,t),{\eta }_{\mu }(x,t){)}_{\mu \ge {\mu }_0}=({\varphi }_{\mu }(x-{\omega }_{\mu }t),{\psi }_{\mu }(x-{\omega }_{\mu }t){)}_{\mu \ge {\mu }_0}$$ of traveling-wave solutions to the Boussinesq system $$u_t+{\eta }_x+u{u}_x+c{\eta }_{xxx}=0,{\eta }_t+u_x+(\eta u{)}_x+a{u}_{xxx}=0$$ in the case $a,c<0$, with non-null speeds ${\omega }_{\mu }$ arbitrarily close to $0$ (${\omega }_{\mu }\underset{\mu \to +\infty }{\to }0$). We show that the $L^2$-size of such traveling-waves satisfies the uniform (in $\mu$) estimate $\Vert {\varphi }_{\mu }{\Vert }_2^2+\Vert {\psi }_{\mu }{\Vert }_2^2\le C\sqrt{|a|+|c|},$ where $C$ is a positive constant. Furthermore, ${\varphi }_{\mu }$ and $-{\psi }_{\mu }$ are smooth, non-negative, radially decreasing functions which decay exponentially at infinity.

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