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

Abstract

We obtain a one-parameter family $$ (u_{\mu}(x,t),\eta_{\mu}(x,t))_{\mu\geq \mu_0} =(\phi_{\mu}(x-\omega_{\mu} t),\psi_{\mu}(x-\omega_{\mu} t))_{\mu\geq \mu_0} $$ of traveling-wave solutions to the Boussinesq system $$ \left\{\begin{array}{llll} u_t+\eta_x+uu_x+c\eta_{xxx}=0\qquad (x,t)\in\mathbb R^2 \\ \eta_t+u_x+(\eta u)_x+au_{xxx}=0, \end{array}\right. $$ in the case $a,c < 0$, with non-null speeds $\omega_{\mu}$ arbitrarily close to $0$ ($\omega_{\mu}\xrightarrow[\mu\to+\infty]{} 0$). We show that the $L^2$-size of such traveling-waves satisfies the uniform (in $\mu$) estimate $\|\phi_{\mu}\|_2^2+\|\psi_{\mu}\|_2^2\leq C\sqrt{|a|+|c|},$ where $C$ is a positive constant. Furthermore, $\phi_{\mu}$ and $-\psi_{\mu}$ are smooth, non-negative, radially decreasing functions which decay exponentially at infinity.