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 $$u_t+\eta_x+uu_x+c\eta{xxx}=0,\eta_t+u_x+(\eta u)x+au{xxx}=0$$ 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.