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.
Categories: math.AP