Existence and linearized stability of solitary waves for a quasilinear Benney system

Abstract

We prove the existence of solitary wave solutions to the quasilinear Benney system $$i{u}_t+u_{xx}=a|u{|}^{p}u+uv,v_t+f(v{)}_x=(|u{|}^2{)}_x$$ where $f(v)=-\gamma v^3$, $-1<p<+\infty$ and $a,\gamma >0$. We establish, in particular, the existence of travelling waves with speed arbitrary large if $p<0$ and arbitrary close to $0$ if $p>\frac{2}{3}$. We also show the existence of standing waves in the case $-1<p\le \frac{2}{3}$, with compact support if $-1<p<0$.\\ Finally, we obtain, under certain conditions, the linearized stability of such solutions.