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

We prove the existence of solitary wave solutions to the quasilinear Benney system $$iu_{t}+u_{xx}=a|u|^pu+uv,\quad v_t+f(v)_x=(|u|^2)_x$$ 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 $p>\frac 23$. We also show the existence of standing waves in the case $-1<p\leq \frac 23$, with compact support if $-1<p<0$.\\ Finally, we obtain, under certain conditions, the linearized stability of such solutions.