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 $$iu_{t}+u_{xx}=a|u|^pu+uv,\quad v_t+f(v)_x=(|u|<sup>2)_x$$ where $f(v)=-\gamma v<sup>3$, $-1&lt;p&lt;+\infty$ and $$a,\gamma\&gt;0$$. We establish, in particular, the existence of travelling waves with speed arbitrary large if $p&lt;0$ and arbitrary close to $0$ if $p&gt;\frac 23$. We also show the existence of standing waves in the case $-1&lt;p\leq \frac 23$, with compact support if $-1&lt;p&lt;0$.\ Finally, we obtain, under certain conditions, the linearized stability of such solutions.