On a quasilinear non-local Benney System

Abstract: We study the quasilinear non-local Benney System $$\left{\begin{array}{llll} iu_t+u_{xx}=|u|^2u+buv\ v_t+a(\int_{\mathbf{R}^{+}v^{2dx)v_x=-b(|u|^2)_x,\quad}} (x,t)\in\mathbf{R}^+\times [0,T],\, T>0. \end{array}\right.$$ We establish the existence and uniqueness of strong local solutions to the corresponding Cauchy problem and show, under certain conditions, the blow-up of such solutions in finite time. Furthermore, we prove the existence of global weak solutions and exhibit bound-state solutions to this system.