On a quasilinear non-local Benney System

Abstract

We study the quasilinear non-local Benney System $$\{\begin{array}{l}i{u}_t+u_{xx}=|u{|}^{2}u+buv\\ v_t+a({\int }_{{\mathbf{R}}^{+}}{v}^{2}dx)v_x=-b(|u{|}^2{)}_x,(x,t)\in {\mathbf{R}}^{+}\times [0,T],T>0.\end{array}$$ 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.