On a quasilinear non-local Benney System

TL;DR

This paper investigates a quasilinear non-local Benney System, establishing local existence, uniqueness, finite-time blow-up conditions, and the existence of global weak and bound-state solutions.

Contribution

It provides the first rigorous analysis of existence, uniqueness, blow-up, and global solutions for this specific non-local Benney System.

Findings

Proved local existence and uniqueness of strong solutions.

Identified conditions leading to finite-time blow-up.

Established existence of global weak and bound-state solutions.

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.