On a coupled system of a Ginzburg-Landau equation with a quasilinear conservation law

TL;DR

This paper investigates a coupled Ginzburg-Landau and quasilinear conservation law system, proving local and global solutions, and establishing the existence of standing wave solutions relevant to laser-fluid interactions.

Contribution

It introduces new existence results for solutions and standing waves in a coupled nonlinear PDE system modeling laser and fluid dynamics.

Findings

Existence of local strong solutions for the coupled system.

Global weak solutions for certain flux functions.

Existence of standing wave solutions in specific cases.

Abstract

We study the Cauchy problem for a coupled system of a complex Ginzburg-Landau equation with a quasilinear conservation law $$ \left\{\begin{array}{rlll} e^{-i\theta}u_t&=&u_{xx}-|u|^2u-\alpha g(v)u& v_t+(f(v))_x&=&\alpha (g'(v)|u|^2)_x& \end{array}\right. \qquad x\in\mathbb{R},\, t \geq 0, $$ which can describe the interaction between a laser beam and a fluid flow (see [Aranson, Kramer, Rev. Med. Phys. 74 (2002)]). We prove the existence of a local in time strong solution for the associated Cauchy problem and, for a certain class of flux functions, the existence of global weak solutions. Furthermore we prove the existence of standing waves of the form $(u(t,x),v(t,x))=(U(x),V(x))$ in several cases.