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

Abstract

We study the Cauchy problem for a coupled system of a complex Ginzburg-Landau equation with a quasilinear conservation law $$\{\begin{array}{rlllccc}{e}^{-i\theta }{u}_t& =& u_{xx}-|u{|}^{2}u-\alpha g(v)u& v_t+(f(v){)}_x& =& \alpha (g^{\prime }(v)|u{|}^2{)}_x& \end{array}x\in \mathbb{R},t\ge 0,$$ 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.

32 pages. The results are the same but the paper has been completely rewritten. Many extra details were provided on the proofs