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

We study the Cauchy problem for a coupled system of a complex Ginzburg-Landau equation with a quasilinear conservation law $$ left{begin{array}{rlll} e^{-itheta}u_t&=&u_{xx}-|u|^2u-alpha g(v)u& v_t+(f(v))_x&=&alpha (g(v)|u|^2)_x& end{array}right. qquad xinmathbb{R},, t geq 0, $$ 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.