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We study the Cauchy problem for a coupled system of a complex Ginzburg-Landauequation with a quasilinear conservation law $$ \left\{\begin{array}{rlll}e^{-i\theta}u_t&=&u_{xx}-|u|^2u-\alpha g(v)u& v_t+(f(v))_x&=&\alpha(g'(v)|u|^2)_x& \end{array}\right. \qquad x\in\mathbb{R},\, t \geq 0, $$ whichcan describe the interaction between a laser beam and a fluid flow (see[Aranson, Kramer, Rev. Med. Phys. 74 (2002)]). We prove the existence of alocal in time strong solution for the associated Cauchy problem and, for acertain 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.

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