  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.
