The following system of nonlinear first-order PDEs describes the one-dimensional incompressible flow of an ideal fluid in an open long channel $$h_t+(hv)_x=0,$$ $$v_t+vv_x+gh_x=0,$$ where $h = h(x, t)$ denotes the free surface elevation, $v = v(x, t)$ is the horizontal flow velocity, $x$ is the spatial coordinate along the channel axis, $t$ is time, and $g$ is the acceleration due to gravity. Suppose $v(x,0)=0$ and $h(x,0)=h_0>0$. Im required to solve it by the method of characteristics. I know well this method for the case where i have only one equation but i couldn't find any information about the case where is a system of PDEs. There are some related question here but all of them cover the case when all the equations are linear, but here we have non linear terms like $vv_x$ and $(hv)_x$. I really don't know how to proceed. Any help will be very appreciated.
Edited: I calculated the Riemann invariants which are $R^{\pm}=v\pm 2\sqrt{hg}.$ But now i don't know what to do to solve the system with this information.