The method of characteristics does not work for my system of equations, it turns out they are too simple for that! I thought my system was hyperbolic but instead it is parabolic. The difference is subtle, but parabolic essentially means that the system cannot be reduced to ordinary differencial equations.
My next attempt was the weakly nonlinear approximation. This approach linearizes the equations, and produces linear PDEs instead of nonlinear, and moves the nonlinearity to the right hand side of the equations and they appear as external forces to the system.
Now I can start using Matlab to solve these equations...
