I would like to know if it is possible to find the solution to the following system of partial differential equations: \begin{equation} \partial_x \mathbf{u}+B(x,y) \cdot \partial_y \mathbf{u}-\mathbf{f}(\mathbf{u},x,y)=0 \end{equation}
Where $\mathbf{f,u}\in \rm{I \!R}^{n}$, $B\in \rm{I \!R}^{(n\times n)}$, $x,y \in \rm{I \!R}$
Especially if this is possible by using the method of characteristics. Related is this answer on mathoverflow. In a comment there the 'the method of Darboux' is stated for some nonlinear equations.
Question are:
- Is the Method of characteristics applicable for coupled semilinear pdes?
- Does the method of Darboux always work for semilinear pdes?
- If neither is applicable in general how would one solve these equations?