I've come across a question which asks me to solve the following non-linear wave equation:
$u_{tt}-c^2\cdot u_{xx}=\frac{\alpha}{u^3} -\beta \cdot u $
I tried using the separation of variables method, and substituting a travelling periodic wave solution, but both methods didn't work for me. Is there a certain substitution for such equations? Or does this equation belong to a specific type of non-linear wave equations?
Would appreciate any hints as how to solve it.