I need to solve the following differential equation:
$\rho \frac{\partial u}{\partial t}+au+bu^3+c(\frac{\partial u}{dx})^2=0$
If I ignore the spatial derivative term "$(\frac{\partial u}{\partial x})^2$", I can numerically solve it using ode45 function of MATLAB. But I don't know how to do with the last term in it.
I considered "method of lines", but there are two problems-
a) The spatial derivative term is 2nd order, and therefore it cannot be discretized
b) There is a $u^3$ term, I think such term cannot be represented in the vector space.