Given $u_t - u u_{xx} - u_{xxx} = 0$ I am looking for solutions through the following vector:
\begin{equation} -t \cosh(\omega x) \frac{\partial}{\partial x} + \left( \cosh(\omega x) +2tu\sinh(\omega x) \right)\frac{\partial}{\partial u} \end{equation}
The method is to find invariant solutions through Lie Symmetry. What is written above is one such Symmetry.
The first invariant is automatically $y = t$
Now to find the second invariant: $v(y)$ I have to solve the characteristic:
\begin{equation} \frac{-dx}{t \cosh(\omega x)} = \frac{du}{\cosh(\omega x) + 2tu \sinh(\omega x))} \end{equation}
having two terms in the denominator with one of them having a $u$ is what is throwing me off. How would I solve this?