# Methods to Analytically Solve a Nonlinear PDE

I'm looking for suggestions to solve: $$u_t = u_{xxxx}-3u(u_x)^2-\frac{3}{2}u^2u_{xx}+\frac{1}{2}u_{xx}+F$$ where $$F$$ is currently an unknown function of unknown type (might be linear, exponential, etc.). Ignoring that ambiguity (pretend it vanishes), I am unsure of any method to solve such a PDE. I attempted separation of variables but that didn't help because of the nonlinearities. I then tried seeking a steady-state solution however that didn't lead to anything fruitful. After that, I tried using the $$1$$D and $$2$$D Fourier Transforms in space and space and time respectively but again that didn't really help. In this case, I got terms that involved self-convolutions of either $$u$$ or $$u_x$$ and this made the equation even harder to solve.

One last thought I had was to maybe use the Cole-Hopf Transform: i.e. introduce some new variable $$w=\phi(u)$$. In doing this, I started calculating partials of $$w$$ that would appear in my PDE and the expressions I got seemed even more complicated. This kind of leads me to believe that this won't be a fruitful avenue either, however, I have never really tried using this method so I am wondering if I'm maybe doing something wrong. Here I found: $$w_t = \phi'(u)u_t$$ $$w_x = \phi'(u)u_x$$ $$w_{xx} = \phi''(u)u_x^2+\phi'(u)u_{xx}$$ $$w_{xxx} = \phi'''(u)u_x^3 + 3\phi''(u)u_xu_{xx}+\phi'(u)u_{xxx}$$ $$w_{xxxx} = \phi^{(4)}(u)u_x^4 + 6\phi'''(u)u_x^2u_{xx}+3\phi''(u)u_{xx}^2+4\phi''(u)u_xu_{xxx}+\phi'(u)u_{xxxx}$$ Now I just don't really see how to use these equations to generate a simplified version of my original PDE or even how to definitively say that this won't help me in reducing the PDE. Any help or advice would be appreciated.

• Even if you do manage to find a "pencil and paper" solution, it will almost certainly be so long and unwieldy that it will be useless in practice. In contrast, the equation is totally integrable in $t$ - it is what's known as a time evolution equation, so numerical solutions are very easily constructed. Oct 18, 2021 at 17:46