Solving long time averaged chaotic nonlinear equations

I have a modified chaotic equation of the form: $$\frac{\partial u}{\partial t} = - (u+c)\frac{\partial u}{\partial x} - \frac{\partial^2u}{\partial x^2} - \frac{\partial^4u}{\partial x^4}$$ I am using Neumann and Dirichlet boundary conditions to solve this chaotic equation. I have run tests with this time accurate simulation and get results similar to other numerical experiments.

I had thought to decompose the equations into mean and fluctuating components and then long time average and solve the long time average equations, similar to RANS. But when you do this you get the equation below where $$R$$ is the residual operator we must drive to 0, and $$\bar{u}$$ is the mean component of the velocity $$u$$:

$$R = - (\bar{u}+c)\frac{\partial \bar{u}}{\partial x} - \frac{\partial^2 \bar{u}}{\partial x^2} - \frac{\partial^4 \bar{u}}{\partial x^4} - \overline{u'\frac{\partial u'}{\partial x}}$$

It turns out that the last term, analogous to the Reynolds stress term is near 0, so we end up with:

$$R = - (\bar{u}+c)\frac{\partial \bar{u}}{\partial x} - \frac{\partial^2 \bar{u}}{\partial x^2} - \frac{\partial^4 \bar{u}}{\partial x^4}$$ which is the same as the chaotic unsteady term. Given that this is just the time-accurate equation again, I have found this to be a difficult nonlinear problem to solve. I tried to use a Newton-Krylov solver, and did not succeed in solving the nonlinear problem. Am I missing something? Is there a better solver to use?