Problem: Find a self-similarity solution (soliton behaviour) for the non-linear partial differential equation $$\frac{\partial u}{\partial t} + u^2 \frac{\partial u}{\partial x} + \frac{\partial^3 u}{\partial x^2 \partial t} = 0$$ with boundary condition: $\lim_{x \to -\infty} u(x,t) = 0,\lim_{x \to \infty} u(x,t) = 0$. Please place your final solution in a $\boxed{}$ LaTeX Environment. If there are multiple solutions, please separate them with a ;.
Solution: $\boxed{u(x,t)= \sqrt{6} \, \mathrm{sech}(x-t)}$