Problem: Find a solution for the non-linear partial differential equation $\frac{\partial u}{\partial t} = -5u\frac{\partial u}{\partial x} -2.5u^2\frac{\partial u}{\partial x} - 0.5\frac{\partial^3 u}{\partial x^3},\lim_{x \to \pm \infty} u(x,t) = 0,u'(x,t) = 0,u''(x,t) = 0$. Please place your final solution in a $\boxed{}$ LaTeX Environment.
Solution: $\boxed{u(x, t) = \tanh\left( \frac{x }{\sqrt{2}} \right)}$