Problem: Find a self similarity solution (soliton behvaiour) for the non-linear partial differential equation $$ \frac{\partial u}{\partial t} + 6 u^2 \frac{\partial u}{\partial x} + \frac{\partial^3 u}{\partial x^3} = 0, \quad u(x, t) > 0, \lim_{x \to \pm \infty} u(x,t) = 0 $$ in the limit $t \rightarrow \infty $. Please place your final solution in a $\boxed{}$ LaTeX Environment.DONT USE \operatorname in your boxed solution. There should be a free parameter $v$. Set $v=0.5$ YOU MUST DO THIS. If there are multiple solutions please separate them with a ;.
Solution: $ \boxed{u_1(x, t) = \frac{1}{\sqrt{2}} \sech\left(\frac{x - 0.5t}{\sqrt{2}}\right)} $