Problem: Find a self similarity solution (soliton behvaiour) for the non-linear partial differential equation $\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} = \frac{\partial^2 u}{\partial x^2} - u^2; x \geq 0, t > 0; u(0, t) = \frac{1}{\sqrt{t}}, u(x,0) = 0$ in the limit $t \rightarrow \infty$. Please place your final solution in a $\boxed{}$ LaTeX Environment.DONT USE \operatorname in your boxed solution. If there are multiple solutions please separate them with a ;.
Solution: $\boxed{u(x,t) = \frac{1}{\sqrt{t}} e^{-x/\sqrt{t}}}$