Problem: Find a self similarity solution (soliton behvaiour) for the non-linear partial differential equation $$\partial_t u = partial_{xx} u + u^2 - u; u(x;0) = 0.1 \quad (|x| \leq 3), u(x;0) = 0 \quad (|x| \gt 3); \lim_{x \to -\infty} u(x,t) = 0, \quad \lim_{x \to \infty} u(x,t) = 0$$ in the limit $t\to\infty$. 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: $$\boxed{\frac{0.6}{\sqrt{4 \pi t}} e^{-t} e^{-\frac{x^2}{4t}}}$$