Problem: Find a self similarity solution (soliton behvaiour) for the non-linear partial differential equation $u_t - u_{xx} - u^3 = (1-2t)\cos(x)e^{-t^2} - \cos^3(x)e^{-3t^2}, u(x,0) = \cos(x)$ in the limit N/A. 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: $\boxed{u(x,t) = \cos(x) e^{-t^2}}$