Problem: Find the self-similar (soliton-like) solution $u(x,t)$ for the nonlinear partial-differential equation $$\partial_t u + u^{1/2}\,\partial_x u = \partial_{xx}u - 2.5\,u^2, u(0,t)=1, u(x,0)=0,\;x\ge0,\;t>0$$ in the limit $t\to\infty$
Solution: $$ \boxed{U(x)=\frac{1}{(1+0.5\,x)^{2}}} $$