Problem: Find the self-similar (soliton-like) solution $u(x,t)$ for the nonlinear partial-differential equation $$ \frac{\partial u}{\partial t} + u\frac{\partial u}{\partial x} = \frac{\partial^2 u}{\partial x^2} - u^3, u(0,t)=1, u(x,0)=0, x\ge 0, t>0 $$ in the limit $t\to\infty$
Solution: $$ \boxed{U(x)=\frac{1}{1+\frac{1}{2}x}} $$