Problem: Find a self similarity solution (soliton behvaiour) for the non-linear partial differential equation $$\partial_{t} u = \partial_{xx} u - 1.5 \, \partial_{x} u + 0.2 \, u, \quad u(x,0) = \frac{1}{\sqrt{0.1}} e^{-\frac{x^2}{0.4}}, \quad u(+\infty,t) = u(-\infty,t) = 0$$.
Solution: $$\boxed{\frac{1}{\sqrt{t+0.1}} e^{-\frac{(x-1.5t)^2}{4(t+0.1)} + 0.2t}}$$