Similarity solution

Summary

A similarity solution, or self-similar solution, of a partial differential equation is a solution that is unchanged by a scaling of its variables. Its graph at one time is a rescaled copy of its graph at any other time. Such a solution has the form $u(x, t) = t^{-\alpha} v(x / t^{\beta})$, so finding it reduces to an equation in fewer variables. Many fundamental solutions, rarefaction waves and blow-up profiles are of this kind. See the definition.