Similarity ansatz ($u(x,t)=t^{\alpha }F(x/t^{\beta })$)

A similarity ansatz assumes a profile invariant under a prescribed rescaling, commonly in the displayed power-law form. Substitution into a partial differential equation fixes compatible exponents and can reduce it to an ordinary differential equation for $F$. Boundary conditions and conserved quantities must also be compatible with the scaling; the ansatz alone does not establish existence of a solution.