Self-similar profile

A field is self-similar for prescribed exponents if it has the form

u(t,x)=(T-t)^{-a}V\bigl(D_{T-t}^{-1}x\bigr),\qquad t<T,

with a profile $V$ independent of time. In similarity variables, this is a stationary rescaled field $U(s,y)=V(y)$ .

Profile equations

If $u$ is also required to solve a PDE, substitution gives an equation for $V$ , provided the time powers balance. A concentrating ansatz is not automatically a solution or a symmetry of the PDE. Slowly varying profiles and additional corrections generally destroy exact self-similarity even when they have a self-similar leading term.