Similarity solutions

Also known as invariant solutions. Although I think that the name similarity solutions refers mainly to one particular group: the scaling group.

Consider, for example the following scalar PDE:

defined by the function $F$ on $J^1({\mathbb{R}}^2,\mathbb{R})$.
Suppose we have a one-parameter subgroup of symmetries of the PDE (see symmetry group of a DE system)

That is, the prolongation of it leaves invariant the hypersurface given by $F=0$. Then it transforms solutions into (possibly other) solutions. We have two cases (I think that this is explained in @blumanlibro page 303):

• Case A (group moves you to a different solution)
  You have a genuine symmetry of the PDE, but you do not impose the invariance condition to the solutions themselves. You simply know that if $u(x,t)$ is a solution then so is $\overline{u}(\overline{x},\overline{t})$. the group can be used to generate new solutions from known ones, but the number of independent variables remains the same.
• Case B (invariant/similarity solutions)
  You additionally impose that $u(x,t)$ itself is fixed by the flow:

In Case B, the infinitesimal generator $X$ of the group, which does satisfy $X(F)=0$ mod $F=0$, can be transformed into ${\partial }_y$ with a coordinate change

such that $X(y)=1,X(s)=0,X(w)=0$ . In this coordinates the PDE is

but since $G$ doesn't depend on $y$ we can assume $w_y=0$, so we obtain the ODE

This change of variables is called similarity variables, and the solutions of the original PDE obtained are called similarity solutions. See @Stephani page 172.

Particular case: travelling wave solution.

Keep an eye: A problem arises if the variable $y$ needs to be the dependent variable. See @Stephani page 172. Even more, I think that invariant solutions have to do with characteristic lines (see transport equation).