Similarity solutions

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

Given a scalar PDE in $J^1({\mathbb{R}}^2,\mathbb{R})$ (wlog)

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 solutions.

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. I think that what is happening here is that you have to distinguish the cases where the flow of the symmetry lies inside solutions $u=u(x,t)$ or transform one solution into a different one. I think that this is explained in @blumanlibro page 303, where they call them invariant solutions. Even more, I think that invariant solutions have to do with characteristic lines (see transport equation).

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Author of the notes: Antonio J. Pan-Collantes

antonio.pan@uca.es