Nonlocal Extensions of Similarity Methods

Similarity methods include the calculation and use of symmetries and conservation laws for a given partial differential equation (PDE). There exists a variety of software to calculate and use local symmetries and local conservation laws. However, it is often the case that a given PDE admits no useful local symmetry or local conservation law. It is shown how to construct systematically trees of nonlocally related but equivalent systems of PDEs. A local symmetry or local conservation law of a PDE in such a tree could yield a useful nonlocal symmetry or nonlocal conservation law for a given PDE within such a tree. If this is the case, one can extend the usefulness of similarity methods for a given PDE. Many examples are given.