Lie group method for PDEs

Summary

The Lie group method studies a differential equation through its symmetries: continuous groups of changes of variables that map solutions to solutions. Sophus Lie showed that such symmetries can be found by solving a linear system of determining equations for their infinitesimal generators. Once found, they produce new solutions from old ones and reduce the equation to one with fewer variables. For ordinary differential equations it is the Lie symmetry method. Similarity solutions and traveling waves are the invariant solutions of scaling and translation symmetries. See the definition.