Potential symmetries and solutions by reduction of partial differential equations. (English) Zbl 0789.35146

Summary: We determine some necessary conditions for a given partial differential equation \({\mathcal E}\), written in conservative form to admit a potential symmetry (PS). A PS of \({\mathcal E}\) is a point symmetry of the auxiliary system \({\mathcal S}_ p\) obtained introducing a potential as further unknown function, then a PS leads to the construction of solutions via the classical reduction method. Given a PS, we introduce an algorithm that allows us to determine a class of \({\mathcal E}\)-solutions which includes the ones obtained as invariant solutions under the related point symmetry of \({\mathcal S}_ p\). As examples, we consider a Fokker-Planck equation, a wave equation in non-homogeneous media and a quasilinear wave equation.


35Q53 KdV equations (Korteweg-de Vries equations)
58J70 Invariance and symmetry properties for PDEs on manifolds
35A30 Geometric theory, characteristics, transformations in context of PDEs
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