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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 ${\cal E}$, written in conservative form to admit a potential symmetry (PS). A PS of ${\cal E}$ is a point symmetry of the auxiliary system ${\cal S}\sb p$ 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 ${\cal E}$-solutions which includes the ones obtained as invariant solutions under the related point symmetry of ${\cal S}\sb p$. As examples, we consider a Fokker-Planck equation, a wave equation in non-homogeneous media and a quasilinear wave equation.

35Q53KdV-like (Korteweg-de Vries) equations
58J70Invariance and symmetry properties
35A30Geometric theory for PDE, characteristics, transformations
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