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Optimal system and group invariant solutions for a nonlinear wave equation. (English) Zbl 1036.35010
In this paper, the equation \[ \frac{\partial\phi}{\partial t}+\frac{\partial}{\partial z} \Biggl[\phi^n \biggl(1-\frac {\partial} {\partial z} \biggl(\frac{1}{\phi^m} \frac{\partial\phi}{\partial t}\biggr) \biggr) \Biggr]=0 \] is considered. An optimal system for the algebra of point symmetries of the equation is investigated. For \(n=0, m=4/3\), it is shown that the equation can be reduced to the special case of Ermakov-Pinney equation. For \(n=0\), \(m\neq 4/3\), \(m\neq 0\), it is shown that the equation can be reduced either to a first-order or a second-order ordinary differential equation.

MSC:
35A30 Geometric theory, characteristics, transformations in context of PDEs
58J70 Invariance and symmetry properties for PDEs on manifolds
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