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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
##### Keywords:
point symmetry; Ermakov-Pinney equation
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##### References:
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