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On invariance properties of the wave equation. (English) Zbl 0662.35065
A complete group classification is given of both the wave equation (I) $c\sp 2(x)u\sb{xx}-u\sb{tt}=0$ and its equivalent system (II) $v\sb t=u\sb x$, $c\sp 2(x)v\sb x=u\sb t$, when the wave speed c(x)$\ne const$. Equations (I) and (II) admit either a two-or four-parameter group. For the exceptional case, $c(x)=(Ax+B)\sp 2$, equation (I) admits an infinite group. Equations (I) and (II) do not always admit the same group for a given c(x): The group for (I) can have more parameters or fewer parameters than that for (II); moreover, the groups can be different with the same number of parameters. Separately for (I) and (II), all possible c(x) that admit a four-parameter group are found explicitly. The corresponding invariant solutions are considered. Some of these wave speeds have realistic physical properties: c(x) varies monotonically from one positive constant to another positive constant as x goes from - $\infty$ to $+\infty$.

35L05Wave equation (hyperbolic PDE)
35B40Asymptotic behavior of solutions of PDE
35A30Geometric theory for PDE, characteristics, transformations
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