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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) ${c}^{2}\left(x\right){u}_{xx}-{u}_{tt}=0$ and its equivalent system (II) ${v}_{t}={u}_{x}$, ${c}^{2}\left(x\right){v}_{x}={u}_{t}$, when the wave speed c(x)$\ne const$. Equations (I) and (II) admit either a two-or four-parameter group. For the exceptional case, $c\left(x\right)={\left(Ax+B\right)}^{2}$, 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 $+\infty$.
##### MSC:
 35L05 Wave equation (hyperbolic PDE) 35B40 Asymptotic behavior of solutions of PDE 35A30 Geometric theory for PDE, characteristics, transformations