In this paper, it is shown how a combination of approximate symmetries of a nonlinear wave equation with small dissipations and singularity analysis provides exact analytic solutions. We perform the analysis using the Lie symmetry algebra of this equation and identify the conjugacy classes of the one-dimensional subalgebras of this Lie algebra. We show that the subalgebra classification of the integro-differential form of the nonlinear wave equation is much larger than the one obtained from the original wave equation. A systematic use of the symmetry reduction method allows us to find new invariant solutions of this wave equation.

中文翻译：

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通过近似对称性获得耗散小的非线性波动方程的不变解

在本文中，显示了如何将非线性波动方程的近似对称性与小耗散和奇异性分析结合起来，提供精确的解析解。我们使用该方程的Lie对称代数进行分析，并确定此Lie代数的一维子代数的共轭类。我们表明，非线性波动方程的积分-微分形式的子代数分类要比从原始波动方程获得的子代分类大得多。对称减少方法的系统使用使我们能够找到该波动方程的新不变解。