Symmetry Solutions of a Non-Linear Elastic Wave Equation With Third Order Anharmonic Corrections
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摘要: 应用Lie对称法,当弹性能具有三阶非调和修正项时,分析纵向变形的非线性弹性波动方程.通过不同对称下的恒等条件,寻找对称代数,并将它简化为二阶常微分方程.对该简化的常微分方程作进一步分析后,获得若干个显式的精确解.分析Apostol的研究成果(Apostol B F.On a non-linear wave equation in elasticity.Phys Lett A,2003,318(6):545-552)发现,非调和修正项通常导致解在有限时间内具有时间相关奇异性.除了得到时间相关奇异性的解外,还得到无法显示时间相关奇异性的解.Abstract: Lie symmetry method was applied to analyze a non-linear elastic wave equation for longitudinal deformations with third order anharmonic corrections to the elastic energy. Symmetry algebra was found and reductions to second order ODEs were obtained through invariance under different symmetries. The reduced ODEs were further analyzed to obtain several exact solutions in explicit form. Apostol(Apostol B F. On a non-linear wave equation in elasticity. Phys Lett A, 2003, 318(6):545552) had observed that anharmonic corrections generally lead t o solutions with time-dependent singularities in finite time. Along with solutions with time-dependent singularities are obtained, also solutions which do not exhibit time-dependent singularities were obtained.
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