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首页> 外文期刊>Journal of Mathematical Physics >On the reduction and the existence of approximate analytic solutions of some basic nonlinear ODEs in mathematical physics and nonlinear mechanics
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On the reduction and the existence of approximate analytic solutions of some basic nonlinear ODEs in mathematical physics and nonlinear mechanics

机译：数学物理和非线性力学中一些基本非线性ODE的约化解析解的存在性

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Using series of admissible functional transformations we reduce the one-dimensional axisymmetric nonlinear Schrodinger (NLS) equation, as well as the forced damped nonlinear Duffing (NLD) equation to equivalent nonlinear first-order integrodifferential equations. The forced undamped (NLD) equation results as a special case. The reduced integrodifferential equations are exact. In the limits of small or large values of the parameters characterizing these nonlinear problems, we prove that further reductions lead to first-order nonlinear ordinary differential equations which, except in case of the (NLS) equation, are of the Abel classes. The approximate reduced (NLS) equation admits exact analytic solutions. On the other hand, taking into account the known exact analytic solutions of the equivalent Abel classes of equations we show that there do not exist analytic solutions of the above two nonlinear Duffing oscillators. However, if further asymptotic approximations take place, new approximate analytic solutions concerning the (NLD) equations are constructed. (C) 2004 American Institute of Physics. [References: 22]
机译：通过使用一系列允许的函数变换，我们将一维轴对称非线性Schrodinger（NLS）方程以及强制阻尼非线性Duffing（NLD）方程简化为等效的非线性一阶积分微分方程。强制无阻尼（NLD）方程是一种特殊情况。简化的积分微分方程是精确的。在表征这些非线性问题的参数的大小的上限或下限中，我们证明了进一步的简化导致了一阶非线性常微分方程，该方程除（NLS）方程以外，均属于Abel类。近似简化（NLS）方程允许使用精确的解析解。另一方面，考虑到等效的Abel类方程组的已知精确解析解，我们表明不存在上述两个非线性Duffing振荡器的解析解。但是，如果进行进一步的渐近逼近，则将构造有关（NLD）方程的新的逼近解析解。 （C）2004年美国物理研究所。 [参考：22]

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