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More common errors in finding exact solutions of nonlinear differential equations. I

机译:寻找非线性微分精确解的更常见错误   方程。一世

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摘要

In the recent paper by Kudryashov [Commun. Nonlinear Sci. Numer. Simulat.,2009, V.14, 3507-3529] seven common errors in finding exact solutions ofnonlinear differential equations were listed and discussed in detail. Weindicate two more common errors concerning the similarity (equivalence withrespect to point transformations) and linearizability of differential equationsand then discuss the first of them. Classes of generalized KdV and mKdVequations with variable coefficients are used in order to clarify ourconclusions. We investigate admissible point transformations in classes ofgeneralized KdV equations, obtain the necessary and sufficient conditions ofsimilarity of such equations to the standard KdV and mKdV equations and carriedout the exhaustive group classification of a class of variable-coefficient KdVequations. Then a number of recent papers on such equations are commented usingthe above results. It is shown that exact solutions were constructed in thesepapers only for equations which are reduced by point transformations to thestandard KdV and mKdV equations. Therefore, exact solutions of such equationscan be obtained from known solutions of the standard KdV and mKdV equations inan easier way than by direct solving. The same statement is true for otherequations which are equivalent to well-known equations with respect to pointtransformations.
机译:在Kudryashov最近的论文中[Commun。非线性科学Numer。 Simulat。,2009,V.14,3507-3529]列出并详细讨论了寻找非线性微分方程精确解的七个常见错误。我们指出了关于微分方程的相似性(关于点变换的等效性)和线性化的两个常见错误,然后讨论其中的第一个。为了阐明我们的结论,使用了具有可变系数的广义KdV和mKdVequations类。我们研究了广义KdV方程类中的允许点变换,获得了与标准KdV和mKdV方程类相似的充要条件,并对一类变系数KdVequations进行了详尽的组分类。然后,使用上述结果对有关此类方程式的最新论文进行评论。结果表明,在这些论文中仅针对方程式构建了精确的解,这些方程式通过点转换简化为标准的KdV和mKdV方程。因此,可以以比直接求解更简单的方式从标准KdV和mKdV方程的已知解中获得此类方程的精确解。对于与点转换等效的公知方程式的其他方程式,也适用相同的说法。

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