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arXiv 2109.06512nlin.SImath-phmath.CAmath.MP
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可积Abel方程与Korteweg-de Vries方程对称解的渐近性

Integrable Abel equation and asymptotics of symmetry solutions of Korteweg-de Vries equation

B. I. Suleimanov, A. M. Shavlukov

备注：
Originally published on Russian in Ufa mathematical journal vol. 13 issue 2 pp. 104-111 (2021) Eglish translation: Ufa Math. J. 13:3, 99-106 (2021)
发表信息：
Ufimskji Matem. Zhurn. 13:3, 104-111 (2021). [Ufa Math. J. 13:3, 99-106 (2021)]
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摘要：

本文给出了一个一阶常微分方程的一般解，该方程在构造Korteweg-de Vries方程及其高阶非自治对称性的稳定部分的同时解的大时间渐近性时得到了一个具有有理右端的一般解.这种对称性是由Korteweg-de Vries方程的第一个更高的自治对称性及其经典伽利略对称性的线性组合决定的。这个一般解依赖于一个任意的参数。由隐函数定理，局部地由第一积分明确地写在超几何函数中确定。一般解的一个特殊情况定义了惠特姆方程的自相似解，这是由G. V. Potemin在1988年发现的。在A. V. Gurevich和L. P. Pitaevsky在20世纪70年代早期的著名作品中，确定了Whitham方程的这些解描述了具有小色散的广泛问题中的非阻尼振荡波的主导项的起源。本文的结果再次支持了一个经验规则，即在各种极限通道下，可积方程只能产生某种意义上的可积方程。我们提出了一个一般性的猜想：在描述其他发展方程的对称解的大时间渐近性时，也应该出现与本文所考虑的可积常微分方程类似的可积常微分方程，从而允许应用逆散射问题的方法。

英文摘要：

We provide a general solution for a first order ordinary differential equation with a rational right-hand side, which arises in constructing asymptotics for large time of simultaneous solutions of the Korteweg-de Vries equation and the stationary part of its higher non-autonomous symmetry. This symmetry is determined by a linear combination of the first higher autonomous symmetry of the Korteweg-de Vries equation and of its classical Galileo symmetry. This general solution depends on an arbitrary parameter. By the implicit function theorem, locally it is determined by the first integral explicitly written in terms of hypergeometric functions. A particular case of the general solution defines self-similar solutions of the Whitham equations, found earlier by G.V. Potemin in 1988. In the well-known works by A.V. Gurevich and L.P. Pitaevsky in early 1970s, it was established that these solutions of the Whitham equations describe the origination in the leading term of non-damping oscillating waves in a wide range of problems with a small dispersion. The result of this article supports once again an empirical rule saying that under various passages to the limits, integrable equations can produce only integrable, in certain sense, equations. We propose a general conjecture: integrable ordinary differential equations similar to that considered in the present paper should also arise in describing the asymptotics at large times for other symmetry solutions to evolution equations admitting the application of the method of inverse scattering problem.

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