Reduction of the Laplace sequence and sine-Gordon type equations

K I Faizulina, A R Khakimova
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Abstract

In this work, we continue the development of methods for constructing Lax pairs and recursion operators for nonlinear integrable hyperbolic equations of soliton type, previously proposed in the work of Habibullin et al. (2016 {\it J. Phys. A: Math. Theor.} {\bf 57} 015203). This approach is based on the use of the well-known theory of Laplace transforms. The article completes the proof that for any known integrable equation of sine-Gordon type, the sequence of Laplace transforms associated with its linearization admits a third-order finite-field reduction. It is shown that the found reductions are closely related to the Lax pair and recursion operators for both characteristic directions of the given hyperbolic equation. Previously unknown Lax pairs and recursion operators were constructed.
拉普拉斯序列和正弦-戈登方程的还原
在这项工作中,我们继续发展之前在哈比布林等人的工作(2016 {\itJ. Phys. A: Math. Theor.} {\bf 57} 015203)中提出的为非线性可积分双曲方程构建拉普拉斯对和递归算子的方法。这种方法基于著名的拉普拉斯变换理论。文章完成了这样一个证明:对于任何已知的正弦-戈登型可积分方程,与其线性化相关的拉普拉斯变换序列都可以进行三阶有限场还原。文章证明,所发现的还原与给定双曲方程两个特征方向的拉克斯对和递归算子密切相关。构建了以前未知的 Lax 对和递归算子。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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