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A new finite-dimensional Hamiltonian systems with a mixed Poisson structure for the KdV equation

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Abstract

A Lax pair for the KdV equation is derived by a transformation of the eigenfunction. By a polynomial expansion of the eigenfunction for the resulting Lax pair, finite-dimensional integrable systems can be obtained from the Lax pair. These integrable systems are proved to be the Hamiltonian and are shown to have a new Poisson structure such that the entries of its structure matrix are a mixture of linear and quadratic functions of coordinates. The odd and even functions of the spectral parameter are introduced to build a generating function for conserved integrals. Based on the generating function, the integrability of these Hamiltonian systems is shown.

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Acknowledgments

The authors express their sincere thanks to the anonymous referees whose comments helped to improve the presentation of the paper.

Funding

This work was supported by National Natural Science Foundation of China (project No. 11271337).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Zhengzhou University, Zhengzhou, Henan, China

    Dianlou Du & Xue Wang

  2. College of Science, Henan Institute of Engineering, Zhengzhou, Henan, China

    Xue Wang

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  1. Dianlou Du
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  2. Xue Wang
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Correspondence to Xue Wang.

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Translated from Teoreticheskaya i Matematicheskaya Fizika, 2022, Vol. 211, pp. 361–374 https://doi.org/10.4213/tmf10147.

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Du, D., Wang, X. A new finite-dimensional Hamiltonian systems with a mixed Poisson structure for the KdV equation. Theor Math Phys 211, 745–757 (2022). https://doi.org/10.1134/S0040577922060010

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