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Poisson structure and action–angle variables for the Hirota equation

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Abstract

In this work, we employ the inverse scattering transform (IST) to study the action–angle variables for the Hirota equation. Based on variational principle, we demonstrate that an Euler–Lagrange equation is equivalent to the Hirota equation. By using the IST, several properties of scattering data for the equation are discussed, and we calculate their Poisson brackets successfully with the help of tensor product. Interestingly, we reveal that the action–angle variables can be constructed by the scattering data. Furthermore, the spectral parameter expressions of conservation laws for the equation are derived, related to the Hamiltonian formulation for the equation.

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Acknowledgements

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Funding

This work was supported by the National Natural Science Foundation of China under Grant Nos. 11975306 and 12371255, the Natural Science Foundation of Jiangsu Province under Grant No. BK20181351, the Six Talent Peaks Project in Jiangsu Province under Grant No. JY-059, and the 333 Project in Jiangsu Province.

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  1. School of Mathematics, China University of Mining and Technology, Xuzhou, 221116, People’s Republic of China

    Yu Zhang & Shou-Fu Tian

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  1. Yu Zhang
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  2. Shou-Fu Tian
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Each of the authors contributed to each part of this study equally. All authors read and proved the final vision of the manuscript. All authors have the same contribution.

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Correspondence to Shou-Fu Tian.

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The work was supported in part by the National Natural Science Foundation of China under Grant Nos. 11975306 and 12371255.

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Zhang, Y., Tian, SF. Poisson structure and action–angle variables for the Hirota equation. Z. Angew. Math. Phys. 74, 236 (2023). https://doi.org/10.1007/s00033-023-02129-z

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