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Quasi-periodic Solutions of the Kaup–Kupershmidt Hierarchy

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Abstract

Based on solving the Lenard recursion equations and the zero-curvature equation, we derive the Kaup–Kupershmidt hierarchy associated with a 3×3 matrix spectral problem. Resorting to the characteristic polynomial of the Lax matrix for the Kaup–Kupershmidt hierarchy, we introduce a trigonal curve \(\mathcal {K}_{m-1}\) and present the corresponding Baker–Akhiezer function and meromorphic function on it. The Abel map is introduced to straighten out the Kaup–Kupershmidt flows. With the aid of the properties of the Baker–Akhiezer function and the meromorphic function and their asymptotic expansions, we arrive at their explicit Riemann theta function representations. The Riemann–Jacobi inversion problem is achieved by comparing the asymptotic expansion of the Baker–Akhiezer function and its Riemann theta function representation, from which quasi-periodic solutions of the entire Kaup–Kupershmidt hierarchy are obtained in terms of the Riemann theta functions.

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (project no. 11171312), Innovation Scientists and Technicians Troop Construction Projects of Henan Province (084200410019).

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Authors and Affiliations

  1. Department of Mathematics, Zhengzhou University, 100 Kexue Road, Zhengzhou, Henan, 450001, People’s Republic of China

    Xianguo Geng, Lihua Wu & Guoliang He

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  1. Xianguo Geng
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  2. Lihua Wu
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  3. Guoliang He
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Correspondence to Xianguo Geng.

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Communicated by P. Newton.

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Geng, X., Wu, L. & He, G. Quasi-periodic Solutions of the Kaup–Kupershmidt Hierarchy. J Nonlinear Sci 23, 527–555 (2013). https://doi.org/10.1007/s00332-012-9160-3

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