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Explicit solutions and Darboux transformations of a generalized D-Kaup–Newell hierarchy

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Abstract

Darboux and Bäcklund transformations on integrable couplings are formulated and generalized. The generalization of the theory pertains to spectral problems where the spatial spectral matrix is a polynomial in \(\lambda \) of any order. A specific application to a generalized D-Kaup–Newell integrable couplings system is worked out, along with an explicit formula for the associated Bäcklund transformation. Solutions are given for the 0, 1, 2, 3-order generalized D-Kaup–Newell integrable coupling system. Formulas for the general m-th-order integrable couplings system are seen. Graphs of explicit solutions to the fourth-order integrable couplings are presented for chosen parameters showing solitons. A brief discussion about open problems and physical implications of the paper concludes the paper.

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Funding

This research was in part supported by NSFC under the Grants 11371326, 11301331, and 11371086, and NSF under the Grant DMS-1664561.

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Authors and Affiliations

  1. Department of Mathematics, The University of Tampa, Tampa, FL, 33606-1490, USA

    Morgan McAnally

  2. Department of Mathematics, Zhejiang Normal University, Jinhua, 321004, Zhejiang, China

    Wen-Xiu Ma

  3. Department of Mathematics, King Abdulaziz University, Jeddah, Saudi Arabia

    Wen-Xiu Ma

  4. Department of Mathematics and Statistics, University of South Florida, Tampa, FL, 33620-5700, USA

    Wen-Xiu Ma

  5. College of Mathematics and Physics, Shanghai University of Electric Power, Shanghai, 200090, China

    Wen-Xiu Ma

  6. College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao, 266590, Shandong, China

    Wen-Xiu Ma

  7. Department of Mathematical Sciences, International Institute for Symmetry Analysis and Mathematical Modeling, North-West University, Mafikeng Campus, Mmabatho, 2735, South Africa

    Wen-Xiu Ma

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  1. Morgan McAnally
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  2. Wen-Xiu Ma
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Correspondence to Morgan McAnally.

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McAnally, M., Ma, WX. Explicit solutions and Darboux transformations of a generalized D-Kaup–Newell hierarchy. Nonlinear Dyn 102, 2767–2782 (2020). https://doi.org/10.1007/s11071-020-06030-6

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  • DOI: https://doi.org/10.1007/s11071-020-06030-6

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