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Soliton and breather solutions on the nonconstant background of the local and nonlocal Lakshmanan–Porsezian–Daniel equations by Bäcklund transformation

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Abstract

Under investigation in this paper is the integrable Lakshmanan–Porsezian–Daniel (LPD) equation, which was proposed as a model for the nonlinear spin excitations in the one-dimensional isotropic biquadratic Heisenberg ferromagnetic spin. Our main purpose was to construct soliton and breather solutions on the nonconstant background for the integrable local and nonlocal LPD equations. Firstly, the Bäcklund transformations are constructed based on the pseudopotential of equations. Secondly, starting from the nonconstant initial solution \(\textrm{sech}\) and applying the obtained transformation, various nonlinear wave solutions of the local LPD equation are provided, including the time-periodic breather, W-shaped soliton, M-type soliton and two-soliton solutions, the elastic interactions between the two-soliton solutions are shown and the relationship between parameters and wave structures is discussed. Thirdly, beginning with the nonconstant initial solutions \(\textrm{sech}\) and \(\textrm{tanh}\), the time-periodic breather, bell-shaped one-soliton and anti-bell-shaped one-soliton solutions of the nonlocal LPD equation are generated and these solutions possess no singularity. What is more, the time-periodic breather solutions exhibit the x-periodic background and double-periodic background, which is different from the previous results. The corresponding dynamics of these solutions related to the integrable local and nonlocal LPD equations are illustrated graphically. The results in this paper might be helpful for us to understand the nonlinear characteristics of magnetic materials.

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Data Availability Statement

All data generated or analyzed during this study are included in this manuscript.

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant No. 12201283) and the Natural Science Foundation of Fujian Province (Grant No. 2022J01892).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Minnan Normal University, Zhangzhou, 363000, People’s Republic of China

    Wei-Kang Xie & Fang-Cheng Fan

  2. School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai, 2002240, People’s Republic of China

    Fang-Cheng Fan

  3. Fujian Key Laboratory of Granular Computing and Applications, Zhangzhou, 363000, People’s Republic of China

    Fang-Cheng Fan

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Xie, WK., Fan, FC. Soliton and breather solutions on the nonconstant background of the local and nonlocal Lakshmanan–Porsezian–Daniel equations by Bäcklund transformation. Z. Angew. Math. Phys. 74, 182 (2023). https://doi.org/10.1007/s00033-023-02082-x

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