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Darboux transformation and loop soliton solutions for the complex space–time-shifted nonlocal short pulse equation

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Abstract

Under investigation is the complex space–time-shifted nonlocal short pulse equation, which is connected with the complex space–time-shifted nonlocal sine-Gordon equation through a covariant reciprocal transformation. The first and the second types of Darboux transformations with respect to N different purely imaginary spectrum and 2N general complex spectrum are constructed by using loop group method, respectively. The generalized Darboux transformation corresponding to fixed number of purely imaginary spectrum with higher-order algebraic poles is proposed through limit technique. As an application, several kinds of analytical solutions including the bell-shaped loop soliton, higher-order loop soliton, breathing loop soliton and hybrid bell-shaped–breathing loop soliton solutions are obtained. It is found that the space–time-shifted parameters \(x_0\) and \(t_0\) can only have trivial effect of translations on the bell-shaped loop solitons, while they can produce nontrivial deformations for the breathing loop solitons. The singular solution traveling with certain space–time line is also given.

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Acknowledgements

The work was supported by the National Natural Science Foundation of China (11705290), the National Natural Science Foundation of Henan Province (232300420119, 232300420350), the Scientific Research Team Development Project of ZUT (K2020TD004), the Independent Innovation Application Research Project of ZUT (K2020YY006), the Key Scientific Research Project of Colleges and Universities in Henan Province (21A140030) and the Natural Science Foundation of ZUT (K2022MS002, K2023MS003).

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Authors and Affiliations

  1. College of Science, Zhongyuan University of Technology, Zhengzhou, 450007, Henan, China

    Xin Wang, Jingfeng Kang, Jianlin Zhang, Tengjin Zhao & Wentao Jin

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  1. Xin Wang
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  2. Jingfeng Kang
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  3. Jianlin Zhang
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Contributions

XW was involved in conceptualization, methodology, design, formal analysis, software, validation, writing—original draft and writing—review and editing. JK was involved in software and writing—review and editing. JZ was involved in formal analysis and editing. TZ and WJ were involved in software and editing.

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Correspondence to Xin Wang.

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Wang, X., Kang, J., Zhang, J. et al. Darboux transformation and loop soliton solutions for the complex space–time-shifted nonlocal short pulse equation. Nonlinear Dyn 111, 13375–13390 (2023). https://doi.org/10.1007/s11071-023-08554-z

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