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Bäcklund transformations for the modified short pulse equation and complex modified short pulse equation

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Abstract

It is demonstrated that under a reciprocal transformation, the modified short pulse (mSP) equation is brought to the associated mSP equation, which is shown to be equivalent to the associated SP equation. This connection allows us to build a Bäcklund transformation (BT) and the general formula of its iterations for the mSP equation. In addition to the (real) mSP equation, we further apply the BT method to the complex modified short pulse (cmSP) equation, and both its BT and nonlinear superposition formula are worked out. As applications, for the cmSP equation we calculate its various solutions, such as the soliton solutions, cuspon solutions, breather solutions and their interaction solutions.

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Data Availability Statement

This manuscript has associated data in a data repository. [Authors’ comment: All data included in this article are available upon request by contacting the corresponding author.]

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Acknowledgements

It is our pleasure to thank the anonymous referees for their useful suggestions and comments. This work is supported by the Natural Science Foundation of Guangxi Zhuang autonomous region, China (Grant No. 2022GXNSFAA035598), the National Natural Science Foundation of China (Grant Nos. 11871471, 11931017, 11905110 and 12171474), the Yue Qi Outstanding Scholar Project, China University of Mining & Technology, Beijing (Grant No. 00-800015Z1177).

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Authors and Affiliations

  1. Department of Mathematics, China University of Mining and Technology, Beijing, 100083, People’s Republic of China

    Min Xue & Q. P. Liu

  2. School of Mathematics and Statistics, Nanning Normal University, Nanning, 530001, Guangxi, People’s Republic of China

    Hui Mao

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  1. Min Xue
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  2. Q. P. Liu
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  3. Hui Mao
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Correspondence to Hui Mao.

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Xue, M., Liu, Q.P. & Mao, H. Bäcklund transformations for the modified short pulse equation and complex modified short pulse equation. Eur. Phys. J. Plus 137, 500 (2022). https://doi.org/10.1140/epjp/s13360-022-02710-x

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