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Inverse Scattering Transform and Dynamics of Soliton Solutions for Nonlocal Focusing Modified Korteweg-de Vries Equation

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Abstract

The Riemann–Hilbert problem is developed to study the general N-soliton solution of the nonlocal modified Korteweg-de Vries equation in this work. The key to proving this result is the analysis of the Riemann–Hilbert problem related to the Cauchy problem of the nonlocal modified Korteweg-de Vries equation. The general N-soliton solution is acquired by solving the Riemann–Hilbert problem of the nonlocal equation under the reflectionless case and the matrix forms of the soliton solutions are given. The abundant phenomena of the solutions corresponding to different eigenvalues are further analyzed, including bounded solutions, singular solutions, degenerate solution and kink solution. It is noticed that many abundant phenomena are not found in the local modified Korteweg-de Vries equation by using RH method. Finally, the propagation behaviors of the solution (including one-, two-, and three soliton) are observed, and the characteristic lines are further used to analyze the continuity and other phenomena of the solution.

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Acknowledgements

The authors acknowledge the anonymous referee for careful reading and corrections of the paper, constructive comments, and suggestions for improving the quality of the paper. This work was supported by the National Natural Science Foundation of China under Grant Nos. 12371255 and 11975306, Xuzhou Basic Research Program Project under Grant No. KC23048, the Six Talent Peaks Project in Jiangsu Province under Grant No. JY-059, the 333 Project in Jiangsu Province, the Postgraduate Research & Practice Program of Education & Teaching Reform of CUMT under Grant No. 2023YJSJG050, the Postgraduate Research & Practice Innovation Program of Jiangsu Province under Grant No. KYCX-232644, and the Graduate Innovation Program of the China University of Mining and Technology under Grant No. 2023WLKXJ118.

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  1. School of Mathematics, China University of Mining and Technology, Xuzhou, 221116, Republic of China

    Xiao-Fan Zhang, Shou-Fu Tian, Jin-Jie Yang & Tian-Tian Zhang

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Zhang, XF., Tian, SF., Yang, JJ. et al. Inverse Scattering Transform and Dynamics of Soliton Solutions for Nonlocal Focusing Modified Korteweg-de Vries Equation. Qual. Theory Dyn. Syst. 23, 113 (2024). https://doi.org/10.1007/s12346-024-00974-5

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