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Data-driven solutions and parameter discovery of the nonlocal mKdV equation via deep learning method

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Abstract

In this paper, we systematically study the integrability and data-driven solutions of the nonlocal mKdV equation. The infinite conservation laws of the nonlocal mKdV equation and the corresponding infinite conservation quantities are given through Riccti equation. The data-driven solutions of the zero boundary for the nonlocal mKdV equation are studied by using the multilayer physical information neural network algorithm, which include kink soliton, complex soliton, bright-bright soliton and the interaction between soliton and kink-type. For the data-driven solutions with nonzero boundary, we study kink, dark, anti-dark and rational solution. By means of image simulation, the relevant dynamic behavior and error analysis of these solutions are given. In addition, we discuss the inverse problem of the integrable nonlocal mKdV equation by applying the physics-informed neural network algorithm to discover the parameters of the nonlinear terms of the equation.

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

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Funding

The project is supported by National Natural Science Foundation of China (No. 12175069 and No. 12235007) and Science and Technology Commission of Shanghai Municipality (No. 21JC1402500 and No. 22DZ2229014).

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  1. School of Mathematical Sciences, Shanghai Key Laboratory of PMMP, East China Normal University, Shanghai, 200241, People’s Republic of China

    Jinyan Zhu & Yong Chen

  2. College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao, 266590, People’s Republic of China

    Yong Chen

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  1. Jinyan Zhu
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The project is supported by National Natural Science Foundation of China (No. 12175069 and No. 12235007) and Science and Technology Commission of Shanghai Municipality (No. 21JC1402500 and No. 22DZ2229014).

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Zhu, J., Chen, Y. Data-driven solutions and parameter discovery of the nonlocal mKdV equation via deep learning method. Nonlinear Dyn 111, 8397–8417 (2023). https://doi.org/10.1007/s11071-023-08287-z

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