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Lump solutions to dimensionally reduced Kadomtsev–Petviashvili-like equations

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Abstract

In this paper, we present a new (3+1)-dimensional KP-like nonlinear partial differential equation and construct the lump solutions, rationally localized in all directions in the space, to its two dimensionally reduced cases. The proposed method in this work is based on a generalized bilinear differential equation, which implies that we can build the lump solutions to the presented KP-like equation from special polynomial solutions to the aforementioned generalized bilinear equation. Since there are totally six free parameters in the resulting lump solutions, we can get the sufficient and necessary conditions guaranteeing analyticity and rational localization of the solutions by using these six free parameters. Two special cases are plotted as illustrative examples, and some contour plots with different determinant values are presented to show that the corresponding lump solution tends to zero when the determinant approaches zero.

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Acknowledgments

All the authors deeply appreciate all the anonymous reviewers for their helpful and constructive suggestions, which can help improve this paper further. This work is supported by the National Natural Science Foundation of China (Nos. 11101029, 11271362 and 11375030), the Fundamental Research Funds for the Central Universities(No. 610806), Beijing City Board of Education Science and Technology Key Project (No. KZ201511232034), Beijing Nova program (No. Z131109000413029) and Beijing Finance Funds of Natural Science Program for Excellent Talents (No. 2014000026833ZK19). All the authors would like to appreciate Prof. Dengshan Wang for his useful remarks and helpful discussions.

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  1. Department of Applied Mathematics, University of Science and Technology Beijing, Beijing, 100083, China

    Jian-Ping Yu

  2. Department of Mathematics, Beijing University of Chemical Technology, Beijing, 100029, China

    Yong-Li Sun

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  1. Jian-Ping Yu
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Correspondence to Yong-Li Sun.

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Yu, JP., Sun, YL. Lump solutions to dimensionally reduced Kadomtsev–Petviashvili-like equations. Nonlinear Dyn 87, 1405–1412 (2017). https://doi.org/10.1007/s11071-016-3122-5

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