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Integrability, bilinearization, exact traveling wave solutions, lump and lump-multi-kink solutions of a \(\varvec{(3 + 1)}\)-dimensional negative-order KdV–Calogero–Bogoyavlenskii–Schiff equation

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Abstract

In this article, we consider a \((3+1)\)-dimensional negative-order KdV–CBS equation which represents interactions of long wave propagation dynamics with remarkable applications in the field of fluid mechanics and quantum mechanics. We investigate the integrability aspect of the considered model in the framework of Hirota bilinear differential calculus, construct infinitely many conservations laws and formulate a Lax pair. At first, we introduce the concept of Bell polynomial theory and utilize it to obtain the Hirota bilinear form. We introduce a two-field condition to determine the bilinear Bäcklund transformation. We use the Cole–Hopf transformation in bilinear Bäcklund transformation and linearize it to obtain the Lax pair formulation. The existence of infinitely many conservation laws has been checked through the Bell polynomial theory. Moreover, we derive one-kink, two-kink and three-kink soliton solution from the Hirota bilinear form. We have successfully investigated the existence of traveling wave solution for the \((3+1)\)-dimensional negative-order KDV–CBS equation and the conditions for the existence of the solution are reported. The traveling wave solutions are extracted in the form of incomplete elliptic integral of second kind and Jacobi elliptic function. Particularly, the use of long wave limit yields kink soliton solutions. Furthermore, we exhibit necessary and sufficient condition for extracting lump solutions of \((3+1)\)-dimensional nonlinear evolution equations, which have few particular types of Hirota bilinear form. The lump solutions are exploited by means of well-known test function in the Hirota bilinear form. This method reduces the number of algebraic equations to solve in deriving lump solutions of variety of NLLEs in comparison with the previously available methods in literature. Finally, two new forms of test functions are chosen and lump-multi-kink solutions have been determined.

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Funding

The author UKM wishes to express his gratitude to the CSIR for providing financial support in the form of an SRF scholarship, as evidenced by letter number: 09/106(0198)/2019-EMR-I. The author BK wishes to express his gratitude to the UGC for providing financial support in the form of a JRF scholarship, as evidenced by letter number: F-1/UGC/Mathematics/2022/S-529. The author AD gratefully acknowledges the financial support provided under the Scheme “Fund for Improvement of S & T Infrastructure (FIST)” of the Department of Science & Technology (DST), Government of India, as evidenced by letter number: SR/FST/MS-I/2019/42 to the Department of Mathematics, University of Kalyani. The research work of AD is also funded by SERB-DST (Govt of India), file no: EEQ/2022/000719.

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Authors and Affiliations

  1. Department of Mathematics, University of Kalyani, Kalyani, 741235, India

    Uttam Kumar Mandal, Biren Karmakar & Amiya Das

  2. Department of Mathematics, Zhejiang Normal University, Jinhua, 321004, Zhejiang, China

    Wen-Xiu Ma

  3. Department of Mathematics, King Abdulaziz University, 21589, Jeddah, Saudi Arabia

    Wen-Xiu Ma

  4. Department of Mathematics and Statistics, University of South Florida, Tampa, FL, 33620-5700, USA

    Wen-Xiu Ma

  5. Material Science Innovation and Modelling, North-West University Mafikeng Campus, Private Bag X2046, Mmabatho, 2735, South Africa

    Wen-Xiu Ma

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  1. Uttam Kumar Mandal
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Mandal, U.K., Karmakar, B., Das, A. et al. Integrability, bilinearization, exact traveling wave solutions, lump and lump-multi-kink solutions of a \(\varvec{(3 + 1)}\)-dimensional negative-order KdV–Calogero–Bogoyavlenskii–Schiff equation. Nonlinear Dyn 112, 4727–4748 (2024). https://doi.org/10.1007/s11071-023-09028-y

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