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A generalized (2+1)-dimensional Hirota bilinear equation: integrability, solitons and invariant solutions

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Abstract

In this paper, we consider an extended form of generalized \((2+1)\)-dimensional Hirota bilinear equation which demonstrates nonlinear wave phenomena in shallow water, oceanography and nonlinear optics. We have successfully studied the integrability characteristic of the nonlinear equation in different aspects. We have applied the Painlevè analysis technique on the equation and found that it is not completely integrable in Painlevè sense. The concept of Bell polynomial form is introduced and the Hirota bilinear form, Bäcklund transformations are obtained. By means of Cole-Hopf transformation, we have derived the Lax pairs by direct linearization of coupled system of binary Bell polynomials. We have also derived infinite conservation laws from two field condition of the generalized \((2+1)\)-dimensional Hirota bilinear equation. We have exploited the expressions of one-soliton, two-soliton and three-soliton solutions directly from Hirota bilinear form and demonstrated them pictorially. Further, Lie symmetry approach is applied to analyze the Lie symmetries and vector fields of the considered problem. The symmetry reductions were then obtained using similarity variables and some closed-form solutions such as parabolic wave solutions and kink wave solutions are secured.

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Abbreviations

KdV:

Korteweg de-vries

KP:

Kadomtsev–Petviashvili

mKdV:

Modified Korteweg de-vries

PDE:

Partial differential equation

ODE:

Ordinary differential equation

WTC:

Weiss–Tabor–Carnevale

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Acknowledgements

The authors Uttam Kumar Mandal and Sandeep Malik wishes to express their 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 and 09/1051(0028)/2018-EMR-I, respectively. The author Sachin Kumar wants to acknowledge 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/2021/104 to the Department of Mathematics and Statistics, Central University of Punjab.

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

    Uttam Kumar Mandal & Amiya Das

  2. Department of Mathematics and Statistics, Central University of Punjab, Bathinda, 151401, Punjab, India

    Sandeep Malik & Sachin Kumar

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  1. Uttam Kumar Mandal
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Mandal, U.K., Malik, S., Kumar, S. et al. A generalized (2+1)-dimensional Hirota bilinear equation: integrability, solitons and invariant solutions. Nonlinear Dyn 111, 4593–4611 (2023). https://doi.org/10.1007/s11071-022-08036-8

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