Linear superposition and interaction of Wronskian solutions to an extended (2+1)-dimensional KdV equation

Li Cheng; Yi Zhang; Ying-Wu Hu; Li Cheng; Yi Zhang; Ying-Wu Hu

doi:10.3934/math.2023864

AIMS Mathematics

2023, Volume 8, Issue 7: 16906-16925. doi: 10.3934/math.2023864

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Research article

Linear superposition and interaction of Wronskian solutions to an extended (2+1)-dimensional KdV equation

1.
Normal School, Jinhua Polytechnic, Jinhua 321007, China
2.
Key Laboratory of Crop Harvesting Equipment Technology, Jinhua Polytechnic, Jinhua 321007, China
3.
Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China

Received: 24 February 2023 Revised: 24 April 2023 Accepted: 26 April 2023 Published: 15 May 2023
MSC : 35C08, 35Q51, 37K40

The main purpose of this work is to discuss an extended KdV equation, which can provide some physically significant integrable evolution equations to model the propagation of two-dimensional nonlinear solitary waves in various science fields. Based on the bilinear Bäcklund transformation, a Lax system is constructed, which guarantees the integrability of the introduced equation. The linear superposition principle is applied to homogeneous linear differential equation systems, which plays a key role in presenting linear superposition solutions composed of exponential functions. Moreover, some special linear superposition solutions are also derived by extending the involved parameters to the complex field. Finally, a set of sufficient conditions on Wronskian solutions is given associated with the bilinear Bäcklund transformation. The Wronskian identities of the bilinear KP hierarchy provide a direct and concise way for proving the Wronskian determinant solution. The resulting Wronskian structure generates $ N $-soliton solutions and a few of special Wronskian interaction solutions, which enrich the solution structure of the introduced equation.
- extended (2+1)-dimensional KdV equation,
- bilinear Bäcklund transformation,
- Lax pair,
- linear superposition solution,
- Wronskian interaction solution
Citation: Li Cheng, Yi Zhang, Ying-Wu Hu. Linear superposition and interaction of Wronskian solutions to an extended (2+1)-dimensional KdV equation[J]. AIMS Mathematics, 2023, 8(7): 16906-16925. doi: 10.3934/math.2023864

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Abstract

The main purpose of this work is to discuss an extended KdV equation, which can provide some physically significant integrable evolution equations to model the propagation of two-dimensional nonlinear solitary waves in various science fields. Based on the bilinear Bäcklund transformation, a Lax system is constructed, which guarantees the integrability of the introduced equation. The linear superposition principle is applied to homogeneous linear differential equation systems, which plays a key role in presenting linear superposition solutions composed of exponential functions. Moreover, some special linear superposition solutions are also derived by extending the involved parameters to the complex field. Finally, a set of sufficient conditions on Wronskian solutions is given associated with the bilinear Bäcklund transformation. The Wronskian identities of the bilinear KP hierarchy provide a direct and concise way for proving the Wronskian determinant solution. The resulting Wronskian structure generates $ N $-soliton solutions and a few of special Wronskian interaction solutions, which enrich the solution structure of the introduced equation.

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