Integrability investigation and lump-type wave structures for higher-dimensional generalized Kadomtsev-Petviashvili equations

Li Cheng; Wen-Xiu Ma; Xiao-Fei Hu; Li Cheng; Wen-Xiu Ma; Xiao-Fei Hu

doi:10.3934/math.20261193

AIMS Mathematics

2026, Volume 11, Issue 9: 30125-30144. doi: 10.3934/math.20261193

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Research article

Integrability investigation and lump-type wave structures for higher-dimensional generalized Kadomtsev-Petviashvili equations

1.
Normal College, Jinhua University of Vocational Technology, Jinhua 321007, China
2.
School of Mathematical Sciences, Zhejiang Normal University, Jinhua 321004, China
3.
Research Center of Astrophysics and Cosmology, Khazar University, 41 Mehseti Street, Baku 1096, Azerbaijan
4.
Department of Mathematics and Statistics, University of South Florida, Tampa, FL 33620-5700, USA
5.
Material Science Innovation and Modelling, North-West University, Mafikeng Campus, Private Bag X2046, Mmabatho 2735, South Africa

Received: 27 July 2026 Revised: 01 September 2026 Accepted: 04 September 2026 Published: 16 September 2026
MSC : 35C08, 35Q51, 37K40

This paper systematically investigates the integrable properties and lump-type wave structures of higher-dimensional generalized Kadomtsev-Petviashvili (KP) equations, which describe nonlinear wave behaviors in physical systems. For the (3+1)-dimensional KP and Boussinesq equations, a three-soliton condition is established. Based on this condition, a bilinear Bäcklund transformation is constructed, from which the corresponding Lax pair and modified evolutionary equation are derived. The integrability of the family of generalized $ (N+1) $-dimensional KP equations is also verified. By means of the long wave limit approach, lump-type solutions to these equations are constructed, and a (3+2)-dimensional example is presented to demonstrate their dynamical features. Through appropriate coefficient simplifications and linear variable transformations, the studied integrable equations can be reduced to the (2+1)-dimensional KP equation, (1+1)-dimensional Boussinesq equation, and their lower-dimensional reductions. This work enriches existing theoretical findings and provides an effective approach for constructing lump-type wave structures to nonlinear evolution equations.
- generalized Kadomtsev-Petviashvili equation,
- Bäcklund transformation,
- Lax pair,
- lump-type wave,
- long wave limit
Citation: Li Cheng, Wen-Xiu Ma, Xiao-Fei Hu. Integrability investigation and lump-type wave structures for higher-dimensional generalized Kadomtsev-Petviashvili equations[J]. AIMS Mathematics, 2026, 11(9): 30125-30144. doi: 10.3934/math.20261193

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Abstract

This paper systematically investigates the integrable properties and lump-type wave structures of higher-dimensional generalized Kadomtsev-Petviashvili (KP) equations, which describe nonlinear wave behaviors in physical systems. For the (3+1)-dimensional KP and Boussinesq equations, a three-soliton condition is established. Based on this condition, a bilinear Bäcklund transformation is constructed, from which the corresponding Lax pair and modified evolutionary equation are derived. The integrability of the family of generalized $ (N+1) $-dimensional KP equations is also verified. By means of the long wave limit approach, lump-type solutions to these equations are constructed, and a (3+2)-dimensional example is presented to demonstrate their dynamical features. Through appropriate coefficient simplifications and linear variable transformations, the studied integrable equations can be reduced to the (2+1)-dimensional KP equation, (1+1)-dimensional Boussinesq equation, and their lower-dimensional reductions. This work enriches existing theoretical findings and provides an effective approach for constructing lump-type wave structures to nonlinear evolution equations.

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