Painlevé analysis and nonlinear wave structures of a variable-coefficient extended KdV equation in $ (3+1) $ dimensions

Majid Madadi; Yakup Yildirim; Udoh Akpan; Lanre Akinyemi; Majid Madadi; Yakup Yildirim; Udoh Akpan; Lanre Akinyemi

doi:10.3934/math.20261073

AIMS Mathematics

2026, Volume 11, Issue 8: 26739-26764. doi: 10.3934/math.20261073

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Research article

Painlevé analysis and nonlinear wave structures of a variable-coefficient extended KdV equation in $ (3+1) $ dimensions

1.
Department of Mathematics, Saveetha School of Engineering, Saveetha Institute of Medical and Technical Sciences, Saveetha University, Chennai 602105, Tamil Nadu, India
2.
Research Center of Applied Mathematics, Khazar University, Baku, Azerbaijan
3.
Mathematics Research Center, Near East University, 99138 Nicosia, Cyprus
4.
Department of Computer Engineering, Biruni University, Istanbul–34010, Turkey
5.
Department of Mathematics, West Texas A & M University, Texas, USA
6.
Department of Mathematics, Prairie View A & M University, Texas, USA

Received: 11 June 2026 Revised: 19 July 2026 Accepted: 23 July 2026 Published: 25 August 2026
MSC : 35Q51, 37K10, 37K40

A variable-coefficient Korteweg–de Vries (KdV)-type equation in $ (3+1) $ dimensions is investigated with emphasis on Painlevé integrability and exact nonlinear wave dynamics. Using the Painlevé analysis, the model is shown to possess the Painlevé property under appropriate smoothness and nondegeneracy assumptions on the spatially varying coefficients. A Hirota bilinear representation is subsequently derived, from which an $ N $-soliton solution is constructed. The model admits a variety of exact nonlinear wave structures, including multi-solitons, resonant Y-type waves, fusion and fission waves, X-type solitons, hybrid interaction patterns, and rational lump-type solutions in the admissible parameter regime. The spatially varying coefficients significantly modify soliton trajectories, resonant interaction geometries, and the locations of localized structures, while the exact solutions retain their asymptotic profiles and exhibit shape-preserving interaction patterns. The coexistence of exponential and rational wave structures highlights the richness of the solution space. These results enlarge the class of spatially inhomogeneous nonlinear evolution equations (NLEEs) exhibiting Painlevé–Hirota coherent-wave structures and provide analytical insight into nonlinear wave propagation in spatially inhomogeneous media.
- extended KdV equation,
- Painlevé analysis,
- Hirota bilinear,
- soliton solution,
- lump wave
Citation: Majid Madadi, Yakup Yildirim, Udoh Akpan, Lanre Akinyemi. Painlevé analysis and nonlinear wave structures of a variable-coefficient extended KdV equation in $ (3+1) $ dimensions[J]. AIMS Mathematics, 2026, 11(8): 26739-26764. doi: 10.3934/math.20261073

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Abstract

A variable-coefficient Korteweg–de Vries (KdV)-type equation in $ (3+1) $ dimensions is investigated with emphasis on Painlevé integrability and exact nonlinear wave dynamics. Using the Painlevé analysis, the model is shown to possess the Painlevé property under appropriate smoothness and nondegeneracy assumptions on the spatially varying coefficients. A Hirota bilinear representation is subsequently derived, from which an $ N $-soliton solution is constructed. The model admits a variety of exact nonlinear wave structures, including multi-solitons, resonant Y-type waves, fusion and fission waves, X-type solitons, hybrid interaction patterns, and rational lump-type solutions in the admissible parameter regime. The spatially varying coefficients significantly modify soliton trajectories, resonant interaction geometries, and the locations of localized structures, while the exact solutions retain their asymptotic profiles and exhibit shape-preserving interaction patterns. The coexistence of exponential and rational wave structures highlights the richness of the solution space. These results enlarge the class of spatially inhomogeneous nonlinear evolution equations (NLEEs) exhibiting Painlevé–Hirota coherent-wave structures and provide analytical insight into nonlinear wave propagation in spatially inhomogeneous media.

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