Soliton solutions and dynamics of the (3+1)-dimensional Kadomtsev-Petviashvili equation via $ \phi^{6} $-model expansion method

Abid Ullah Khan; Asif Khan; Salma Trabelsi; Marwa Balti; Abid Ullah Khan; Asif Khan; Salma Trabelsi; Marwa Balti

doi:10.3934/math.20251017

AIMS Mathematics

2025, Volume 10, Issue 10: 22883-22910. doi: 10.3934/math.20251017

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Research article

Soliton solutions and dynamics of the (3+1)-dimensional Kadomtsev-Petviashvili equation via $ \phi^{6} $-model expansion method

1.
Department of Mathematics, University of Malakand, Chakdara, Dir Lower, Khyber Pakhtunkhwa, Pakistan
2.
Department of Mathematics and statistics, College of Science, King Faisal University, Al Ahsa 31982, Saudi Arabia

Received: 18 August 2025 Revised: 17 September 2025 Accepted: 22 September 2025 Published: 10 October 2025
MSC : 35Q55, 35C08, 37K10, 35A20

In this work, we explored the three-dimensional Kadomtsev-Petviashvili equation by the $ \phi^{6} $-model expansion method to generate precise solutions, such as periodic waves and singular structures, kink-type solitons, and dark solitons. The periodic solutions are the repeating patterns of a wave, and the singular solutions are the profiles that have unlimited amplitude at a certain point. The kink-type solitons depict sudden transitions, and the dark solitons represent local amplitude depressions in optical systems. Two- and three-dimensional plots were used to show the dynamic behavior and interactions of these solutions. The $ \phi^{6} $-model expansion technique is highly effective to study nonlinear systems, and can potentially be applied to fluid mechanics, plasma physics, and optical communication. The obtained results contribute to the understanding of the soliton propagation in the multi-dimensional systems.
- (3+1)-dimensional Kadomtsev-Petviashvili (KP) equation,
- $ \phi^{6} $-model expansion method,
- exact periodic,
- singular soliton,
- dark soliton
Citation: Abid Ullah Khan, Asif Khan, Salma Trabelsi, Marwa Balti. Soliton solutions and dynamics of the (3+1)-dimensional Kadomtsev-Petviashvili equation via $ \phi^{6} $-model expansion method[J]. AIMS Mathematics, 2025, 10(10): 22883-22910. doi: 10.3934/math.20251017

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Abstract

In this work, we explored the three-dimensional Kadomtsev-Petviashvili equation by the $ \phi^{6} $-model expansion method to generate precise solutions, such as periodic waves and singular structures, kink-type solitons, and dark solitons. The periodic solutions are the repeating patterns of a wave, and the singular solutions are the profiles that have unlimited amplitude at a certain point. The kink-type solitons depict sudden transitions, and the dark solitons represent local amplitude depressions in optical systems. Two- and three-dimensional plots were used to show the dynamic behavior and interactions of these solutions. The $ \phi^{6} $-model expansion technique is highly effective to study nonlinear systems, and can potentially be applied to fluid mechanics, plasma physics, and optical communication. The obtained results contribute to the understanding of the soliton propagation in the multi-dimensional systems.

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