A Korteweg–de Vries–Sawada–Kotera–Ramani-type equation: Its integrability and multi-wave solutions

Majid Madadi; Kamyar Hosseini; Farzaneh Alizadeh; Evren Hincal; Sekson Sirisubtawee; Majid Madadi; Kamyar Hosseini; Farzaneh Alizadeh; Evren Hincal; Sekson Sirisubtawee

doi:10.3934/math.2026529

AIMS Mathematics

2026, Volume 11, Issue 5: 12866-12894. doi: 10.3934/math.2026529

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A Korteweg–de Vries–Sawada–Kotera–Ramani-type equation: Its integrability and multi-wave solutions

1.
Mathematics Research Center, Near East University, Mersin 10, Nicosia, 99138, Turkey
2.
Research Center of Applied Mathematics, Khazar University, Baku, Azerbaijan
3.
Faculty of Engineering and Natural Sciences, Istanbul Okan University, Istanbul, Turkey
4.
Department of Mathematical Sciences, Saveetha School of Engineering, SIMATS, Chennai-602105, Tamilnadu, India
5.
Department of Mathematics, Faculty of Applied Science, King Mongkut's University of Technology North Bangkok, Bangkok 10800, Thailand
6.
Centre of Excellence in Mathematics, CHE, Si Ayutthaya Road, Bangkok 10400, Thailand

Received: 16 March 2026 Revised: 20 April 2026 Accepted: 27 April 2026 Published: 11 May 2026
MSC : 35C08, 35Qxx, 35Q51, 37K40

This paper studies the integrability and nonlinear multi-wave solutions of a Korteweg–de Vries–Sawada–Kotera–Ramani-type (KdV–SKR-type) equation arising in shallow-water wave theory, which is important for modeling nonlinear wave interactions. The integrability of the equation is confirmed by Painlevé analysis, and its bilinear form is subsequently derived using the Bell polynomial (BP) method. Based on the resulting formulation, multi-soliton solutions are constructed via the simplified Hirota method, and the corresponding multi-lump solutions are obtained through the long-wave limit. Furthermore, a Pfaffian framework is developed to construct compact general $ N $-soliton solutions and to systematically generate multi-lump waves within a unified algebraic structure. Various interaction phenomena, including resonant and breather waves, are also derived. The results confirm the integrable nature of the model and reveal rich nonlinear dynamics. The novelty of this work lies in the introduction of the Pfaffian formulation for this equation and the unified construction of its solutions, extending previous studies.
- KdV–SKR-type equation,
- bilinearization,
- integrability,
- multi-soliton waves,
- multi-lump waves
Citation: Majid Madadi, Kamyar Hosseini, Farzaneh Alizadeh, Evren Hincal, Sekson Sirisubtawee. A Korteweg–de Vries–Sawada–Kotera–Ramani-type equation: Its integrability and multi-wave solutions[J]. AIMS Mathematics, 2026, 11(5): 12866-12894. doi: 10.3934/math.2026529

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Abstract

This paper studies the integrability and nonlinear multi-wave solutions of a Korteweg–de Vries–Sawada–Kotera–Ramani-type (KdV–SKR-type) equation arising in shallow-water wave theory, which is important for modeling nonlinear wave interactions. The integrability of the equation is confirmed by Painlevé analysis, and its bilinear form is subsequently derived using the Bell polynomial (BP) method. Based on the resulting formulation, multi-soliton solutions are constructed via the simplified Hirota method, and the corresponding multi-lump solutions are obtained through the long-wave limit. Furthermore, a Pfaffian framework is developed to construct compact general $ N $-soliton solutions and to systematically generate multi-lump waves within a unified algebraic structure. Various interaction phenomena, including resonant and breather waves, are also derived. The results confirm the integrable nature of the model and reveal rich nonlinear dynamics. The novelty of this work lies in the introduction of the Pfaffian formulation for this equation and the unified construction of its solutions, extending previous studies.

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