Analytic Investigation of a Generalized Variable-Coefficient KdV Equation with External-Force Term

Gongxun Li, Zhiyan Wang, Ke Wang, Nianqin Jiang and Guangmei Wei ()
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Gongxun Li: LMIB and School of Mathematical Sciences, Beihang University, Beijing 100191, China
Zhiyan Wang: LMIB and School of Mathematical Sciences, Beihang University, Beijing 100191, China
Ke Wang: LMIB and School of Mathematical Sciences, Beihang University, Beijing 100191, China
Nianqin Jiang: School of Physics, Beihang University, Beijing 100191, China
Guangmei Wei: LMIB and School of Mathematical Sciences, Beihang University, Beijing 100191, China

Mathematics, 2025, vol. 13, issue 10, 1-15

Abstract: This paper investigates integrable properties of a generalized variable-coefficient Korteweg–de Vries (gvcKdV) equation incorporating dissipation, inhomogeneous media, and an external-force term. Based on Painlevé analysis, sufficient and necessary conditions for the equation’s Painlevé integrability are obtained. Under specific integrability conditions, the Lax pair for this equation is successfully constructed using the extended Ablowitz–Kaup–Newell–Segur system (AKNS system). Furthermore, the Riccati-type Bäcklund transformation (R-BT), Wahlquist–Estabrook-type Bäcklund transformation (WE-BT), and the nonlinear superposition formula are derived. In utilizing these transformations and the formula, explicit one-soliton-like and two-soliton-like solutions are constructed from a seed solution. Moreover, the infinite conservation laws of the equation are systematically derived. Finally, the influence of variable coefficients and the external-force term on the propagation characteristics of a solitory wave is discussed, and soliton interaction is illustrated graphically.

Keywords: generalized variable-coefficient KdV equation; Painlevé analysis; lax pair; auto-Bäcklund transformation; conservation law (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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