In this paper, we studied a $ (2+1) $-dimensional variable-coefficient nonlinear evolution equation that generalizes several well-known models, including the Kadomtsev–Petviashvili, generalized second-order Benjamin–Ono, and Boussinesq equations. The integrability of the equation was investigated using Painlevé analysis, leading to explicit conditions on the variable coefficients. These conditions were shown to be consistent with the existence of multi-soliton solutions. By applying Hirota's bilinear method, we explicitly derived and verified one-, two-, three-, and four-soliton solutions. Their pairwise interaction structure motivates a proposed general $ N $-soliton form. In addition, breather waves, lump solutions, and hybrid lump–soliton interactions were derived using appropriate ansatz methods. The effects of variable coefficients on wave behavior were illustrated through different solution structures. The results showed that variable coefficients play an important role in shaping wave dynamics and interaction patterns. This work extends existing results on nonlinear evolution equations and provides useful insight into wave propagation in inhomogeneous media.
Citation: Majid Madadi, Yakup Yildirim, Lanre Akinyemi, Solomon Manukure. Integrability criteria and multi-wave solutions of a generalized variable-coefficient nonlinear evolution equation[J]. AIMS Mathematics, 2026, 11(8): 26677-26698. doi: 10.3934/math.20261070
In this paper, we studied a $ (2+1) $-dimensional variable-coefficient nonlinear evolution equation that generalizes several well-known models, including the Kadomtsev–Petviashvili, generalized second-order Benjamin–Ono, and Boussinesq equations. The integrability of the equation was investigated using Painlevé analysis, leading to explicit conditions on the variable coefficients. These conditions were shown to be consistent with the existence of multi-soliton solutions. By applying Hirota's bilinear method, we explicitly derived and verified one-, two-, three-, and four-soliton solutions. Their pairwise interaction structure motivates a proposed general $ N $-soliton form. In addition, breather waves, lump solutions, and hybrid lump–soliton interactions were derived using appropriate ansatz methods. The effects of variable coefficients on wave behavior were illustrated through different solution structures. The results showed that variable coefficients play an important role in shaping wave dynamics and interaction patterns. This work extends existing results on nonlinear evolution equations and provides useful insight into wave propagation in inhomogeneous media.
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Majid Madadi, Yakup Yildirim, Lanre Akinyemi, Solomon Manukure. Integrability criteria and multi-wave solutions of a generalized variable-coefficient nonlinear evolution equation[J]. AIMS Mathematics, 2026, 11(8): 26677-26698. doi: 10.3934/math.20261070
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