In this study, we focused on the exploration of the Painlevé-type auto-Bäcklund transformation (aBT) and several novel analytical solutions to the fourth-order potential Ito equation describing water wave dynamics. By implementing the truncated Painlevé expansion, the Painlevé-type aBT was rigorously constructed, from which the kink soliton, singular solutions, and rational solutions were systematically derived. Through the Painlevé analysis, the bilinear form of the target equation was explicitly obtained, and the new homoclinic test technique was further adopted to derive the breather-type kink soliton and singular periodic wave solutions. Additionally, the interaction solutions constructed with the cos-cosh ansatz were analyzed in detail. Ultimately, a series of distinct traveling wave solutions were successfully developed by employing four robust analytical approaches, namely the variational method, energy balance method, direct mapping method, and the Kudryashov method. The obtained solutions covered kink-type, anti-kink-type, periodic, singular periodic, and rational forms, which were explicitly expressed via exponential, trigonometric, hyperbolic, arc tangent, and rational functions. The spatiotemporal profiles of some obtained solutions were visualized by selecting appropriate physical parameters to intuitively illustrate their inherent physical properties. Our findings of this work are expected to provide valuable references for theoretical research and contribute to a deeper and more systematic understanding of the fundamental properties and intrinsic mechanisms of the studied equation.
Citation: Jing Yang, Peng Xu, Feng Shi, Jing-Hua Liu, Kang-Jia Wang. Unveiling nonlinear wave dynamics: auto-Bäcklund transformation and multiple-distinct analytical solutions to the fourth-order potential Ito equation for water waves[J]. AIMS Mathematics, 2026, 11(10): 35718-35736. doi: 10.3934/math.20261384
In this study, we focused on the exploration of the Painlevé-type auto-Bäcklund transformation (aBT) and several novel analytical solutions to the fourth-order potential Ito equation describing water wave dynamics. By implementing the truncated Painlevé expansion, the Painlevé-type aBT was rigorously constructed, from which the kink soliton, singular solutions, and rational solutions were systematically derived. Through the Painlevé analysis, the bilinear form of the target equation was explicitly obtained, and the new homoclinic test technique was further adopted to derive the breather-type kink soliton and singular periodic wave solutions. Additionally, the interaction solutions constructed with the cos-cosh ansatz were analyzed in detail. Ultimately, a series of distinct traveling wave solutions were successfully developed by employing four robust analytical approaches, namely the variational method, energy balance method, direct mapping method, and the Kudryashov method. The obtained solutions covered kink-type, anti-kink-type, periodic, singular periodic, and rational forms, which were explicitly expressed via exponential, trigonometric, hyperbolic, arc tangent, and rational functions. The spatiotemporal profiles of some obtained solutions were visualized by selecting appropriate physical parameters to intuitively illustrate their inherent physical properties. Our findings of this work are expected to provide valuable references for theoretical research and contribute to a deeper and more systematic understanding of the fundamental properties and intrinsic mechanisms of the studied equation.
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Figures(9)
Jing Yang, Peng Xu, Feng Shi, Jing-Hua Liu, Kang-Jia Wang. Unveiling nonlinear wave dynamics: auto-Bäcklund transformation and multiple-distinct analytical solutions to the fourth-order potential Ito equation for water waves[J]. AIMS Mathematics, 2026, 11(10): 35718-35736. doi: 10.3934/math.20261384
The shapes of Eq (2.14) with $ m = 1 $, $ \alpha = 1, $ and $ \beta = -1 $
The shapes of Eq (2.18) with $ \alpha = 1 $, $ \beta = -1 $, $ m = 1, $ and $ s = 1 $
The shapes of Eq (2.19) with $ \alpha = 1 $, $ \beta = -1 $, $ m = 1, $ and $ s = -1 $
The shapes of breather-type kink soliton solution in Eq (3.5) with $ \alpha = 1 $, $ \beta = 1 $, $ {s}_{5} = 2 $, $ {s}_{6} = 1, $ and $ {s}_{7} = 2 $
The shapes of interaction solution in Eq (4.3) with $ {\mathfrak{R}}_{1} = 1 $, $ {\mathfrak{R}}_{2} = 2 $, $ \alpha = 1 $, and $ \beta = 2 $
The shapes of Eq (5.22) with $ \alpha = -1 $, $ \beta = 1 $, $ \vartheta = 1 $, and $ \varpi = -1 $
The shapes of Eq (5.36) with $ \varLambda = 3 $, $ \alpha = 1 $, $ \beta = 1 $, $ \vartheta = 1, $ and $ \varpi = -2 $
The shapes of Eq (5.58) with $ \alpha = 1 $, $ \beta = 1 $, $ {u}_{0} = 1 $, and $ \vartheta = 1 $
The shapes of Eq (5.58) with $ \alpha = 2 $, $ \beta = -1 $, $ {u}_{0} = 1 $, and $ \vartheta = 1 $
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