https://doi.org/10.1140/epjp/s13360-026-07731-4
Regular Article
Dynamics of localized waves and interactions for the (3+1)-Dimensional variable-coefficient Kairat-II equation
1
School of Mathematics, Xi’an University of Technology, 710054, Xi’an, China
2
School of Science, Xi’an University of Architecture and Technology, 710055, Xi’an, China
3
School of Artificial Intelligence and Computer Science, Shaanxi Normal University, 710062, Xi’an, China
a
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b
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Received:
11
February
2026
Accepted:
20
April
2026
Published online:
4
May
2026
Abstract
This paper investigates the localized wave solutions and their nonlinear dynamical properties of the (3+1)-dimensional variable-coefficient Kairat-II (3DVCK-II) equation. Starting from the constant-coefficient version, we introduce time-dependent coefficients and establish the integrability of the resulting equation via the Painlevé test. Using the Hirota bilinear method, we derive higher-order kink solutions, breather solutions, lump-type solutions, and several hybrid solutions. By considering different functional forms of the time-varying coefficients—such as piecewise trigonometric, hyperbolic and polynomial functions, we systematically analyze the physical effects of these variable coefficients on wave propagation trajectories, symmetry and energy dissipation. Graphical illustrations including three-dimensional and density plots are provided to visualize the dynamic behaviors. Furthermore, to rigorously verify the accuracy of the analytical solutions and the physical stability of the localized waves, deep learning based numerical simulations are conducted using Physics-Informed Neural Networks (PINN). The results demonstrate that variable coefficients significantly enrich the solution structures, offering deeper insights into the nonlinear wave phenomena described by the model.
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© The Author(s), under exclusive licence to Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2026
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
