https://doi.org/10.1140/epjd/s10053-026-01132-z
Research - Photon
Soliton, breather, and mixed interaction solutions of the (3+1)-dimensional shallow water wave equation with time-dependent coefficients
1
Department of Mathematics, Faculty of Science, Soran University, Erbil, Iraq
2
Department of Mathematics, College of Science, University of Duhok, Duhok, Iraq
a
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Received:
2
December
2025
Accepted:
9
February
2026
Published online:
23
March
2026
Abstract
This study investigates the (3+1)-dimensional shallow water wave equation with time-dependent coefficients, a widely accepted approach for water surface dynamics applied to coastal and hydraulic systems from which they can be seen as temporally dependent environmental conditions. Although the Hirota bilinear method has been used with the constant-coefficient shallow water equations, its systematic implementation on broader systems with general time-dependent coefficients has been under investigated. To fill the same gap, we introduce the Hirota bilinear method to formulate and classify several different solution families: single solitons, multi-solitons, breather waves, and mixed soliton–breather interactions. This work novelty is: (i) systematic treatment of the general time-dependent coefficient function (polynomial, trigonometric, hyperbolic) in a (3+1)-dimensional approach, (ii) comprehensive classification of all solution types under a unified analytical scheme, and (iii) thorough characterization of wave interaction dynamics using detailed graphical analysis. We present rich nonlinear phenomena like elastic collisions and their predictable phase shifts, amplitude modulation structures, and energy exchange processes. The detailed, two-dimensional and three-dimensional visualizations illustrate the differing solution behaviors for different coefficient regimes, revealing new insights into how waves propagate around shallow water with time-varying properties. These results are of tremendous significance to coastal engineering as tsunami modeling, oceanography as wave prediction in different ocean settings, and nonlinear physics as wave systems. The theoretical framework of dynamic and nonlinear wave equations is expanded, and new analytic tools for engineering are presented in the systematic methodological solution developed in this work.
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© The Author(s), under exclusive licence to EDP Sciences, SIF 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.
