https://doi.org/10.1140/epjp/s13360-025-07255-3
Regular Article
Lump solutions and various soliton structures constructed of the integrable multi-dimensional Boussinesq-type equation
1
State Key Laboratory of Coal Resources and Safe Mining, China University of Mining and Technology (Beijing), 100083, Beijing, China
2
School of Science, China University of Mining and Technology (Beijing), 100083, Beijing, China
3
School of Mathematics and Physics, China University of Geosciences, Wuhan, 430074, Hubei, China
a
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Received:
17
October
2025
Accepted:
23
December
2025
Published online:
24
January
2026
Abstract
This paper is concerned with the integrable multi-dimensional Boussinesq-type equation formulated by Wazwaz, a model capable of describing shallow-water and other nonlinear propagation phenomena, with applicability to simulating wave propagation dynamics in oceanography and atmospheric science. Initially, in the long-wave limit we construct lump solutions, which describe algebraically localized solitary wave patterns in shallow water, identify and quantify the key parameters that control the peak amplitude of shallow-water waves, and characterize the associated propagation dynamics. Subsequently, by introducing a complex conjugation method, we derive breathing wave solutions. To generate soliton molecules, a velocity resonance condition is applied to the N-soliton solutions, which reveals stable configurations arising in the propagation of shallow-water waves described by this model. Furthermore, by imposing appropriate constraints on the N-soliton solutions, we derive Y-type soliton solutions, thereby broadening the collection of explicit solutions available for this equation.
Key words: Lump solutions / Breathing wave solution / Soliton molecule / The integrable multi-dimensional Boussinesq-type equation
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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.
