https://doi.org/10.1140/epjd/s10053-025-01094-8
Research - Plasmas
Fractal dimension and structural evolution in a modified dispersive water wave system
1
Department of Mathematics, Siksha Bhavana, Visva-Bharati, 731235, Santiniketan, India
2
Department of Mathematics, Sidho-Kanho-Birsha University, 723104, Purulia, India
3
Department of Mathematics, Cooch Behar Panchanan Barma University, 736101, Coochbehar, India
a
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Received:
8
September
2025
Accepted:
10
November
2025
Published online:
2
December
2025
Abstract
The emergence of fractal waveforms in the modified dispersive water wave (MDWW) system, a nonlinear dispersive extension of shallow-water dynamics with recognized analogies in plasma wave theory, is investigated. Through the application of a Riccati-type transformation, the coupled equations are reduced to analytically tractable forms, yielding distinct families of analytic solutions. Subsequently, by imposing auxiliary functional structures involving trigonometric, logarithmic and Jacobi elliptic expressions, these solutions are shown to generate recursive, self-similar waveforms, whose scale-invariant character is established through systematic fractal diagnostics. Successive magnification and voxel and grid-based box-counting computations confirm non-integer fractal dimensions, with robust convergence and statistical validation via relative error, standard error and bootstrap standard deviation. Such consistency across refinement levels establishes stable scaling exponents and rules out numerical artifacts, thereby rigorously demonstrating intrinsic fractal geometry in the MDWW dynamics. From a physical standpoint, such multiscale structures are indicative of how nonlinear dispersive interactions may generate complex spatial organization relevant to turbulent cascades, energy localization and anomalous transport in plasma environments, nonlinear optical media and shallow-water flows. The principal novelty of the work lies in the unified integration of Riccati-based analytic solution construction with quantitative fractal dimension diagnostics and convergence assessment, thereby providing a new analytical–computational framework for probing fine-scale, self-similar behavior in multidimensional dispersive systems of contemporary interest in plasma physics, nonlinear optics, fluid mechanics and wave propagation theory. The results further demonstrate that analytic fractal waveforms can serve as model proxies for structured energy cascades, offering new analytical pathways toward turbulence-inspired modeling and multiscale transport analysis.
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© The Author(s), under exclusive licence to EDP Sciences, SIF and Springer-Verlag GmbH Germany, part of Springer Nature 2025
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.
