article · AIMS Mathematics
The two-mode Nizhnik-Novikov-Veselov equation is a mathematical model used in physics and engineering to describe nonlinear soliton propagation. This research carries out a bifurcation analysis of a (2+1)-dimensional conformable time-fractional version of the model. By deriving explicit analytical solutions, the study reveals several distinctive wave patterns, including periodic waves, bright solitons, dark solitons, and combination bright-dark bell waves, which have relevance in optical systems. The dynamics of the model are further evaluated through planar dynamical systems, mapping out phase portraits and multistability across varying parameters. Additionally, the investigation identifies quasi-periodic and chaotic behaviours under distinct conditions, supported by two-dimensional, three-dimensional, and density visualisations of the resulting wave phenomena.
Solitons are stable wave packets essential for understanding wave transport in physical media, notably in optical systems. By mapping bifurcation, chaotic dynamics, and explicit wave solutions in a fractional mathematical model, this research helps scientists and engineers better understand complex wave propagation, energy localisation, and stability limits in nonlinear physical systems.
The abstract notes that the identified bright, dark, and combo soliton solutions have relevance to optical applications. However, this is early-stage theoretical and mathematical physics research. Any practical adoption by engineers working on optical devices or wave-based systems would require experimental validation, as the abstract does not indicate a direct commercialisation pathway or prototype development.
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The two-mode Nizhnik-Novikov-Veselov (TMNNV) equation finds wide-ranging utility across engineering and scientific fields. It stands as a notable nonlinear physical model for explaining nonlinear soliton propagation. This study explores bifurcation analysis for the (2+1)-dimensional conformable time-fractional TMNNV model for the first time. Also, we have derived the explicit solutions of this model, and these solutions exhibit some unique dynamical patterns: combo bright-dark bell wave, periodic wave, bright soliton, and dark soliton, which are used in several optical applications. 2-D plots and combined 3-D with density plots are presented with the impacts of different parameters. Later, phase portraits and the multistability of this dynamical model are analyzed via intersecting figures with the help of the planner dynamical system. We also examine the quasi-periodic and chaotic behaviors of the governing model under different conditions. Finally, conclusions are drawn based on the results.
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DOI: 10.3934/math.2025211
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