Novel Quasi-Periodic Type Optical Solitons and the Formation of Fractal Structures in Non-integrable Nonlinear Helmholtz Equations with Phase Portraits and Chaotic Analysis (European Journal of Pure and Applied Mathematics, 2025)
In this study, we consider new optical soliton solutions of one of the most important non-integrable model arising in optical fibres, namely Nonlinear Helmholtz equations (NHEs) that describes transverse interactions, transmission of coupled waves and optical solitons’ propagation in the field of fiber optics. We apply an adapted method to obtain some novel plethora of optical quasi-periodic soliton solutions. These solutions are presented in the shape of exponential, hyperbolic, trigonometric and rational functions. A set of 3D visualization, contour plots and 2D curves of these solutions physical relevance are presented with implications for the nonlinear optics. These figures also reveal that the established optical solitons exhibit quasi-periodicity due to the combination of linear periodic and axial perturbations, and that the presence of quasi-periodical perturbations of the solitons leads to the formation of the fractal-like structures. We also study the chaotic/periodic and bifurcation behavior, associated with the model, in the light of Hamiltonian analysis, as a consequence, we find positive results of the quasi-periodicity and fractal-like structures in the systems under consideration. Apart from offering novel analytical perspectives for dealing with the coupled NHEs, the present results would also be a concrete contribution to the understanding the soliton wave dynamics in complicated nonlinear media.
Keywords: Complex structured partial differential equations; Unified method; Nonlinear Helmholtz equations; Quasi-Periodic soliton; Optical fractals; Chaotic analysis