Soliton Dynamics in Fractional and q-Deformed Sinh-Gordon Models: A Unified Numerical Framework

Abstract

This study presents a comparative numerical analysis of four variants of the nonlinear sinh-Gordon equation: the classical, q-deformed, time-fractional, and fractional q-deformed models. These formulations incorporate memory effects via Caputo fractional derivatives and structural asymmetries through q-deformation, providing a unified framework to explore soliton dynamics in complex media. All models are solved using the finite difference method under consistent initial and boundary conditions, enabling direct comparison of their spatiotemporal behavior, full width at half maximum (FWHM), and total energy. Small values of the fractional order (α) and deformation parameter (q) lead to delocalized, oscillatory waveforms due to dominant memory and asymmetry effects. Larger values restore localized kink-type solitons, often accompanied by low-amplitude edge oscillations attributed to internal mode excitations or residual radiation. For q in the range 0.0001–0.5, significant waveform asymmetries emerge, resulting in notable differences in amplitude and FWHM between positive and negative lobes. As q approaches 1, the solution converges to the classical case. In the time-fractional model, the highest energy and FWHM values occur at α = 0.1, both decreasing as α increases toward 1. For α > 1, the solution gradually recovers classical behavior, fully restored at α = 2. A unified initial condition is introduced and applied consistently across all models, enabling reliable benchmarking and revealing internal mode dynamics. These findings provide new insights into nonlinear wave propagation and energy localization in media with memory and structural deformation, demonstrating that α and q act as effective control parameters for soliton structure and stability.