Algebraic reduction of fractional systems: Analytical solutions and memory-induced soliton deformation

This study presents a novel algebraic methodology that transforms systems of nonlinear Caputo fractional differential equations into equivalent first-order ordinary differential equations. By applying the properties of fractional power series, the proposed framework maps non-local fractional operators directly into a local integer-order domain, embedding infinite memory effects into a deterministic, time-dependent polynomial forcing term. The physical applicability of this reduction technique is investigated through the analysis of a multiplicatively coupled fractional Riccati system, specifically focusing on the deformation of solitary wave solutions induced by fractional memory. Computational experiments indicate that non-zero fractional initial conditions function as continuous shape modulators that fundamentally alter the amplitude, width, and asymptotic decay rates of solitary waves, thereby breaking the exact symmetries of standard localized behavior. Despite this memory-induced deformation and the associated loss of general integrability, novel exact analytical solutions are successfully constructed for the perturbed system. By identifying specific parameter constraints that compensate for the non-autonomous memory effects, the nonlinear ordinary differential equation system is mapped into a higher-dimensional linear space and resolved using Bessel functions. The resulting closed-form analytical solutions, represented as exact integrals, are validated against direct numerical approximation methods, confirming the accuracy and practical utility of the proposed mathematical framework for analyzing complex fractional dynamics.