Analytical resolution of nonlinear fractional equations using the GERDFM method: Application to nonlinear Schrödinger and truncated Boussinesq-Burgers equations

Abstract

In this paper we develop the Generalized Exponential Rational Differential Function Method (GERDFM) for analytically solving complex nonlinear fractional partial differential equations, with application to the fractional nonlinear Schr¨odinger equation (NLSE) and the M-fractional truncated Boussinesq-Burgers equation. Our approach transforms these PDEs into adapted ordinary differential equations (ODEs), generating exact solutions for various nonlinear laws (Kerr, power, double power, parabolic) while explicitly incorporating the fractional Caputo derivatives of order M ? [0, 1]. The solitonic profiles obtained, illustrated by 2D/3D visualizations, reveal the crucial impact of non-linearity and fractional order on their dynamics, particularly in long memory optical systems and viscoelastic media. A rigorous numerical validation combining a fractional Runge-Kutta method and an L1 scheme confirms the superiority of our solutions, with a relative error < 10?8 (error < 2% near the solitonic peak) and a reduced computation time compared to conventional methods (Tanh-Coth, Sine-Cosine). These results open up concrete prospects for controlling solitons in anomalous dispersion optical fibers and modelling extreme waves in coastal hydrodynamics, while suggesting promising extensions to coupled and stochastic systems in nonlinear optics, fluid dynamics and plasma physics. This work provides significant advances in modeling wave propagation in complex media with memory effects. The GERDFM method’s ability to handle diverse nonlinearities while maintaining computational efficiency makes it particularly valuable for designing optical communication systems and predicting extreme wave phenomena in coastal engineering. Our analytical framework bridges a critical gap between classical soliton theory and fractional calculus applications.