New computational approaches to the fractional coupled nonlinear Helmholtz equation

Purpose The main aim of this paper is to investigate the fractional coupled nonlinear Helmholtz equation by two new analytical methods. Design/methodology/approach This article takes an inaugural look at the fractional coupled nonlinear Helmholtz equation by using the conformable derivative. It successfully finds new fractional periodic solutions and solitary wave solutions by employing methods such as the fractional method and the fractional simple equation method. The dynamics of these fractional periodic solutions and solitary wave solutions are then graphically represented in 3D with appropriate parameters and fractal dimensions. This research contributes to a deeper comprehension and detailed exploration of the dynamics involved in high dimensional solitary wave propagation. Findings The proposed two mathematical approaches are simple and efficient to solve fractional evolution equations. Originality/value The fractional coupled nonlinear Helmholtz equation is described by using the conformable derivative for the first time. The obtained fractional periodic solutions and solitary wave solutions are completely new.