The concept of derivatives with non-integral orders has been studied for centuries and has found numerous applications in physics and engineering. Recently, there has been growing interest in exploring fractional versions of popular differential equations. This study aims at finding stationary solutions to the fractional generalisation of the Nonlinear Schrödinger equation. Analytical stationary solutions to the nonlinear Schrödinger equation (NLSE) exist and are well known in literature [1] to have their amplitudes related to the Jacobi elliptic trigonometric functions. However, the spectrum of linearisation around the Jacobi elliptic solution is known to be purely imaginary, making it impossible to obtain the solution with direct numerical attempts. We present a novel approach attempting to stabilise the numerical scheme and obtaining the Jacobi elliptic solutions by replacing the NLSE with a pair of coupled (auxiliary) PDEs inspired by the Gierer-Meinhardt (GM) model, which is known to have stable spike-type solutions that have negative real parts in their stability spectrum[2]. The stable nature of the solution persists even when the Laplacian is replaced with a derivative of a non-integral order[3], which makes it a suitable inspiration for our purposes. Results of the numerical trial of the auxiliary PDE approach are presented. We use a pseudo-spectral scheme based on Fourier transforms to simplify the problem from a system of partial differential equations to ordinary differential equations and to allow for generalisation to non-integral order derivatives. Despite extensive experimentation with various parameters, our method fails to converge to the Jacobi elliptic solutions. Lastly, we also present a comprehensive review of the literature on fractional calculus, the Nonlinear Schrödinger equation and the stability analysis of stationary solutions to the NLSE.
