Fractional Nonlinear Schrodinger Equation Revisited

TL;DR

This paper explores the fractional nonlinear Schrödinger equation using analytical methods, revealing diverse solution structures, stability properties, and connections to stochastic processes and quantum statistics.

Contribution

It introduces new analytical solutions and classifications for the fractional nonlinear Schrödinger equation, including solitons and plane waves, and links it to fractional Fokker-Planck equations.

Findings

Bright and dark solitons are constructed and their stability analyzed.

Solutions transition from bright to dark as memory effects vary.

Connections established between FNLSE and fractional stochastic processes.

Abstract

We investigate the space time fractional nonlinear Schrodinger equation (FNLSE) incorporating the modified Riemann Liouville derivative introduced by Jumari. The equation is characterized by two parameters: the fractional derivative parameter (alpha, which captures the memory effects) and the non linearity parameter a. We present analytical solutions via three complementary approaches: the fractional Riccati method, the Adomian decomposition method, and the scaling method. The FNLSE is formulated in terms of generalized Hamiltonian and momentum operators, allowing a unified framework to explore various solution structures. A continuity equation is derived, and a general class of solutions based on Mittag Leffler (ML) plane waves is proposed, from which generalized momentum and energy eigenvalues are systematically classified. Utilizing a generalized Wick rotation, we establish a connection between the FNLSE and a fractional Fokker Planck equation governing a dual stochastic process, revealing links to Q Gaussian statistics. Through separation of variables, we classify a family of solutions including chiral ML plane waves. Additionally, we construct Riccati type bright and dark solitons and assess their stability using a dynamical distance metric. As alpha changes, i.e. the memory effects are tuned, the bright solitons transform to dark solitons, which is equivalent to focusing defocusing transition. A series solution is developed via the Adomian decomposition technique, applied to both chiral and plane wave cases. Finally, we reduce the dimensionality of the FNLSE using the scaling arguments, leading to self similar solutions, and find pure phase as well as Adomian type solutions. Our results highlight the rich analytical landscape of the FNLSE and provide insights into its underlying physical and mathematical structure.