Another formulation of the fractional nonlinear Schrödinger equation

¹ National Committee for Development of Technologies (CNDT), Ministry of Scientific Research and Innovation Yaounde, 1457, Cameroon
² Pure Physics Laboratory: Group of Nonlinear Physics and Complex Systems, Department of Physics, Faculty of Science, University of Douala - Douala, 24157, Cameroon
³ Laboratoire de Physique, Ecole Normale Supérieure de Lyon - Lyon, 69364, France

Received: 30 October 2023
Accepted: 16 January 2024

Abstract

In this letter, we pave the way to establish the formulation of a non-trivial fractional Nonlinear Schrödinger (NLS) equation, which is different from the formulations known so far that consist in directly replacing the integer orders of the derivatives by non-integer ones. Thereafter, we set up some formulations, adapted to some particular physical cases, namely, the cases where the nonlinearity is stronger than the dispersion, in addition to one for which the dispersion strongly dominates the nonlinearity and also the case where the system displays a nonlinearity which is compensated with the dispersion. These formulations highlight the fact that the transition from a formal classical analysis to a fractional one could lead changes in the initial model of a given system. The research for solutions of the equations resulting from this study will undoubtedly reveal new phenomena in the different physical, biological and other systems described by the NLS equation.