A Nonlinear $q$-Deformed Schr\"odinger Equation

TL;DR

This paper introduces a new nonlinear q-deformed Schr"odinger equation derived from a nonlinear derivative operator, which recovers the standard equation as q approaches 1, and explores its analytical and numerical solutions.

Contribution

The paper constructs a novel nonlinear q-deformed Schr"odinger equation using a parameter-dependent derivative operator, extending quantum mechanics with a deformed kinetic term and solitonic solutions.

Findings

Analytical solution for zero potential and q near 1.

Conservation of energy, momentum, and interaction with electromagnetic fields.

Numerical solitonic solutions in one-dimensional free particle case.

Abstract

We construct a new nonlinear deformed Schr\"odinger structure using a nonlinear derivative operator which depends on a parameter $q$. This operator recovers Newton derivative when $q \rightarrow 1$. Using this operator we propose a deformed Lagrangian which gives us a deformed nonlinear Schr\"odinger equation with a nonlinear kinetic energy term and a standard potential $V(\vec{x})$. We analytically solve the nonlinear deformed Schr\"odinger equation for $V(\vec{x}) = 0$ and $q \simeq1$. This model has a continuity equation, the energy is conserved, as well as the momentum and also interacts with electromagnetic field. Planck relation remains valid and in all steps we easily recover the undeformed quantities when the deformation parameter goes to 1. Finally, we numerically solve the equation of motion for the free particle in any spatial dimension, which shows a solitonic pattern when the space is equal to one for particular values of $q$.