Spectral and Dynamical Properties of the Fractional Nonlinear Schr\"odinger Equation under Harmonic Confinement

TL;DR

This paper explores how fractional dispersion affects the spectral stability and dynamics of the nonlinear Schrödinger equation with harmonic confinement, revealing $lpha$-dependent bifurcations, stability changes, and dynamical behaviors.

Contribution

It introduces a comprehensive numerical analysis of the fractional nonlinear Schr46dinger equation with harmonic trapping, highlighting the effects of fractional order on stability and dynamics.

Findings

Decreasing lpha shifts bifurcation curves and destabilizes excited states.

Focusing regime shows increased instability with lower lpha.

Defocusing regime maintains robust coherence despite fractional dispersion.

Abstract

We investigate the spectral and dynamical properties of the fractional nonlinear Schr\"odinger (fNLS) equation with harmonic confinement. In this setting, the classical Laplacian is replaced by its fractional power $(-\partial_x^2)^{\alpha/2}$ with $\alpha\in(1,2]$, introducing nonlocal, L\'evy-type dispersion. This modification fundamentally alters the balance between nonlinearity, dispersion, and trapping, reshaping both the structure and stability of stationary states. Using a Fourier pseudo-spectral discretization, we compute stationary branches as functions of the temporal frequency $\Omega$ in focusing ($\sigma=+1$) and defocusing ($\sigma=-1$) regimes, and assess spectral stability via the linearized eigenvalue problem. Direct simulations, performed with split-step and exponential time-differencing integrators, confirm these predictions and reveal $\alpha$-dependent transitions between coherent oscillations, bounded breathing dynamics, and decoherence or fragmentation. Our results show that decreasing $\alpha$ systematically shifts bifurcation curves, fragments stability windows for excited states, and amplifies instability in the focusing regime, while supporting robust coherence in the defocusing case. Beyond clarifying how harmonic confinement mediates the interplay between nonlinearity and fractional dispersion, the study also provides benchmarks for numerical treatments of fractional operators and points toward potential applications in nonlinear optics, Bose--Einstein condensates, and anomalous transport phenomena.