Spectral and Dynamical Properties of the Fractional Nonlinear Schrödinger Equation under Harmonic Confinement

Abstract

We investigate the spectral and dynamical properties of the fractional nonlinear Schrödinger (fNLS) equation with harmonic confinement. In this setting, the classical Laplacian is replaced by its fractional power $(-\partial_x^2)^{α/2}$ with $α\in(1,2]$, introducing nonlocal, Lévy-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 $Ω$ in focusing ($σ=+1$) and defocusing ($σ=-1$) 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 $α$-dependent transitions between coherent oscillations, bounded breathing dynamics, and decoherence or fragmentation. Our results show that decreasing $α$ 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.

Figures (12)

• Figure 1: Bifurcation diagrams showing the $L^2$-norm $Q$ as a function of $\Omega$ for different values of the fractional parameter $\alpha$. Stable branches are shown in blue, unstable ones in red. The focusing regime corresponds to the branches extending to negative $\Omega$, while the defocusing regime corresponds to the upward branches bifurcating on the right.
• Figure 2: Representative stationary solution profiles for $\alpha=2$ (top row), $\alpha=1.5$ (middle row), and $\alpha=1.1$ (bottom row). Within each group, the first three panels correspond to focusing solutions with $\Omega=-2,-10$ for modes $n=0,1,2$, and the next three to defocusing solutions with $\Omega=7,15$ for the same modes. Decreasing $\alpha$ enhances nonlocal effects: focusing states become thinner and more sensitive to $\Omega$, while defocusing states broaden and develop irregular structures compared to the classical case $\alpha=2$.
• Figure 3: Real spectrum for mode $n=2$ at $\alpha=2$ in focusing (magenta, thick) and defocusing (black, thin) cases. Stability changes occur when $Re(\lambda)$ crosses zero. The green dashed line indicates the value of $\Omega$, at which the nonlinear states bifurcate.
• Figure 4: Real spectrum for $\alpha=1.5$ in focusing (magenta, thick) and defocusing (black, thin) cases. Moderate nonlocality shifts bifurcation points and introduces additional unstable intervals, especially for $n=2$.
• Figure 5: Real spectrum for $\alpha=1.1$ in focusing (magenta, thick) and defocusing (black, thin) cases. Strong nonlocality fragments stability windows and enlarges unstable regions.