Numerical study of fractional Nonlinear Schrödinger equations

Abstract

Using a Fourier spectral method, we provide a detailed numerically investigation of dispersive Schrödinger type equations involving a fractional Laplacian. By an appropriate choice of the dispersive exponent, both mass and energy sub- and supercritical regimes can be computed in one spatial dimension, only. This allows us to study the possibility of finite time blow-up versus global existence, the nature of the blow-up, the stability and instability of nonlinear ground states, and the long time dynamics of solutions. The latter is also studied in a semiclassical setting. Moreover, we numerically construct ground state solutions to the fractional nonlinear Schrödinger equation.

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References in corpus (2)

• Nonlinear fractional Schrödinger equations in one dimension
• On the orbital stability of fractional Schrödinger equations

Cited by in corpus (6)

• A numerical approach to Blow-up issues for Davey-Stewartson II type systems
• Stationary solutions of a fractional Laplacian with singular perturbation
• Nondispersive solutions to the mass critical half-wave equation in two dimensions
• Multi-domain spectral approach for the Hilbert transform on the real line
• Spectral and Dynamical Properties of the Fractional Nonlinear Schrödinger Equation under Harmonic Confinement
• Blowup dynamics for mass critical Half-wave equation in 2D