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 (13)

• Fractional Quantum Mechanics and Levy Path Integrals
• Boson Stars as Solitary Waves
• On the continuum limit for discrete NLS with long-range lattice interactions
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• A numerical approach to Blow-up issues for dispersive perturbations of Burgers' equation
• Numerical study of shock formation in the dispersionless Kadomtsev-Petviashvili equation and dispersive regularizations
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• Nonlinear fractional Schrödinger equations in one dimension
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Cited by in corpus (25)

• Fractional quantum mechanics in polariton condensates with velocity dependent mass
• Symmetry breaking of spatial Kerr solitons in fractional dimension
• Unveiling the link between fractional Schrödinger equation and light propagation in honeycomb lattice
• On the ground states and dynamics of space fractional nonlinear Schrödinger/Gross-Pitaevskii equations with rotation term and nonlocal nonlinear interactions
• Metastable soliton necklaces supported by fractional diffraction and competing nonlinearities
• Normalized solutions for fractional nonlinear scalar field equations via Lagrangian formulation
• Global dynamics of small solutions to the modified fractional Korteweg-de Vries and nonlinear Schrödinger equations
• On fractional Schrödinger equations with Hartree type nonlinearities
• Blowup dynamics for Mass Critical Half-wave equation in 3D
• On soliton (in-)stability in multi-dimensional cubic-quintic nonlinear Schrödinger equations
• Fast Fourier-like Mapped Chebyshev Spectral-Galerkin Methods for PDEs with Integral Fractional Laplacian in Unbounded Domains
• Two-dimensional fractional discrete NLS equations: dispersion relations, rogue waves, fundamental and vortex solitons
• A numerical approach to Blow-up issues for Davey-Stewartson II type systems
• Stationary solutions of a fractional Laplacian with singular perturbation
• Dynamics of solutions in the 1d bi-harmonic nonlinear Schrödinger equation
• On a Fractional Schrödinger equation in the presence of Harmonic potential
• On a collocation method for the time-fractional convection-diffusion equation with variable coefficients
• On the blow-up solutions for the nonlinear Schrödinger equation with combined power-type nonlinearities
• Spectral and Dynamical Properties of the Fractional Nonlinear Schrödinger Equation under Harmonic Confinement
• Multi-domain spectral approach for the Hilbert transform on the real line
• Nondispersive solutions to the mass critical half-wave equation in two dimensions
• Blowup dynamics for mass critical Half-wave equation in 2D
• Vortex solitons in fractional nonlinear Schrödinger equation with the cubic-quintic nonlinearity
• Blow-up criteria for fractional nonlinear Schrödinger equation
• Blow-up dynamics for $L^2$-critical fractional Schrödinger equations