Numerical study of fractional nonlinear Schrödinger equations

Area of Science:

• Mathematical Physics
• Nonlinear Dynamics
• Quantum Mechanics

Background:

• Fractional Schrödinger equations model various physical phenomena, including wave propagation and quantum systems.
• Understanding the behavior of solutions, such as blow-up and stability, is crucial for theoretical and applied sciences.
• Numerical methods are essential for analyzing complex nonlinear partial differential equations that lack analytical solutions.

Purpose of the Study:

• To numerically investigate one-dimensional dispersive Schrödinger-type equations with a fractional Laplacian.
• To analyze the regimes of mass and energy (sub- and supercritical) based on the dispersive exponent.
• To study finite-time blow-up, global existence, blow-up nature, stability of nonlinear ground states, and long-time dynamics, including a semiclassical setting.

Main Methods:

• Fourier spectral method for detailed numerical investigation.
• Analysis of dispersive and fractional Laplacian terms.
• Identification and study of sub- and supercritical regimes.

Main Results:

• Numerical evidence for finite-time blow-up versus global existence depending on parameters.
• Characterization of the nature of blow-up phenomena.
• Investigation of stability and instability of nonlinear ground states.
• Analysis of long-time solution dynamics, including semiclassical behavior.
• Numerical construction of ground state solutions for the fractional nonlinear Schrödinger equation.

Conclusions:

• The study provides a comprehensive numerical understanding of fractional Schrödinger equations.
• Parameter-dependent regimes dictate the existence and behavior of solutions.
• Numerical findings offer insights into the stability and dynamics of nonlinear states, crucial for understanding complex physical systems.