Dynamics of wave packets for the nonlinear Schrödinger equation with a random potential

Area of Science:

• Condensed matter physics
• Quantum mechanics
• Nonlinear dynamics

Background:

• Anderson localization describes the suppression of wave function motion in disordered systems.
• Nonlinear Schrödinger equation models wave propagation in various physical systems.
• Disorder in quantum systems can lead to unique transport phenomena.

Purpose of the Study:

• To investigate the dynamics of an initially localized Anderson mode.
• To analyze the behavior of wave functions in a disordered nonlinear system.
• To determine the long-time dynamics and transport properties.

Main Methods:

• Utilizing the nonlinear Schrödinger equation framework.
• Employing the Liouville operator for dynamic description.
• Applying a perturbation approach to find analytical solutions for initial dynamics.
• Characterizing long-time behavior using a phenomenological probabilistic approach and probability distribution functions.

Main Results:

• The initially localized Anderson mode wave function remains localized over time.
• An analytical expression for the initial wave function dynamics was derived.
• The long-time dynamics are described by a probability distribution function.
• This probability distribution function follows a fractional Fokker-Planck equation, indicating subdiffusion.

Conclusions:

• The study confirms the persistence of localization for Anderson modes in this nonlinear disordered system.
• The derived analytical solutions provide insights into the initial transient dynamics.
• The long-time subdiffusion behavior is quantitatively described by the fractional Fokker-Planck equation, highlighting anomalous transport properties.