Spreading for the generalized nonlinear Schrödinger equation with disorder.

Area of Science:

• Quantum Mechanics
• Nonlinear Dynamics
• Statistical Physics

Background:

• The behavior of wave packets in complex potentials is crucial for understanding quantum systems.
• Nonlinear Schrödinger equations describe phenomena across various fields, including optics and quantum mechanics.
• Random potentials introduce disorder, significantly altering system dynamics.

Purpose of the Study:

• To analyze the dynamics of localized wave packets governed by the generalized nonlinear Schrödinger equation.
• To investigate the influence of a random potential and varying nonlinearity parameter (p) on wave packet spread.
• To explore subdiffusive behavior and identify potential critical phenomena.

Main Methods:

• Numerical simulations were employed to accurately track wave packet evolution over short and long timescales.
• The average second moment (m2) of the wave packet was computed to quantify its spread.
• The exponent (alpha) in the subdiffusive relation m2 ~ t^alpha was determined as a function of p.

Main Results:

• Subdiffusive behavior (m2 ~ t^alpha) was observed and quantified.
• Contrary to prior hypotheses, no critical behavior was found as a function of the nonlinearity parameter p.
• A scaling property for alpha(p) was identified, along with a maximal value around p = 1/2.

Conclusions:

• The study provides numerical evidence for subdiffusive wave packet spreading in disordered nonlinear systems.
• The absence of critical behavior challenges existing theoretical predictions.
• The discovered scaling property and maximal value for alpha(p) offer new insights into nonlinear wave dynamics.