Stability of one-dimensional array solitons.

Area of Science:

• Nonlinear Dynamics
• Mathematical Physics
• Condensed Matter Theory

Background:

• The discrete nonlinear Schrödinger equation (DNLS) models various physical phenomena, including light propagation in optical arrays.
• Understanding the stability of array solitons is crucial for controlling wave propagation in such systems.
• Dispersion and periodic boundary conditions significantly influence soliton behavior.

Purpose of the Study:

• To investigate the stability of array solitons in the DNLS equation with dispersion.
• To determine the dependence of the linear growth rate on discrete wave number and soliton amplitude.
• To compare instability thresholds in circular arrays with quasicollapse thresholds in open arrays.

Main Methods:

• Linearized eigenvalue problem formulation.
• Variational method for analytical calculation of growth rates.
• Numerical solution of the eigenvalue problem using the shooting method.
• Numerical verification of instability and quasicollapse thresholds.

Main Results:

• The linear growth rate of array solitons was analytically derived.
• Numerical solutions showed excellent agreement with analytical predictions.
• Instability thresholds for circular arrays were found to coincide with quasicollapse thresholds for open arrays under specific conditions.

Conclusions:

• The study provides a comprehensive analysis of array soliton stability in the DNLS equation.
• Both analytical and numerical methods confirm the calculated stability properties.
• The findings offer insights into the transition from stable soliton propagation to collapse in discrete nonlinear systems.