Stability of multiple pulses in discrete systems

Area of Science:

• Nonlinear dynamics
• Quantum physics
• Mathematical modeling

Background:

• Discrete nonlinear Schrödinger equation (DNLS) models various physical phenomena.
• Understanding the stability of multi-pulse solutions is crucial for predicting system behavior.
• Previous studies have explored single-pulse dynamics, but multi-pulse stability remains an active research area.

Purpose of the Study:

• To rigorously analyze the linear stability of multiple-pulse solutions in the DNLS.
• To identify conditions under which widely separated pulses remain stable.
• To investigate the role of phase shifts between pulses on overall stability.

Main Methods:

• Rigorous mathematical analysis of the linearized system.
• Identification of instability through positive real eigenvalues.
• Variational approach for two-pulse and N-pulse systems.
• Supplementation with detailed numerical stability analysis.

Main Results:

• Widely separated single pulses in a bound state are unstable.
• Instability is linked to positive real eigenvalues in the linearized DNLS.
• A phase shift of pi between adjacent pulses is a critical condition for potential stability.
• Variational analysis predicts linear stability for two pulses with a pi phase shift and specific separation.
• The variational method is generalizable to N-pulse systems.

Conclusions:

• The stability of multiple-pulse solutions in the DNLS is highly sensitive to inter-pulse phase shifts.
• A phase shift of pi is necessary but not sufficient for stability, requiring specific pulse separations.
• Numerical and analytical methods confirm the instability of certain multi-pulse configurations and predict stability under specific conditions.