Instabilities and bifurcations of nonlinear impurity modes

Area of Science:

• Nonlinear dynamics
• Condensed matter physics
• Mathematical physics

Background:

• The discrete nonlinear Schrödinger equation (DNLS) models various physical systems, including optical lattices and Bose-Einstein condensates.
• Understanding localized modes (discrete solitons and breathers) and their interaction with impurities is crucial for controlling wave phenomena.

Purpose of the Study:

• To investigate the structure and stability of nonlinear impurity modes in the DNLS equation with a single on-site nonlinear impurity.
• To analyze the interplay between discreteness, nonlinearity, and disorder in shaping these modes.
• To explore the generation of stationary states and transitions between localized states near impurity sites.

Main Methods:

• Theoretical analysis of nonlinear localized modes.
• Numerical simulations of the discrete nonlinear Schrödinger equation.
• Bifurcation analysis to identify transitions between states.

Main Results:

• A repulsive impurity interacting with a discrete soliton or breather generates a family of stationary states near the impurity.
• The interplay of discreteness, nonlinearity, and disorder significantly influences the behavior of impurity modes.
• Theoretical and numerical criteria were established for transitions between different localized states via bifurcations.

Conclusions:

• Nonlinear impurities can induce complex localized structures in discrete systems.
• Bifurcation cascades provide a mechanism for the emergence of diverse stationary states.
• The findings offer insights into controlling and predicting wave localization in disordered nonlinear systems.