Solitons in two-dimensional lattices possessing defects, dislocations, and quasicrystal structures

Area of Science:

• Nonlinear physics
• Condensed matter theory
• Computational physics

Background:

• The two-dimensional nonlinear Schrödinger equation (2D NLSE) is a fundamental model for describing wave phenomena in various physical systems.
• Understanding the behavior of localized nonlinear modes (solitons) in the presence of complex potentials is crucial for predicting system dynamics.
• External potentials with large variations from periodicity, such as vacancy defects, edge dislocations, and quasicrystal structures, present significant challenges to theoretical analysis.

Purpose of the Study:

• To obtain localized nonlinear modes (solitons) for the 2D NLSE with highly irregular external potentials.
• To investigate the stability and evolution of these solitons in non-periodic lattice structures.
• To characterize the behavior of solitons in the presence of defects like vacancies and dislocations, and in quasicrystalline potentials.

Main Methods:

• A spectral fixed-point computational scheme was employed to derive the solitons.
• Direct numerical simulations were used to track the evolution of the obtained solitons.
• Analysis focused on systems with vacancy defects, edge dislocations, and quasicrystal structures.

Main Results:

• Localized nonlinear modes (solitons) were successfully obtained for the 2D NLSE in the presence of irregular potentials.
• Numerical simulations revealed that these irregular-lattice solitons can exhibit diverse behaviors: stability, instability, or collapse.
• The study demonstrates the existence and dynamics of solitons in complex, non-periodic potential landscapes.

Conclusions:

• Solitons in the 2D NLSE are not limited to periodic potentials and can exist in highly irregular environments.
• The stability of these solitons is strongly dependent on the specific nature of the potential's irregularity.
• This work provides insights into the robustness and potential fragility of nonlinear localized modes in disordered systems.