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Two-dimensional dissipative solitons supported by localized gain.

Yaroslav V Kartashov¹, Vladimir V Konotop, Victor A Vysloukh

¹ICFO-Institut de Ciencies Fotoniques, and Universitat Politecnica de Catalunya, Mediterranean Technology Park, 08860 Castelldefels (Barcelona), Spain. yaroslav.kartashov@icfo.es

|January 7, 2011

Summary

Stable solitons emerge from a balance of localized gain and nonlinear dissipation in the 2D nonlinear Schrödinger equation. These localized modes require sufficient gain and energy flow, leading to asymmetric forms above a critical gain threshold.

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Area of Science:

Nonlinear physics
Optical solitons
Mathematical physics

Background:

The nonlinear Schrödinger equation (NLSE) is a fundamental model in various fields, including optics and Bose-Einstein condensates.
Understanding the behavior of localized modes (solitons) in dissipative systems is crucial for applications.
The interplay between gain and dissipation can lead to complex phenomena in nonlinear systems.

Purpose of the Study:

To investigate the existence and stability of localized modes in the two-dimensional NLSE with localized gain and nonlinear cubic dissipation.
To identify the conditions under which stable solitons can form in this dissipative system.
To explore the emergence of asymmetric solitons due to symmetry breaking.

Main Methods:

Analytical and numerical methods were employed to study the two-dimensional nonlinear Schrödinger equation.
The balance between localized gain and nonlinear cubic dissipation was analyzed.
The energy flow and gain threshold for soliton existence were investigated.

Main Results:

Stable localized modes, identified as solitons, exist when there is a balance between localized gain and nonlinear cubic dissipation.
Soliton formation requires the gain to be sufficiently strong and the energy flow to exceed a critical threshold.
Above a critical gain value, symmetry breaking leads to the emergence of asymmetric dissipative solitons.

Conclusions:

The study demonstrates the possibility of stable soliton formation in a dissipative 2D NLSE model.
The findings highlight the critical role of gain and energy flow in stabilizing these localized modes.
The emergence of asymmetric solitons signifies a transition to more complex nonlinear dynamics.