Amplitude death criteria for coupled complex Ginzburg-Landau systems

Abstract: Amplitude death, which occurs in a system when one or more macroscopic wavefunctions collapse to zero, has been observed in mutually coupled solid-state lasers, analog circuits, and thermoacoustic oscillators, to name a few applications. While studies have considered amplitude death on oscillator systems and in externally forced complex Ginzburg-Landau systems, a route to amplitude death has not been studied in autonomous continuum systems. We derive simple analytic conditions for the onset of amplitude death of one macroscopic wavefunction in a system of two coupled complex Ginzburg-Landau equations with general nonlinear self- and cross-interaction terms. Our results give a more general theoretical underpinning for recent amplitude death results reported in the literature, and suggest an approach for tuning parameters in such systems so that they either permit or prohibit amplitude death of a wavefunction (depending on the application). Numerical simulation of the coupled complex Ginzburg-Landau equations, for examples including cubic, cubic-quintic, and saturable nonlinearities, is used to illustrate the analytical results.

• The paper derives simple, analytic conditions for the onset of amplitude death in systems of two coupled complex Ginzburg-Landau equations with general nonlinear interaction terms.
• The study shows that nonlinear self- and cross-interaction terms and cross-phase modulation parameters play a crucial role, which can be tuned to induce or avert amplitude death.
• Supported by numerical simulations, the findings detail specific conditions for amplitude death in systems with various nonlinearities, highlighting the necessity of dissipative dynamics.
• The analysis underscores the absence of amplitude death in simpler related systems like coupled nonlinear Schrödinger equations, where only conservative dynamics prevail. This highlights the necessity of dissipative dynamics inherent to the GL framework for amplitude death to occur.

Analysis of Amplitude Death in Coupled Complex Ginzburg-Landau Systems

The study of amplitude death in coupled complex Ginzburg-Landau (GL) systems, as undertaken by Van Gorder, Krause, and Kwiecinski, provides essential insights into the mechanisms and conditions under which amplitude death can occur in autonomous continuum systems. Amplitude death, defined as the scenario when one or more macroscopic wavefunctions in a system collapse to zero, has significant implications across various domains, including solid-state lasers, analog circuits, and thermoacoustic oscillators. The primary contribution of this paper is the derivation of simple, analytic conditions for the onset of amplitude death in systems governed by two coupled GL equations with general nonlinear self- and cross-interaction terms.

Key Contributions and Methodology

1. Analytical and Numerical Approach: The authors derive conditions that specify when amplitude death of one wavefunction occurs in a system of two coupled complex GL equations. This derivation involves examining a general system form, taking into account nonlinear functions $f$ and $g$, which are central to the system's kinetics. These functions are assumed to be monotone increasing—aligning with many physical applications.
2. Role of Nonlinearity and Cross-Phase Modulation (XPM): Through this analytical framework, the authors extend the theoretical understanding of amplitude death, emphasizing the role of nonlinear terms and cross-phase modulation parameters. By exploring examples with cubic, cubic-quintic, and saturable nonlinear interaction terms, the paper expounds upon how these terms can be tuned to induce or avert amplitude death.
3. Numerical Simulations: The study employs numerical simulations to substantiate the theoretical findings, examining cases with specific forms of $f$ and $g$. The simulations, performed using discretization techniques and adaptive Runge-Kutta schemes, illustrate the theoretical predictions and offer practical verification.
4. Extensions beyond Simple Systems: The comprehensive analysis in this paper underscores the absence of amplitude death in simpler related systems like coupled nonlinear Schrödinger equations, where only conservative dynamics prevail. This highlights the necessity of dissipative dynamics inherent to the GL framework for amplitude death to occur.

Numerical and Analytical Findings

• Cubic Kinetics: The authors demonstrate the conditions under which amplitude death occurs in a cubic GL system, specifically when asymmetry in XPM parameters exceeds certain thresholds. The study reveals that amplitude death is facilitated by these asymmetries, indicating potential control mechanisms.
• Cubic-Quintic and Saturable Nonlinearities: Through further exploration, it is shown that additional complexity in nonlinearity, such as cubic-quintic terms, can induce differential dynamics, leading one solution to exhibit amplitude death while the other persists. Saturable nonlinearities adjust these dynamics, influencing the timescales of amplitude death.

Implications for Future Research and Applications

The findings have significant ramifications for theoretical and applied research. In nonlinear optics, for instance, the results suggest methodologies for controlling wavefunction dynamics via parameter tuning, potentially leading to experimental verification. Furthermore, the study proposes extending the analysis to systems with more than two wavefunctions, raising the possibility of new dynamic behaviors and classification of amplitude death regimes.

Conclusion

This paper effectively advances the understanding of amplitude death in complex dissipative systems, providing robust theoretical conditions supported by numerical evidence. The research sets the stage for broader explorations into high-dimensional systems and invites cross-disciplinary applications across physics, engineering, and potentially other areas where GL systems are relevant. The clear delineation of parameter spaces where amplitude death is viable could inform both future experimental setups and novel computational methods in fields engaging with nonlinear wave phenomena.