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The Diffusion of Passive Tracers in Laminar Shear Flow

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Breathing dissipative solitons in three-component reaction-diffusion system.

S V Gurevich¹, Sh Amiranashvili, H-G Purwins

¹Institut für Angewandte Physik, Corrensstr. 2/4, D-48149 Münster, Germany.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics

|February 7, 2007

Summary

Changes in inhibitor time constants can destabilize stationary solutions in reaction-diffusion systems. This study analyzes the bifurcation to a breathing state using analytical and numerical methods for reaction-diffusion systems.

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Area of Science:

Chemical kinetics
Nonlinear dynamics
Mathematical modeling

Background:

Reaction-diffusion systems are fundamental to understanding pattern formation.
Localized stationary solutions, like dissipative solitons, are crucial in various physical and chemical processes.
The stability of these solutions dictates their persistence and behavior.

Purpose of the Study:

To investigate the stability of localized stationary solutions in a three-component reaction-diffusion system.
To analyze the destabilization mechanism caused by changes in inhibitor time constants.
To characterize the bifurcation from a stationary to a breathing state.

Main Methods:

Employing a two-time-scale expansion near the bifurcation point.
Deriving an amplitude equation to describe the system's dynamics.
Conducting numerical simulations to validate analytical predictions.

Main Results:

Demonstrated that altering inhibitor time constants can destabilize stationary solutions.
Identified the breathing mode instability as the primary destabilization pathway.
Observed a bifurcation from a stationary to a breathing state for the dissipative soliton.
Analytical predictions showed good agreement with numerical simulation results.

Conclusions:

The stability of localized stationary solutions in reaction-diffusion systems is sensitive to parameter variations.
The transition to oscillatory behavior (breathing state) is a key phenomenon driven by specific parameter changes.
The developed amplitude equation provides a valuable tool for understanding these bifurcations in reaction-diffusion systems.