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Published on: September 5, 2019
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Fundamental- and first-order localized states in a cubic-quintic reaction-diffusion system
1TU Berlin, Institute for Theoretical Physics, Hardenbergstrasse 36, Sekr EW 7-1, 10623 Berlin, Germany and Carl-von-Ossietzky University of Oldenburg, Institute for Chemistry and Biology of the Marine Environment (ICBM), Carl-von-Ossietzky-Strasse 9-11, 26111 Oldenburg, Germany.
Summary
This study explores self-localized solutions in reaction-diffusion systems. The cubic-quintic model allows for stable states, including complex structures like clusters, unlike simpler models.
Area of Science:
- Reaction-diffusion systems
- Nonlinear dynamics
- Mathematical modeling
Background:
- Reaction-diffusion systems are fundamental in modeling spatially extended phenomena.
- Self-localized solutions, or "patterns," are crucial for understanding complex behaviors.
- Investigating solutions with varying radial quantum numbers is key to classifying system behaviors.
Purpose of the Study:
- To analyze rotationally symmetric self-localized solutions in one- and two-component reaction-diffusion systems.
- To compare the existence of solutions with zero and one radial profile intersections across different nonlinearities.
- To explore the stability and variety of solutions in cubic-quintic reaction-diffusion systems.
Main Methods:
- Analysis of rotationally symmetric solutions in one and two dimensions.
- Focus on fundamental and first higher-order solutions.
- Investigation of quadratic-cubic and cubic-quintic nonlinearities.
Main Results:
- Solutions with one radial profile intersection do not exist for quadratic-cubic nonlinearity.
- Cubic-quintic nonlinearity supports the existence of solutions with one radial profile intersection.
- The cubic-quintic system exhibits stable states, antistates, pairs, and clusters, interpretable as states with nonzero azimuthal quantum numbers.
Conclusions:
- The choice of nonlinearity significantly impacts the existence of self-localized solutions.
- Cubic-quintic reaction-diffusion systems offer a richer variety of stable localized states.
- These findings advance the understanding of pattern formation in nonlinear systems.
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