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An amplitude equation for the conserved-Hopf bifurcation-Derivation, analysis, and assessment
Daniel Greve1, Uwe Thiele1,2,3
1Institut für Theoretische Physik, Universität Münster, Wilhelm-Klemm-Str. 9, 48149 Münster, Germany.
Chaos (Woodbury, N.Y.)
|December 5, 2024
Summary
We derived a new amplitude equation for conserved-Hopf instability, crucial for understanding oscillatory systems with two conservation laws. This equation reveals universal suppression of coarsening in oscillatory phase separation.
Area of Science:
- Physics
- Applied Mathematics
- Materials Science
Background:
- Hopf bifurcation describes oscillatory instabilities in dynamical systems.
- Conserved systems with multiple conservation laws exhibit complex behaviors.
- Amplitude equations simplify the analysis of instabilities near bifurcation points.
Purpose of the Study:
- Derive a generic amplitude equation for conserved-Hopf instability.
- Analyze oscillatory phase separation in a Cahn-Hilliard model.
- Investigate the suppression of coarsening in such systems.
Main Methods:
- Weakly nonlinear theory to derive amplitude equations.
- Analysis of a symmetric two-component Cahn-Hilliard model.
- Analytical solutions for amplitude equation stability and dynamics.
Main Results:
- A nonlinear nonlocal amplitude equation with real coefficients was derived for a specific model.
- The derived equation accurately predicts the bifurcation diagram and time evolution.
- Universal suppression of coarsening was demonstrated in oscillatory phase separation.
- A generic amplitude equation with complex coefficients was obtained for unrestricted cases.
Conclusions:
- The derived amplitude equations are effective tools for studying conserved-Hopf instabilities.
- Oscillatory phase separation universally suppresses coarsening.
- The generic amplitude equation accurately models transient dynamics and traveling wave states.
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