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Traveling waves in coupled reaction-diffusion models with degenerate sources
Jonathan J Wylie1, Robert M Miura
1Department of Mathematics, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong. wylie@math.cityu.edu.hk
Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|October 10, 2006
Summary
Degenerate source terms in nonlinear diffusion equations can trigger unusual traveling waves. These waves may exhibit surprising behaviors, including outward propagation, slowing, and eventual self-annihilation.
Area of Science:
- Mathematical Physics
- Nonlinear Dynamics
- Partial Differential Equations
Background:
- Coupled nonlinear diffusion equations are fundamental in modeling various physical phenomena.
- Systems with degenerate source terms lack isolated rest states, leading to complex dynamics.
- Understanding the behavior of traveling waves is crucial for predicting system evolution.
Purpose of the Study:
- To derive conditions for triggering traveling waves in systems with degenerate source terms.
- To investigate the unique properties of traveling waves arising from source term degeneracy.
- To analyze the emergence and interaction of traveling waves in such systems.
Main Methods:
- Analysis of coupled nonlinear diffusion equations with general, physically relevant source terms.
- Derivation of conditions for wave initiation from localized perturbations of stable steady states.
- Mathematical investigation of the properties and behavior of degenerate traveling waves.
Main Results:
- Established conditions for triggering traveling waves in degenerate systems.
- Identified novel properties of traveling waves not observed in nondegenerate systems.
- Demonstrated the occurrence of outward propagating waves that slow, reverse direction, and annihilate.
Conclusions:
- Degenerate source terms introduce unique and counterintuitive behaviors in traveling waves.
- The derived conditions provide a framework for understanding wave dynamics in these systems.
- The phenomenon of wave reversal and annihilation highlights the complexity of nonlinear diffusion.
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