Mean field effects for counterpropagating traveling wave solutions of reaction-diffusion systems
- Northwestern Univ., Evanston, IL (United States). Dept. of Engineering Sciences and Applied Mathematics
In many problems, one observes traveling waves that propagate with constant velocity and shape in the {chi} direction, say, are independent of y, and z and describe transitions between two equilibrium states. As parameters of the system are varied, these traveling waves can become unstable and give rise to waves having additional structure, such as traveling waves in the y and z directions, which can themselves be subject to instabilities as parameters are further varied. To investigate this scenario the authors consider a system of reaction-diffusion equations with a traveling wave solution as a basic state. They determine solutions bifurcating from the basic state that describe counterpropagating traveling wave in directions orthogonal to the direction of propagation of the basic state and determine their stability. Specifically, they derive long wave modulation equations for the amplitudes of the counterpropagating traveling waves that are coupled to an equation for a mean field, generated by the translation of the basic state in the direction of its propagation. The modulation equations are then employed to determine stability boundaries to long wave perturbations for both unidirectional and counterpropagating traveling waves. The stability analysis is delicate because the results depend on the order in which transverse and longitudinal perturbation wavenumbers are taken to zero. For the unidirectional wave they demonstrate that it is sufficient to consider the cases of (1) purely transverse perturbations, (2) purely longitudinal perturbations, and (3) longitudinal perturbations with a small transverse component. These yield Eckhaus type, zigzag type, and skew type instabilities, respectively.
- Sponsoring Organization:
- USDOE
- DOE Contract Number:
- FG02-87ER25027
- OSTI ID:
- 46114
- Journal Information:
- SIAM Journal of Applied Mathematics, Vol. 55, Issue 2; Other Information: PBD: Apr 1995
- Country of Publication:
- United States
- Language:
- English
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