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CH08_FRONTS - Chapter 8 Front Propagation 8.1...

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Chapter 8 Front Propagation 8.1 Reaction-Diffusion Systems We’ve studied simple N = 1 dynamical systems of the form du dt = R ( u ) . (8.1) Recall that the dynamics evolves u ( t ) monotonically toward the first stable fixed point encountered. Now let’s extend the function u ( t ) to the spatial domain as well, i.e. u ( x ,t ), and add a diffusion term: ∂u ∂t = D 2 u + R ( u ) , (8.2) where D is the diffusion constant. This is an example of a reaction-diffusion system . If we extend u ( x ,t ) to a multicomponent field u ( x ,t ), we obtain the general reaction-diffusion equation (RDE) ∂u i ∂t = D ij 2 u i + R i ( u 1 ,...,u N ) . (8.3) Here, u is interpreted as a vector of reactants , R ( u ) describes the nonlinear local reaction kinetics , and D ij is the diffusivity matrix . If diffusion is negligible, this PDE reduces to decoupled local ODEs of the form ˙ u = R ( u ), which is to say a dynamical system at each point in space. Thus, any fixed point u of the local reaction dynamics also describes a spatially homogeneous, time-independent solution to the RDE. These solutions may be characterized as dynamically stable or unstable, depending on the eigenspectrum of the Jacobian matrix M ij = i R j ( u ). At a stable fixed point, Re ( λ i ) < 0 for all eigenvalues. 8.1.1 Single component systems We first consider the single component system, ∂u ∂t = D 2 u + R ( u ) . (8.4) 1
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2 CHAPTER 8. FRONT PROPAGATION Note that the right hand side can be expressed as the functional derivative of a Lyapunov functional , L [ u ] = integraldisplay d d x bracketleftbigg 1 2 D ( u ) 2 U ( u ) bracketrightbigg , (8.5) where U ( u ) = u integraldisplay 0 du R ( u ) . (8.6) (The lower limit in the above equation is arbitrary.) Thus, eqn. 8.4 is equivalent to ∂u ∂t = δL δu ( x ,t ) . (8.7) Thus, the Lyapunov functional runs strictly downhill, i.e. ˙ L< 0, except where u ( x ,t ) solves the RDE, at which point ˙ L = 0. 8.1.2 Propagating front solutions Suppose the dynamical system ˙ u = R ( u ) has two or more fixed points. Each such fixed point represents a static, homogeneous solution to the RDE. We now seek a dynamical, inhomogeneous solution to the RDE in the form of a propagating front , described by u ( x,t ) = u ( x Vt ) , (8.8) where V is the (as yet unknown) front propagation speed. With this Ansatz , the PDE of eqn. 8.4 is converted to an ODE, D d 2 u 2 + V du + R ( u ) = 0 , (8.9) where ξ = x Vt . With R ( u ) U ( u ) as in eqn. 8.6, we have the following convenient interpretation. If we substitute u q , ξ t , D m , and v γ , this equation describes the damped motion of a massive particle under friction: m ¨ q + γ ˙ q = U ( q ). The fixed points q satisfy U ( q ) = 0 and are hence local extrema of U ( q ). The propagating front solution we seek therefore resembles the motion of a massive particle rolling between extrema of the potential U ( q ). Note that the stable fixed points of the local reaction kinetics have R ( q ) = U ′′ ( q ) < 0, corresponding to unstable mechanical equilibria. Conversely, unstable fixed points of the local reaction kinetics have R ( q ) = U ′′ ( q ) > 0, corresponding to stable mechanical equilibria.
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