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Unformatted text preview: 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 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 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 − V t ) , (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 dξ 2 + V du dξ + R ( u ) = 0 , (8.9) where ξ = x − V t . 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...
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This note was uploaded on 09/30/2011 for the course PHYS 221a taught by Professor Staff during the Fall '11 term at UCSD.
- Fall '11