Transient Dynamics of Pulled Fronts

Substitution of this expansion into (4.16) yields the hierarchy of ordinary differential equations  

Each t)i is determined by its differential equation, the requirement (0) = 0, and the appropriate boundary conditions. The equations for rj /Ci and V312/C212 are precisely the differential equations for j(t) in (4.16). 

In the far edge, where 0(VF) 1, a different expansion is needed, as the transient profile falls off faster than p, so that t] -p. Linearizing about p = 0 one finds 

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