Linear spreading of perturbations

Let s consider first the one-dimensional case (Ebert and van Saarloos, 2000 van Saarloos, 2003)  

If the initial condition is sufficiently localized around the origin then, after a transient (the time needed for (4.3) to become substantially broader than the extent of the initial condition), the full solution (4.2) is well approximated by (4.3). Thus, in this onedimensional case with localized initial condition we have 

The dominant terms do not depend on the level g used to define the front. Thus, associated to a uniform state which is unstable with growth rate g, there is a front controlling the spreading of the concentration that approaches at long times the asymptotic velocity 

This asymptotic speed is reached from below and after a rather slow algebraic transient. To focus on the front region Eq. (4.5) can be written in terms of = x — 2ct  

This function has different parts. In the region close to the moving leading edge of the front (defined as 4Dt) the solution approaches an exponential of the form e a°with ac = Jg/D. Careful 

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