Variational Principles and the Cutoff Problem

Recent studies have applied improved variational principles to deal with the velocity shift due to a cutoff in the reaction term, both for fronts propagating into unstable and metastable states [284, 36]. They confirm the results by Brunet and Derrida and improve the results by Kessler et al. The variational principle given in (4.68) implies that for any admissible trial function a lower bound for the velocity can be found by (4.64). The trial function for which equality in (4.64) holds diverges at p = 0, and it is convenient to consider trial functions that in addition to the requirements g 0 and 0 also satisfy g(0) oo. Such trial functions allow us to obtain accurate lower bounds for the front velocity. We perform a change of variables p = p(s), where i — 1/g, and consider s as the independent variable in (4.68). With this change of variables, the variational principle reads 

For small , the leading order of the trial function g(p, e) is g(p, 0) = goip), which is exactly the optintizing function for the case without a cutoff. On the other hand, for small e we can write v(e) = Vq + (dv/ ) o 4-------- so that 

Vofo (-go/8o)dp where we have made use of the variational result for the case without cutoff  

These results are in agreement with the previous results obtained from a different variational principle and also with the results obtained by Kessler et al. The dependence of the second result is exactly that given in (4.86). 

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