The critical wave speed for the Fisher–Kolmogorov–Petrowskii–Piscounov equation with cut-off

Abstract

The Fisher-Kolmogorov-Petrowskii-Piscounov (FKPP) equation with cut-off was introduced in (Brunet and Derrida 1997 Shift in the velocity of a front due to a cut-off Phys. Rev. E 56 2597-604) to model N-particle systems in which concentrations less than ε = 1/N are not attainable. It was conjectured that the cut-off function, which sets the reaction terms to zero if the concentration is below the small threshold ε, introduces a substantial shift in the propagation speed of the corresponding travelling waves. In this paper, we prove the conjecture of Brunet and Derrida, showing that the speed of propagation is given by c crit (ε) = 2 − π 2 /(ln ε) 2 + O((ln ε) −3 ), as ε → 0, for a large class of cut-off functions. Moreover, we extend this result to a more general family of scalar reaction-diffusion equations with cut-off. The main mathematical techniques used in our proof are the geometric singular perturbation theory and the blow-up method, which lead naturally to the identification of the reasons for the logarithmic dependence of c crit on ε as well as for the universality of the corresponding leading-order coefficient (π 2 ).

A geometric approach to bistable front propagation in scalar reaction–diffusion equations with cut-off

“…The present article builds on the results obtained in [3], in that we show how the geometric approach developed there, in the context of the FKPP equation with cut-off in (1.1), can be generalized to study front propagation in the broad class of cut-off reaction-diffusion equations that is given by φ t = φ xx + f (φ)Θ φ, ε, φ ε .…”

“…In [3], we proved the existence of traveling front solutions that propagate between the rest states at 1 and 0 in Eq. (1.1), and we established the validity of the conjecture of Brunet and Derrida, including a generalization of it.…”

“…(1.1), and we established the validity of the conjecture of Brunet and Derrida, including a generalization of it. In particular, we considered the general class of cut-off functions Θ that satisfy Θ φ, ε, φ ε < 1 if φ < ε and Θ φ, ε, φ ε ≡ 1 if φ > ε, (1.4) where, moreover, Θ is bounded at φ = ε and 0 < ε for φ < ε; see [3] for details.) We gave a rigorous derivation of the leading-order ε-asymptotics of c FKPP in (1.3), and we showed that the coefficient π 2 in that expansion is universal within the class of cut-off functions that satisfy (1.4).…”

“…in [5,6], the front speed c 0 is unique in the bistable case, whereas the FKPP equation supports traveling front solutions for a continuum of speeds c in the absence of a cut-off, with c ≥ c FKPP (0) and c FKPP (0) the 'critical' front speed; cf. [3] for a detailed discussion.) Finally, the cut-off Θ is as defined in (1.4); for clarity of exposition, we will only discuss the case where Θ = H (the Heaviside cutoff) in detail here.…”

Upper and lower bounds for the speed of pulled fronts with a cut-off

“…In recent work it has been show that the Brunet-Derrida formula for the speed is correct to O((ln ǫ) −3 ) for a wider class of reaction terms [10].…”