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Publications [#318326] of James H. Nolen

Papers Published

  1. Nolen, J; Roquejoffre, J-M; Ryzhik, L, Refined long-time asymptotics for Fisher–KPP fronts, Communications in Contemporary Mathematics (January, 2018), pp. 1850072-1850072, World Scientific Pub Co Pte Lt [doi]
    (last updated on 2019/04/24)

    Abstract:
    © 2018 World Scientific Publishing Company. We study the one-dimensional Fisher-KPP equation, with an initial condition u0(x) that coincides with the step function except on a compact set. A well-known result of Bramson in [Maximal displacement of branching Brownian motion, Comm. Pure Appl. Math. 31 (1978) 531-581; Convergence of Solutions of the Kolmogorov Equation to Travelling Waves (American Mathematical Society, Providence, RI, 1983)] states that, as t → +∞, the solution converges to a traveling wave located at the position X(t) = 2t - (3/2)logt + x0 + o(1), with the shift x0 that depends on u0. Ebert and Van Saarloos have formally derived in [Front propagation into unstable states: Universal algebraic convergence towards uniformly translating pulled fronts, Phys. D 146 (2000) 1-99; Front propagation into unstable states, Phys. Rep. 386 (2003) 29-222] a correction to the Bramson shift, arguing that X(t) = 2t - (3/2)logt + x0 - 3π/t + O(1/t). Here, we prove that this result does hold, with an error term of the size O(1/t1-γ), for any γ > 0. The interesting aspect of this asymptotics is that the coefficient in front of the 1/t-term does not depend on u0.

 

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