Math @ Duke

Publications [#318326] of James H. Nolen
Papers Published
 Nolen, J; Roquejoffre, JM; Ryzhik, L, Refined longtime asymptotics for Fisher–KPP fronts,
Communications in Contemporary Mathematics
(January, 2018),
pp. 18500721850072, World Scientific Pub Co Pte Lt [doi]
(last updated on 2019/02/20)
Abstract: © 2018 World Scientific Publishing Company. We study the onedimensional FisherKPP equation, with an initial condition u0(x) that coincides with the step function except on a compact set. A wellknown result of Bramson in [Maximal displacement of branching Brownian motion, Comm. Pure Appl. Math. 31 (1978) 531581; 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) 199; Front propagation into unstable states, Phys. Rep. 386 (2003) 29222] 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/t1γ), for any γ > 0. The interesting aspect of this asymptotics is that the coefficient in front of the 1/tterm does not depend on u0.


dept@math.duke.edu
ph: 919.660.2800
fax: 919.660.2821
 
Mathematics Department
Duke University, Box 90320
Durham, NC 277080320

