Generalized Transition Waves and Their Properties

By Henri Berestycki, François Hamel. 2010

205 citations recorded by the source corpus.

In this paper, we generalize the usual notions of waves, fronts, and propagation speeds in a very general setting. These new notions, which cover all usual situations, involve uniform limits, with respect to the geodesic distance, to a family of hypersurfaces that are parametrized by time. We prove the existence of new such waves for some time‐dependent reaction‐diffusion equations, as well as general intrinsic properties, some monotonicity properties, and some uniqueness results for almost‐planar fronts. The classification results, which are obtained under some appropriate assumptions, show the robustness of our general definitions. © 2012 Wiley Periodicals, Inc.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/cb9ac85aa0cb7329beadadb6d73057acaa828dcd

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Transition waves for Fisher-KPP equations with general time-heterogeneous and space-periodic coefficients (2015, background): “A generalization of the notion of travelling waves has been given by Berestycki and Hamel [4, 5]. uniformly with respect to t ∈ R 1 .”
Critical travelling waves for general heterogeneous one-dimensional reaction–diffusion equations (2015, unknown): “This is why we use the name “spatial transition wave” in the present paper instead of “generalized transition waves” as in the original articles [4, 5].”
Travelling Waves in a Nonlocal Reaction-Diffusion Equation as a Model for a Population Structured by a Space Variable and a Phenotypic Trait (2012, background): “For very general reaction diffusion equations, we refer to [7] for a definition of generalized transition waves and their properties.”
Traveling waves for a generalized Holling-Tanner predator-prey model (2017, background): “It is shown in [7] that the invasion profile can be approximated by certain generalized transition waves, a notion introduced by Berestycki and Hamel [2] to describe traveling-wave-like transition phenomena.”
Bistable pulsating fronts for reaction-diffusion equations in a periodic habitat (2014, background): “In [9], qualitative properties of transition fronts, which are more general than pulsating fronts, were investigated in unstructured heterogeneous media.”
Semi-wave and spreading speed for the diffusive competition model with a free boundary (2015, background): “In space dimension N = 1, more accurate estimate of the spreading speed is obtained, and the existence of an almost planar “generalized transition wave” (see [1]) is established.”