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We propose a novel method for establishing the convergence rates of solutionsto reaction-diffusion equations to traveling waves. The analysis is based onthe study of the traveling wave shape defect function introduced in [2]. Itturns out that the convergence rate is controlled by the distance between the``phantom front location'' for the shape defect function and the true frontlocation of the solution. Curiously, the convergence to a traveling wave itselfhas a pulled nature, regardless of whether the traveling wave is of pushed,pulled, or pushmi-pullyu type. In addition to providing new results, thisapproach simplifies dramatically the proof in the Fisher-KPP case and gives aunified, succinct explanation for the known algebraic rates of convergence inthe Fisher-KPP case and the exponential rates in the pushed case.

Front location determines convergence rate to traveling waves