Speed selection for traveling-wave solutions to the diffusion-reaction equation with cubic reaction term and Burgers nonlinear convection

V. A. Sabelnikov and A. N. Lipatnikov
Phys. Rev. E 90, 033004 – Published 9 September 2014

Abstract

The problem of traveling wave (TW) speed selection for solutions to a generalized Murray-Burgers-KPP-Fisher parabolic equation with a strictly positive cubic reaction term is considered theoretically and the initial boundary value problem is numerically solved in order to support obtained analytical results. Depending on the magnitude of a parameter inherent in the reaction term (i) the term is either a concave function or a function with the inflection point and (ii) transition from pulled to pushed TW solution occurs due to interplay of two nonlinear terms; the reaction term and the Burgers convection term. Explicit pushed TW solutions are derived. It is shown that physically observable TW solutions, i.e., solutions obtained by solving the initial boundary value problem with a sufficiently steep initial condition, can be determined by seeking the TW solution characterized by the maximum decay rate at its leading edge. In the Appendix, the developed approach is applied to a non-linear diffusion-reaction equation that is widely used to model premixed turbulent combustion.

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  • Received 4 June 2014

DOI:https://doi.org/10.1103/PhysRevE.90.033004

©2014 American Physical Society

Authors & Affiliations

V. A. Sabelnikov

  • ONERA - The French Aerospace Laboratory, F-91761 Palaiseau, France

A. N. Lipatnikov

  • Department of Applied Mechanics, Chalmers University of Technology, Gothenburg, 412 96, Sweden

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Vol. 90, Iss. 3 — September 2014

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