Numerical computation of an evans function for travelling waves

K. Harley; P. Van Heijster; R. Marangell; G.J. Pettet; M. Wechselberger

doi:10.1016/j.mbs.2015.05.009

Numerical computation of an evans function for travelling waves

K. Harley, P. Van Heijster, R. Marangell^*, G.J. Pettet, M. Wechselberger

^*Corresponding author for this work

Research output: Contribution to journal › Article › Academic › peer-review

9 Citations (Scopus)

Abstract

We demonstrate a geometrically inspired technique for computing Evans functions for the linearised operators about travelling waves. Using the examples of the F-KPP equation and a Keller-Segel model of bacterial chemotaxis, we produce an Evans function which is computable through several orders of magnitude in the spectral parameter and show how such a function can naturally be extended into the continuous spectrum. In both examples, we use this function to numerically verify the absence of eigenvalues in a large region of the right half of the spectral plane. We also include a new proof of spectral stability in the appropriate weighted space of travelling waves of speed c ≥ 2δ in the F-KPP equation.

Original language	English
Pages (from-to)	36-51
Number of pages	16
Journal	Mathematical Biosciences
Volume	266
DOIs	https://doi.org/10.1016/j.mbs.2015.05.009
Publication status	Published - Aug 2015
Externally published	Yes

Keywords

Evans functions
Fisher-KPP equation
Keller-segel equations
Spectral stability
Travelling waves

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@article{261ade0abdd44343a2d72bd794ce1fc4,

title = "Numerical computation of an evans function for travelling waves",

abstract = "We demonstrate a geometrically inspired technique for computing Evans functions for the linearised operators about travelling waves. Using the examples of the F-KPP equation and a Keller-Segel model of bacterial chemotaxis, we produce an Evans function which is computable through several orders of magnitude in the spectral parameter and show how such a function can naturally be extended into the continuous spectrum. In both examples, we use this function to numerically verify the absence of eigenvalues in a large region of the right half of the spectral plane. We also include a new proof of spectral stability in the appropriate weighted space of travelling waves of speed c ≥ 2δ in the F-KPP equation.",

keywords = "Evans functions, Fisher-KPP equation, Keller-segel equations, Spectral stability, Travelling waves",

author = "K. Harley and {Van Heijster}, P. and R. Marangell and G.J. Pettet and M. Wechselberger",

note = "Funding Information: R.M., G.J.P. and M.W. gratefully acknowledge the partial support of Australian Research Council grant ARC DP110102775 . P.vH. gratefully acknowledges support under the Australian Research Council{\textquoteright}s Discovery Early Career Researcher Award funding scheme DE140100741. K.H. also gratefully acknowledges support from an Australian Mathematical Society Lift-off Fellowship. Publisher Copyright: {\textcopyright} 2015 Elsevier Inc. Copyright: Copyright 2019 Elsevier B.V., All rights reserved.",

year = "2015",

month = aug,

doi = "10.1016/j.mbs.2015.05.009",

language = "English",

volume = "266",

pages = "36--51",

journal = "Mathematical Biosciences",

issn = "0025-5564",

publisher = "Elsevier",

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TY - JOUR

T1 - Numerical computation of an evans function for travelling waves

AU - Harley, K.

AU - Van Heijster, P.

AU - Marangell, R.

AU - Pettet, G.J.

AU - Wechselberger, M.

N1 - Funding Information: R.M., G.J.P. and M.W. gratefully acknowledge the partial support of Australian Research Council grant ARC DP110102775 . P.vH. gratefully acknowledges support under the Australian Research Council’s Discovery Early Career Researcher Award funding scheme DE140100741. K.H. also gratefully acknowledges support from an Australian Mathematical Society Lift-off Fellowship. Publisher Copyright: © 2015 Elsevier Inc. Copyright: Copyright 2019 Elsevier B.V., All rights reserved.

PY - 2015/8

Y1 - 2015/8

N2 - We demonstrate a geometrically inspired technique for computing Evans functions for the linearised operators about travelling waves. Using the examples of the F-KPP equation and a Keller-Segel model of bacterial chemotaxis, we produce an Evans function which is computable through several orders of magnitude in the spectral parameter and show how such a function can naturally be extended into the continuous spectrum. In both examples, we use this function to numerically verify the absence of eigenvalues in a large region of the right half of the spectral plane. We also include a new proof of spectral stability in the appropriate weighted space of travelling waves of speed c ≥ 2δ in the F-KPP equation.

AB - We demonstrate a geometrically inspired technique for computing Evans functions for the linearised operators about travelling waves. Using the examples of the F-KPP equation and a Keller-Segel model of bacterial chemotaxis, we produce an Evans function which is computable through several orders of magnitude in the spectral parameter and show how such a function can naturally be extended into the continuous spectrum. In both examples, we use this function to numerically verify the absence of eigenvalues in a large region of the right half of the spectral plane. We also include a new proof of spectral stability in the appropriate weighted space of travelling waves of speed c ≥ 2δ in the F-KPP equation.

KW - Evans functions

KW - Fisher-KPP equation

KW - Keller-segel equations

KW - Spectral stability

KW - Travelling waves

U2 - 10.1016/j.mbs.2015.05.009

DO - 10.1016/j.mbs.2015.05.009

M3 - Article

C2 - 26048189

AN - SCOPUS:84951913600

VL - 266

SP - 36

EP - 51

JO - Mathematical Biosciences

JF - Mathematical Biosciences

SN - 0025-5564

ER -