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A stability index for travelling waves in activator-inhibitor systems

Part of: Symplectic geometry, contact geometry Representations of solutions Partial differential equations

Published online by Cambridge University Press:  26 January 2019

Paul Cornwell  and
Christopher K. R. T. Jones
Paul Cornwell
Affiliation:
Department of Mathematics, UNC Chapel Hill, Phillips Hall CB #3250, Chapel Hill, NC2 7516, USA (pcorn@live.unc.edu; ckrtj@email.unc.edu)
Christopher K. R. T. Jones
Affiliation:
Department of Mathematics, UNC Chapel Hill, Phillips Hall CB #3250, Chapel Hill, NC2 7516, USA (pcorn@live.unc.edu; ckrtj@email.unc.edu)
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Abstract

We consider the stability of nonlinear travelling waves in a class of activator-inhibitor systems. The eigenvalue equation arising from linearizing about the wave is seen to preserve the manifold of Lagrangian planes for a nonstandard symplectic form. This allows us to define a Maslov index for the wave corresponding to the spatial evolution of the unstable bundle. We formulate the Evans function for the eigenvalue problem and show that the parity of the Maslov index determines the sign of the derivative of the Evans function at the origin. The connection between the Evans function and the Maslov index is established by a ‘detection form,’ which identifies conjugate points for the curve of Lagrangian planes.

Keywords

Maslov indexstability of travelling wavesEvans functionnon-self-adjoint eigenvalue problemshomoclinic orbitssolitary wavestopological indices

MSC classification

Primary: 35B35: Stability
Secondary: 53D12: Lagrangian submanifolds; Maslov index 35C07: Traveling wave solutions
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 1 , February 2020 , pp. 517 - 548
Copyright
Copyright © Royal Society of Edinburgh 2019

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