A stability index for travelling waves in activator-inhibitor systems. Issue 1 (February 2020)

• Record Type: Journal Article
• Authors: Cornwell, Paul; Jones, Christopher K. R. T.
• Is Part Of: Proceedings. Volume 150:Issue 1(2020)
• Page Start: 517
• Page End: 548
• Publication Date: 2020-02
• Subjects: Maslov index, -- stability of travelling waves, -- Evans function, -- non-self-adjoint eigenvalue problems, -- homoclinic orbits, -- solitary waves, -- topological indices; 35B35, -- 53D12, -- 35C07; Mathematics -- Periodicals; 510
• DOI: 10.1017/prm.2018.92
• Languages: English
• ISSNs: 0308-2105

Abstract

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.