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A stability index for travelling waves in activator-inhibitor systems

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

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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References

1Alexander, J., Gardner, R. A. and Jones, C. K. R. T.. A topological invariant arising in the stability analysis of travelling waves. J. Reine Angew. Math 410 (1990), 143.Google Scholar
2Alexander, J. C. and Jones, C. K. R. T.. Existence and stability of asymptotically oscillatory double pulses. J. Reine Angew. Math 446 (1994), 49–79.Google Scholar
3Arnol'd, V. I.. Characteristic class entering in quantization conditions. Funct. Anal.Appl. 1 (1967), 1–13.CrossRefGoogle Scholar
4Arnol'd, V. I.. The Sturm theorems and symplectic geometry. Funct. Anal. Appl. 19 (1985), 251–259.CrossRefGoogle Scholar
5Bates, P. W. and Jones, C. K. R. T.. Invariant manifolds for semilinear partial differential equations. In: Kirchgraber, U., Walther, H.O. (eds) Dynamics Reported. Dynamics Reported, vol 2 (Wiesbaden: Vieweg+Teubner Verlag, 1989).CrossRefGoogle Scholar
6Beck, M. and Malham, S.. Computing the Maslov index for large systems. Proc. Am. Math. Soc. 143 (2015), 2159–2173.CrossRefGoogle Scholar
7Beck, M., Cox, G., Jones, C., Latushkin, Y., McQuighan, K. and Sukhtayev, A.. Instability of pulses in gradient reaction-diffusion systems: a symplectic approach, to appear in Philosophal Transactions of the Royal Society A.Google Scholar
8Bose, A. and Jones, C. K. R..T.. Stability of the in-phase travelling wave solution in a pair of coupled nerve fibers. Indiana Univ. Math. J. 44 (1995), 189–220.CrossRefGoogle Scholar
9Bridges, T. J. and Derks, G.. Unstable eigenvalues and the linearization about solitary waves and fronts with symmetry In Proceedings of the royal society of London A: mathematical, Physical and Engineering Sciences, vol. 455, pp. 2427–2469 (The Royal Society, no. 1987, 1999).Google Scholar
10Bridges, T. J. and Derks, G.. The symplectic Evans matrix, and the instability of solitary waves and fronts. Arch. Ration. Mech. Anal. 156 (2001), 1–87.CrossRefGoogle Scholar
11Bridges, T. J. and Derks, G.. Constructing the symplectic Evans matrix using maximally analytic individual vectors. Proc. R. Soc. Edinb.: Sec. Math. 133 (2003), 505–526.CrossRefGoogle Scholar
12Chardard, F. and Bridges, T. J.. Transversality of homoclinic orbits, the Maslov index and the symplectic Evans function. Nonlinearity 28 (2014), 77.CrossRefGoogle Scholar
13Chardard, F., Dias, F. and Bridges, T. J.. Computing the Maslov index of solitary waves, part 1: Hamiltonian systems on a four-dimensional phase space. Phys. D: Nonlin. Phenom. 238 (2009), 1841–1867.CrossRefGoogle Scholar
14Chen, C.-N. and Choi, Y. S.. Traveling pulse solutions to FitzHugh–Nagumo equations. Calc. Var. Partial. Differ. Equ. 54 (2015), 1–45.CrossRefGoogle Scholar
15Chen, C.-N. and Hu, X.. Maslov index for homoclinic orbits of Hamiltonian systems, vol. 24, pp. 589–603 (Annales de l'Institut Henri Poincaré, Analyse non Linéare, no. 4, 2007).Google Scholar
16Chen, C.-N. and Hu, X.. Stability analysis for standing pulse solutions to FitzHugh–Nagumo equations. Calc. Var. Partial. Differ. Equ. 49 (2014), 827–845.CrossRefGoogle Scholar
17Cornwell, P., Opening the Maslov box for traveling waves in skew-gradient systems: counting eigenvalues and proving (in)stability, Preprint–arXiv:1709.01908 (2017a). (to appear in the Indiana University Mathematics Journal)Google Scholar
18Cornwell, P. and Jones, C. K. R. T.. On the existence and stability of fast traveling waves in a doubly diffusive FitzHugh-Nagumo System. SIAM J. Appl. Dyn. Syst. 17 (2018), 754–787.CrossRefGoogle Scholar
19Cox, G., Jones, C. K. R. T. and Marzuola, J. L.. A Morse index theorem for elliptic operators on bounded domains. Commun. Partial Diff. Equ. 40 (2015), 1467–1497.CrossRefGoogle Scholar
20Crampin, M. and Pirani, F. A. E.. Applicable differential geometry, vol. 59 (Cambridge, UK: Cambridge University Press, 1986).Google Scholar
21Duistermaat, J. J.. Symplectic geometry (Spring School, Utrecht, NL, June 7–14, 2004).Google Scholar
22Evans, J. W.. Nerve axon equations: iv. The stable and the unstable impulse. Indiana Univ. Math. J. 24 (1975), 1169–1190.CrossRefGoogle Scholar
23Fenichel, N.. Geometric singular perturbation theory for ordinary differential equations. J. Differ. Equ. 31 (1979), 53–98.CrossRefGoogle Scholar
24Fife, P. C. and McLeod, J. B.. The approach of solutions of nonlinear diffusion equations to travelling front solutions. Arch. Ration. Mech. Anal. 65 (1977), 335–361.CrossRefGoogle Scholar
25Flores, G.. Stability analysis for the slow travelling pulse of the FitzHugh–Nagumo system. SIAM. J. Math. Anal. 22 (1991), 392–399.CrossRefGoogle Scholar
26Gardner, R. A. and Zumbrun, K.. The gap lemma and geometric criteria for instability of viscous shock profiles. Commun. Pure. Appl. Math. 51 (1998), 797–855.3.0.CO;2-1>CrossRefGoogle Scholar
27Hassett, B.. Introduction to algebraic geometry (New York: Cambridge University Press, 2007).CrossRefGoogle Scholar
28Henry, D.. Geometric theory of semilinear parabolic equations, Lecture notes in mathematics (Berlin, New York: Springer-Verlag, 1981).CrossRefGoogle Scholar
29Homburg, A. J. and Sandstede, B.. Homoclinic and heteroclinic bifurcations in vector fields. Handbook of dynamical systems 3 (2010), 379–524.Google Scholar
30Howard, P., Latushkin, Y., and Sukhtayev, A.. The Maslov and Morse indices for Schrödinger operators on $\open {R}$, arXiv preprint arXiv:1608.05692 (2016).Google Scholar
31Howard, P. and Sukhtayev, A.. The Maslov and Morse indices for Schr”odinger operators on [0, 1]. J. Differ. Equ. 260 (2016), 4499–4549.CrossRefGoogle Scholar
32Jones, C., Latushkin, Y. and Sukhtaiev, S.. Counting spectrum via the Maslov index for one dimensional θ-periodic Schr”odinger operators. Proc. Am. Math. Soc. 145 (2017), 363–377.CrossRefGoogle Scholar
33Jones, C. K. R. T.. Stability of the travelling wave solution of the FitzHugh–Nagumo system. Trans. Am. Math. Soc. 286 (1984), 431–469.CrossRefGoogle Scholar
34Jones, C. K. R. T., Geometric singular perturbation theory, pp. 44–118, Dynamical systems (Springer, 1995).Google Scholar
35Jones, C. K. R. T. and Kopell, N.. Tracking invariant manifolds with differential forms in singularly perturbed systems. J. Differ. Equ. 108 (1994), 64–88.CrossRefGoogle Scholar
36Jones, C. K. R. T., Latushkin, Y. and Marangell, R.. The Morse and Maslov indices for matrix Hills equations. Spectral analysis, differential equations and mathematical physics: a festschrift in honor of Fritz Gesztesys 60th birthday 87 (2013), 205–233.Google Scholar
37Jones, C. K. R. T., Kopell, N. and Langer, R.. Construction of the FitzHugh-Nagumo pulse using differential forms. Patterns and dynamics in reactive media 37 (1991), 101–116.CrossRefGoogle Scholar
38Kapitula, T. and Promislow, K.. Spectral and dynamical stability of nonlinear waves, vol. 457 (Springer, 2013).CrossRefGoogle Scholar
39Kato, T., Perturbation theory for linear operators (New York: Springer, 1966).Google Scholar
40Kuehn, C.. Multiple time scale dynamics, vol. 191 (Berlin: Springer, 2015).CrossRefGoogle Scholar
41Magnus, J. R. and Neudecker, H.. Matrix differential calculus with applications in statistics and econometrics, Wiley series in probability and mathematical statistics (Chichester, 1988).Google Scholar
42McKean, H. P.. Nagumo's equation. Adv. Math. (N. Y) 4 (1970), 209–223.CrossRefGoogle Scholar
43Murray, J. D.. Mathematical biology. II spatial models and biomedical applications (New York: Springer-Verlag, Incorporated, 2001).Google Scholar
44Pego, R. L. and Weinstein, M. I.. Eigenvalues, and instabilities of solitary waves. Philos. Trans. R. Soc. London A: Math. Phys. Eng. Sci. 340 (1992), 47–94.Google Scholar
45Robbin, J. and Salamon, D.. The Maslov index for paths. Topology 32 (1993), 827–844.CrossRefGoogle Scholar
46Sandstede, B.. Stability of travelling waves. Handbook Dyn. Syst. 2 (2002), 983–1055.CrossRefGoogle Scholar
47Schecter, S.. Exchange lemmas 1: deng's lemma. J. Differ. Equ. 245 (2008), 392–410.CrossRefGoogle Scholar
48Souriau, J.-M.. Construction explicite de l'indice de Maslov. Applications, pp. 117–148 Group theoretical methods in physics (Berlin: Springer, 1976).Google Scholar
49Turing, A. M.. The chemical basis of morphogenesis. Bull. Math. Biol. 52 (1990), 153–197.CrossRefGoogle ScholarPubMed
50Vinberg, Ė. B.. A course in algebra, no. 56 (American Mathematical Soc., 2003).Google Scholar
51Yanagida, E.. Stability of travelling front solutions of the Fitzhugh–Nagumo equations. Math. Comput. Model. 12 (1989), 289–301.CrossRefGoogle Scholar
52Yanagida, E.. Standing pulse solutions in reaction-diffusion systems with skew-gradient structure. J. Dyn. Differ. Equ. 14 (2002), 189–205.CrossRefGoogle Scholar