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Computing the Maslov index for large systems


Authors: Margaret Beck and Simon J. A. Malham
Journal: Proc. Amer. Math. Soc. 143 (2015), 2159-2173
MSC (2010): Primary 35P05
DOI: https://doi.org/10.1090/S0002-9939-2014-12575-5
Published electronically: December 15, 2014
MathSciNet review: 3314123
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Abstract: We address the problem of computing the Maslov index for large linear symplectic systems on the real line. The Maslov index measures the signed intersections (with a given reference plane) of a path of Lagrangian planes. The natural chart parameterization for the Grassmannian of Lagrangian planes is the space of real symmetric matrices. Linear system evolution induces a Riccati evolution in the chart. For large order systems this is a practical approach as the computational complexity is quadratic in the order. The Riccati solutions, however, also exhibit singularites (which are traversed by changing charts). Our new results involve characterizing these Riccati singularities and two trace formulae for the Maslov index as follows. First, we show that the number of singular eigenvalues of the symmetric chart representation equals the dimension of intersection with the reference plane. Second, the Cayley map is a diffeomorphism from the space of real symmetric matrices to the manifold of unitary symmetric matrices. We show the logarithm of the Cayley map equals the arctan map (modulo $ 2\mathrm {i}$) and its trace measures the angle of the Langrangian plane to the reference plane. Third, the Riccati flow under the Cayley map induces a flow in the manifold of unitary symmetric matrices. Using the natural unitary action on this manifold, we pullback the flow to the unitary Lie algebra and monitor its trace. This avoids singularities, and is a natural robust procedure. We demonstrate the effectiveness of these approaches by applying them to a large eigenvalue problem. We also discuss the extension of the Maslov index to the infinite dimensional case.


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  • [1] J. Alexander, R. Gardner, and C. Jones, A topological invariant arising in the stability analysis of travelling waves, J. Reine Angew. Math. 410 (1990), 167-212. MR 1068805 (92d:58028)
  • [2] A.Y. Aljasser, Computing the Evans function for the stability of combustion waves, PhD thesis, 2012.
  • [3] V. I. Arnold, On a characteristic class entering into conditions of quantization, Funkcional. Anal. i Priložen. 1 (1967), 1-14 (Russian). MR 0211415 (35 #2296)
  • [4] V. I. Arnold, Sturm theorems and symplectic geometry, Funktsional. Anal. i Prilozhen. 19 (1985), no. 4, 1-10, 95 (Russian). MR 820079 (87j:58033)
  • [5] Amitabha Bose and Christopher K. R. T. Jones, Stability of the in-phase travelling wave solution in a pair of coupled nerve fibers, Indiana Univ. Math. J. 44 (1995), no. 1, 189-220. MR 1336438 (96k:92005), https://doi.org/10.1512/iumj.1995.44.1984
  • [6] Jonathan Bougie, Asim Gangopadhyaya, Jeffry Mallow, and Constantin Rasinariu, Supersymmetric quantum mechanics and solvable models, Symmetry 4 (2012), no. 3, 452-473. MR 2979407, https://doi.org/10.3390/sym4030452
  • [7] Frédéric Chardard, Frédéric Dias, and Thomas J. Bridges, Computing the Maslov index of solitary waves, Part 2: Phase space with dimension greater than four, Phys. D 240 (2011), no. 17, 1334-1344. MR 2831770 (2012k:37130), https://doi.org/10.1016/j.physd.2011.05.014
  • [8] M. A. de Gosson, The principles of Newtonian and quantum mechanics, Imperial College Press, London, 2001. The need for Planck's constant, $ h$; With a foreword by Basil Hiley. MR 1897416 (2003k:81004)
  • [9] Jian Deng and Christopher Jones, Multi-dimensional Morse index theorems and a symplectic view of elliptic boundary value problems, Trans. Amer. Math. Soc. 363 (2011), no. 3, 1487-1508. MR 2737274 (2011k:35156), https://doi.org/10.1090/S0002-9947-2010-05129-3
  • [10] Jian Deng and Shunsau Nii, An infinite-dimensional Evans function theory for elliptic boundary value problems, J. Differential Equations 244 (2008), no. 4, 753-765. MR 2391343 (2009a:35180), https://doi.org/10.1016/j.jde.2007.10.037
  • [11] Kenro Furutani, Fredholm-Lagrangian-Grassmannian and the Maslov index, J. Geom. Phys. 51 (2004), no. 3, 269-331. MR 2079414 (2005g:53150), https://doi.org/10.1016/j.geomphys.2004.04.001
  • [12] Leon Greenberg and Marco Marletta, Numerical methods for higher order Sturm-Liouville problems, Numerical analysis 2000, Vol. VI, Ordinary differential equations and integral equations, J. Comput. Appl. Math. 125 (2000), no. 1-2, 367-383. MR 1803203 (2001k:65123), https://doi.org/10.1016/S0377-0427(00)00480-5
  • [13] Jeffrey Humpherys and Kevin Zumbrun, An efficient shooting algorithm for Evans function calculations in large systems, Phys. D 220 (2006), no. 2, 116-126. MR 2253406 (2007e:35006), https://doi.org/10.1016/j.physd.2006.07.003
  • [14] Christopher K. R. T. Jones, Yuri Latushkin, and Robert Marangell, The Morse and Maslov indices for matrix Hill's equations, Spectral analysis, differential equations and mathematical physics: a festschrift in honor of Fritz Gesztesy's 60th birthday, Proc. Sympos. Pure Math., vol. 87, Amer. Math. Soc., Providence, RI, 2013, pp. 205-233. MR 3087908
  • [15] Christopher K. R. T. Jones, Instability of standing waves for nonlinear Schrödinger-type equations, Ergodic Theory Dynam. Systems 8$ ^*$ (1988), Charles Conley Memorial Issue, 119-138. MR 967634 (90d:35267), https://doi.org/10.1017/S014338570000938X
  • [16] V. Ledoux and S.J.A. Malham, Spectral shooting is Schubert calculus, working paper 2009.
  • [17] Veerle Ledoux, Simon J. A. Malham, Jitse Niesen, and Vera Thümmler, Computing stability of multidimensional traveling waves, SIAM J. Appl. Dyn. Syst. 8 (2009), no. 1, 480-507. MR 2496765 (2010a:65127), https://doi.org/10.1137/080724009
  • [18] Veerle Ledoux, Simon J. A. Malham, and Vera Thümmler, Grassmannian spectral shooting, Math. Comp. 79 (2010), no. 271, 1585-1619. MR 2630004 (2011c:65139), https://doi.org/10.1090/S0025-5718-10-02323-9
  • [19] Simon J. A. Malham and Anke Wiese, Stochastic Lie group integrators, SIAM J. Sci. Comput. 30 (2008), no. 2, 597-617. MR 2385877 (2009b:37154), https://doi.org/10.1137/060666743
  • [20] Carsten R. Maple and Marco Marletta, Solving Hamiltonian systems arising from ODE eigenproblems, Numer. Algorithms 22 (1999), no. 3-4, 263-284 (2000). MR 1749487 (2001c:65158), https://doi.org/10.1023/A:1019171110743
  • [21] Hans Munthe-Kaas, High order Runge-Kutta methods on manifolds, Proceedings of the NSF/CBMS Regional Conference on Numerical Analysis of Hamiltonian Differential Equations (Golden, CO, 1997), 1999, pp. 115-127. MR 1662814 (99i:65075), https://doi.org/10.1016/S0168-9274(98)00030-0
  • [22] Andrew Pressley and Graeme Segal, Loop groups, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1986. Oxford Science Publications. MR 900587 (88i:22049)
  • [23] Joel Robbin and Dietmar Salamon, The spectral flow and the Maslov index, Bull. London Math. Soc. 27 (1995), no. 1, 1-33. MR 1331677 (96d:58021), https://doi.org/10.1112/blms/27.1.1
  • [24] Björn Sandstede, Stability of travelling waves, Handbook of dynamical systems, Vol. 2, North-Holland, Amsterdam, 2002, pp. 983-1055. MR 1901069 (2004e:37121), https://doi.org/10.1016/S1874-575X(02)80039-X
  • [25] Myunghyun Oh and Björn Sandstede, Evans functions for periodic waves on infinite cylindrical domains, J. Differential Equations 248 (2010), no. 3, 544-555. MR 2557905 (2011d:35257), https://doi.org/10.1016/j.jde.2009.08.003
  • [26] Jeremy Schiff and S. Shnider, A natural approach to the numerical integration of Riccati differential equations, SIAM J. Numer. Anal. 36 (1999), no. 5, 1392-1413 (electronic). MR 1706774 (2000d:34024), https://doi.org/10.1137/S0036142996307946

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Additional Information

Margaret Beck
Affiliation: Maxwell Institute for Mathematical Sciences and School of Mathematical and Computer Sciences, Heriot-Watt University, Edinburgh EH14 4AS, United Kingdom; Department of Mathematics, Boston University, Boston, Massachusetts 02215
Email: mabeck@math.bu.edu

Simon J. A. Malham
Affiliation: Maxwell Institute for Mathematical Sciences and School of Mathematical and Computer Sciences, Heriot-Watt University, Edinburgh EH14 4AS, United Kingdom
Email: S.J.Malham@ma.hw.ac.uk

DOI: https://doi.org/10.1090/S0002-9939-2014-12575-5
Received by editor(s): November 5, 2013
Published electronically: December 15, 2014
Additional Notes: The work of the first author was partially supported by NSF DMS 1007450 and a Sloan Fellowship
Communicated by: Ken Ono
Article copyright: © Copyright 2014 American Mathematical Society

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