Advertisement

The Evans Function for Sturm–Liouville Operators on the Real Line

  • Todd Kapitula
  • Keith Promislow
Chapter
Part of the Applied Mathematical Sciences book series (AMS, volume 185)

Abstract

In this chapter we construct the Evans function for exponentially asymptotic differential operators on unbounded domains. While several key elements of the construction naturally carry over from the bounded domain construction, important new subtleties arise. To focus on these key issues, we restrict our attention to second-order Sturm–Liouville operators. The extension to higher-order linear operators in addressed in Chapter 10.

Keywords

Eigenvalue Problem Riemann Surface Branch Point Essential Spectrum Point Spectrum 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. [3]
    M. Ablowitz and P. Clarkson. Solitons, Nonlinear Evolution Equations, and Inverse Scattering, volume 149 of London Math. Soc. Lecture Note Series. Cambridge University Press, Cambridge, 1991.Google Scholar
  2. [8]
    S. Albeverio and R. Høegh-Krohn. Perturbation of resonances in quantam mechanics. J. Math. Anal. Appl., 101:491–513, 1984.MathSciNetzbMATHCrossRefGoogle Scholar
  3. [9]
    J. Alexander and C.K.R.T. Jones. Existence and stability of asymptotically oscillatory triple pulses. Z. Angew. Math. Phys., 44:189–200, 1993.MathSciNetzbMATHCrossRefGoogle Scholar
  4. [10]
    J. Alexander and C.K.R.T. Jones. Existence and stability of asymptotically oscillatory double pulses. J. Reine Angew. Math., 446:49–79, 1994.MathSciNetzbMATHGoogle Scholar
  5. [31]
    S. Benzoni-Gavage, D. Serre, and K. Zumbrun. Alternate Evans functions and viscous shock waves. SIAM J. Math. Anal., 32(5):929–962, 2001.MathSciNetzbMATHCrossRefGoogle Scholar
  6. [32]
    W.-J. Beyn and J. Lorenz. Stability of traveling waves: dichotomies and eigenvalue conditions on finite intervals. Num. Funct. Anal. Opt., 20: 201–244, 1999.MathSciNetzbMATHCrossRefGoogle Scholar
  7. [33]
    W.-J. Beyn and J. Rottmann-Matthes. Resolvent estimates for boundry-value problems on large intervals via the theory of discrete approximations. Num. Funct. Anal. Opt., 28:603–629, 2007.MathSciNetzbMATHCrossRefGoogle Scholar
  8. [34]
    W.-J. Beyn, Y. Latushkin, and J. Rottmann-Matthes. Finding eigenvalues of holomorphic Fredholm operator pencils using boundary value problems and contour integrals. arXiv:1210.3952, 2013.Google Scholar
  9. [38]
    A. Bose and C.K.R.T. Jones. Stability of the in-phase travelling wave solution in a pair of coupled nerve fibres. Indiana U. Math. J., 44(1): 189–220, 1995.MathSciNetzbMATHCrossRefGoogle Scholar
  10. [47]
    K. Chadan and P. Sabatier. Inverse Problems in Quantum Scattering Theory. Springer-Verlag, New York, second edition, 1989.zbMATHCrossRefGoogle Scholar
  11. [48]
    F. Chardard. Maslov index for solitary waves obtained as a limit of the Maslov index for periodic waves. C.R. Acad. Sci. Paris, Ser. I, 345: 689–694, 2007.Google Scholar
  12. [49]
    F. Chardard, F. Dias, and T. Bridges. Fast computation of the Maslov index for hyperbolic periodic orbits. J. Phys. A: Math. Gen., 39:14545–14557, 2006.MathSciNetzbMATHCrossRefGoogle Scholar
  13. [50]
    F. Chardard, F. Dias, and T. Bridges. On the Maslov index of multipulse homoclinic orbits. Proc. Royal. Soc. London A, 465:2897–2910, 2009.MathSciNetzbMATHCrossRefGoogle Scholar
  14. [51]
    F. Chardard, F. Dias, and T. Bridges. Computing the Maslov index of solitary waves. Part 1: Hamiltonian systems on a 4-dimensional phase space. Physica D, 238:1841–1867, 2010.Google Scholar
  15. [52]
    F. Chardard, F. Dias, and T. Bridges. Computing the Maslov index of solitary waves. Part 2: Phase space with dimension greater than four. Physica D, 240:1334–1344, 2011.Google Scholar
  16. [59]
    W.A. Coppel. Dichotomies in stability theory. In Lecture Notes in Mathematics 629. Springer-Verlag, New York, 1978.Google Scholar
  17. [74]
    P. Drazin and R. Johnson. Solitons: An Introduction. Cambridge University Press, Cambridge, 1989.zbMATHCrossRefGoogle Scholar
  18. [75]
    M. Eastham. The Asymptotic Solution of Linear Differential Systems: Applications of the Levinson Theorem. Clarendon Press, Oxford, 1989.zbMATHGoogle Scholar
  19. [86]
    R. Froese. Asymptotic distribution of resonances in one dimension. J. Diff. Eqs., 137(2):251–272, 1997.MathSciNetzbMATHCrossRefGoogle Scholar
  20. [87]
    R. Froese. Upper bounds for the resonance-counting function of Schrödinger operators in odd dimensions. Canad. J. Math., 50(3):538–546, 1998.MathSciNetzbMATHCrossRefGoogle Scholar
  21. [92]
    R. Gardner. Spectral analysis of long-wavelength periodic waves and applications. J. Reine Angew. Math., 491:149–181, 1997.MathSciNetzbMATHGoogle Scholar
  22. [96]
    R. Gardner and K. Zumbrun. The gap lemma and geometric criteria for instability of viscous shock profiles. Comm. Pure Appl. Math., 51(7): 797–855, 1998.MathSciNetCrossRefGoogle Scholar
  23. [98]
    F. Gesztesy and H. Holden. A unified approach to eigenvalues and resonances of Schrödinger operators using Fredholm determinants. J. Math. Anal. Appl., 123: 181–198, 1987.MathSciNetzbMATHCrossRefGoogle Scholar
  24. [131]
    A. Jensen and M. Melgaard. Perturbation of eigenvalues embedded at a threshold. Proc. Roy. Soc. Edinburgh, 132A:163–179, 2002.MathSciNetCrossRefGoogle Scholar
  25. [134]
    C.K.R.T. Jones. Stability of the travelling wave solutions of the Fitzhugh–Nagumo system. Trans. AMS, 286(2):431–469, 1984.zbMATHCrossRefGoogle Scholar
  26. [140]
    C.K.R.T. Jones, Y. Latushkin, and R. Marangell. The Morse and Maslov indices for matrix Hills equations. preprint, 2013.Google Scholar
  27. [143]
    T. Kapitula. Stability criterion for bright solitary waves of the perturbed cubic–quintic Schrödinger equation. Physica D, 116(1–2):95–120, 1998.MathSciNetzbMATHCrossRefGoogle Scholar
  28. [151]
    T. Kapitula and B. Sandstede. Instability mechanism for bright solitary wave solutions to the cubic–quintic Ginzburg–Landau equation. J. Opt. Soc. Am. B, 15(11):2757–2762, 1998b.MathSciNetCrossRefGoogle Scholar
  29. [153]
    T. Kapitula and B. Sandstede. Edge bifurcations for near-integrable systems via Evans function techniques. SIAM J. Math. Anal., 33(5):1117–1143, 2002.MathSciNetzbMATHCrossRefGoogle Scholar
  30. [179]
    S. Lafortune and J. Lega. Instability of local deformations of an elastic rod. Physica D, 182(1–2):103–124, 2003.MathSciNetzbMATHCrossRefGoogle Scholar
  31. [184]
    Y. Latushkin and A. Sukhtayev. The Evans function and the Weyl–Titchmarsh function. Disc. Cont. Dyn. Sys. Ser. S, 5:939–970, 2012.MathSciNetzbMATHCrossRefGoogle Scholar
  32. [193]
    Y. Li and K. Promislow. The mechanism of the polarization mode instability in birefringent fiber optics. SIAM J. Math. Anal., 31(6):1351–1373, 2000.MathSciNetzbMATHCrossRefGoogle Scholar
  33. [195]
    Z.-Q. Ma. The Levinson theorem. J. Phys. A: Math. Gen., 39:R625–R659, 2006.zbMATHCrossRefGoogle Scholar
  34. [203]
    A. Markushevich. Theory of Functions. Chelsea Publishing, New York, 1985.Google Scholar
  35. [207]
    P. Miller. Applied Asymptotic Analysis, volume 75 of Graduates Studies in Mathematics, American Mathematical Society, Providence, RI, 2006.Google Scholar
  36. [220]
    R. Pego and M. Weinstein. Eigenvalues, and instabilities of solitary waves. Phil. Trans. R. Soc. Lond. A, 340:47–94, 1992.MathSciNetzbMATHCrossRefGoogle Scholar
  37. [240]
    A. Ramm. Perturbation of resonances. J. Math. Anal. Appl., 88:1–7, 1982.MathSciNetzbMATHCrossRefGoogle Scholar
  38. [243]
    M. Reed and B. Simon. Methods of Modern Mathematical Physics III: Scattering Theory. Academic Press, New York, 1979.zbMATHGoogle Scholar
  39. [252]
    B. Sandstede and A. Scheel. Absolute and convective instabilities of waves on unbounded and large bounded domains. Physica D, 145:233–277, 2000a.MathSciNetzbMATHCrossRefGoogle Scholar
  40. [254]
    B. Sandstede and A. Scheel. On the stability of travelling waves with large spatial period. J. Diff. Eq., 172:134–188, 2001a.MathSciNetzbMATHCrossRefGoogle Scholar
  41. [261]
    B. Simon. Notes on infinite determinants of Hilbert space operators. Adv. Math., 24:244–273, 1977a.zbMATHGoogle Scholar
  42. [263]
    B. Simon. Resonances in one dimension and Fredholm determinants. J. Func. Anal., 178:396–420, 2000.zbMATHCrossRefGoogle Scholar
  43. [296]
    M. Zworski. Resonances in physics and geometry. Notices Amer. Math. Soc., 46(3):319–328, 1999.MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Todd Kapitula
    • 1
  • Keith Promislow
    • 2
  1. 1.Department of Mathematics and StatisticsCalvin CollegeGrand RapidsUSA
  2. 2.Department of MathematicsMichigan State UniversityEast LansingUSA

Personalised recommendations