Point Spectrum: Linear Hamiltonian Systems

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


Hamiltonian systems are about balance, with the energy and other invariants preserved under the flow. For a spatially localized critical point of a Hamiltonian system, the balance is reflected in the symmetry of the spectrum, which typically pins the essential spectrum to the imaginary axis in unweighted spaces. The mechanism for bifurcation in Hamiltonian systems thus falls upon the point spectrum.


Hamiltonian System Imaginary Axis Essential Spectrum Hamiltonian Structure Constraint Matrix 
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  1. [13]
    C. De Angelis. Self-trapped propagation in the nonlinear cubic-quintic Schrödinger equation: a variational approach. IEEE J. Quantum Elect., 30(3):818–821, 1994.CrossRefGoogle Scholar
  2. [36]
    J. Bona, P. Souganidis, and W. Strauss. Stability and instability of solitary waves of Korteweg–de Vries type. Proc. R. Soc. London A, 411:395–412, 1987.MathSciNetzbMATHCrossRefGoogle Scholar
  3. [44]
    J. Bronski, M. Johnson, and T. Kapitula. An instability index theory for quadratic pencils and applications. preprint, 2012.Google Scholar
  4. [54]
    P. Chossat and R. Lauterbach. Methods in Equivariant Bifurcations and Dynamical Systems, volume 15 of Advanced Series in Nonlinear Dynamics. World Scientific, Singapore, 2000.CrossRefGoogle Scholar
  5. [55]
    M. Chugunova and D. Pelinovsky. On quadratic eigenvalue problems arising in stability of discrete vortices. Lin. Alg. Appl., 431:962–973, 2009.MathSciNetzbMATHCrossRefGoogle Scholar
  6. [56]
    M. Chugunova and D. Pelinovsky. Count of eigenvalues in the generalized eigenvalue problem. J. Math. Phys., 51(5):052901, 2010.Google Scholar
  7. [66]
    B. Deconinck and T. Kapitula. On the spectral and orbital stability of spatially periodic stationary solutions of generalized Korteweg–de Vries equations. submitted.Google Scholar
  8. [97]
    S. Gatz and J. Herrmann. Soliton propagation in materials with saturable nonlinearity. J. Opt. Soc. Am. B, 8(11):2296–2302, 1991.CrossRefGoogle Scholar
  9. [107]
    M. Grillakis. Linearized instability for nonlinear Schrödinger and Klein–Gordon equations. Comm. Pure Appl. Math., 46:747–774, 1988.MathSciNetCrossRefGoogle Scholar
  10. [108]
    M. Grillakis. Analysis of the linearization around a critical point of an infinite-dimensional Hamiltonian system. Comm. Pure Appl. Math., 43: 299–333, 1990.MathSciNetzbMATHCrossRefGoogle Scholar
  11. [110]
    M. Grillakis, J. Shatah, and W. Strauss. Stability theory of solitary waves in the presence of symmetry, II. J. Funct. Anal., 94:308–348, 1990.MathSciNetzbMATHCrossRefGoogle Scholar
  12. [127]
    M. Hǎrǎguş and T. Kapitula. On the spectra of periodic waves for infinite-dimensional Hamiltonian systems. Physica D, 237(20):2649–2671, 2008.MathSciNetCrossRefGoogle Scholar
  13. [130]
    R. Jackson and M. Weinstein. Geometric analysis of bifurcation and symmetry breaking in a Gross–Pitaevskii equation. J. Stat. Phys., 116: 881–905, 2004.MathSciNetzbMATHCrossRefGoogle Scholar
  14. [132]
    M. Johansson and Y. Kivshar. Discreteness-induced oscillatory instabilities of dark solitons. Phys. Rev. Lett., 82(1):85–88, 1999.CrossRefGoogle Scholar
  15. [135]
    C.K.R.T. Jones. Instability of standing waves for nonlinear Schrödinger-type equations. Ergod. Th. & Dynam. Sys., 8:119–138, 1988.zbMATHCrossRefGoogle Scholar
  16. [136]
    C.K.R.T. Jones and J. Moloney. Instability of standing waves in nonlinear optical waveguides. Phys. Lett. A, 117:175–180, 1986a.CrossRefGoogle Scholar
  17. [137]
    C.K.R.T. Jones and J. Moloney. Stability and instability of nonlinear standing waves in planar optical waveguides. In H. Gibbs, P. Mandel, N. Peyghambarian, and S. Smith, editors, Optical Bistability III, volume 8 of Proceedings in Physics. Springer-Verlag, New York, 1986b.Google Scholar
  18. [147]
    T. Kapitula and P. Kevrekidis. Linear stability of perturbed Hamiltonian systems: theory and a case example. J. Phys. A: Math. Gen., 37(30): 7509–7526, 2004.MathSciNetzbMATHCrossRefGoogle Scholar
  19. [148]
    T. Kapitula and K. Promislow. Stability indices for constrained self-adjoint operators. Proc. Amer. Math. Soc., 140(3):865–880, 2012.MathSciNetzbMATHCrossRefGoogle Scholar
  20. [150]
    T. Kapitula and B. Sandstede. A novel instability mechanism for bright solitary-wave solutions to the cubic–quintic Ginzburg–Landau equation. J. Opt. Soc. Am. B, 15:2757–2762, 1998a.MathSciNetCrossRefGoogle Scholar
  21. [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
  22. [155]
    T. Kapitula and A. Stefanov. A Hamiltonian–Krein (instability) index theory for KdV-like eigenvalue problems. preprint, 2013.Google Scholar
  23. [156]
    T. Kapitula, P. Kevrekidis, and B. Malomed. Stability of multiple pulses in discrete systems. Phys. Rev. E, 63(036604), 2001.Google Scholar
  24. [157]
    T. Kapitula, P. Kevrekidis, and B. Sandstede. Counting eigenvalues via the Krein signature in infinite-dimensional Hamiltonian systems. Physica D, 195(3&4): 263–282, 2004a.MathSciNetzbMATHCrossRefGoogle Scholar
  25. [159]
    T. Kapitula, P. Kevrekidis, and B. Sandstede. Addendum: Counting eigenvalues via the Krein signature in infinite-dimensional Hamiltonian systems. Physica D, 201(1&2):199–201, 2005.MathSciNetzbMATHCrossRefGoogle Scholar
  26. [161]
    T. Kapitula, E. Hibma, H.-P. Kim, and J. Timkovich. Instability indices for matrix pencils. preprint, 2013.Google Scholar
  27. [170]
    P. Kevrekidis and D. Pelinovsky. Discrete vector on-site vortices. Proc. Royal Soc. A, 462:2671–2694, 2006.MathSciNetzbMATHCrossRefGoogle Scholar
  28. [174]
    Y. Kivshar, D. Pelinovsky, T. Cretegny, and M. Peyrard. Internal modes of solitary waves. Phys. Rev. Lett., 80(23):5032–5035, 1998.CrossRefGoogle Scholar
  29. [175]
    Y. Kodama, M. Romagnoli, and S. Wabnitz. Soliton stability and interactions in fibre lasers. Elect. Lett., 28(21):1981–1983, 1992.CrossRefGoogle Scholar
  30. [176]
    R. Kollár. Homotopy method for nonlinear eigenvalue pencils with applications. SIAM J. Math. Anal., 43(2):612–633, 2011.MathSciNetzbMATHCrossRefGoogle Scholar
  31. [177]
    R. Kollár and P. Miller. Graphical Krein signature theory and Evans–Krein functions. preprint, 2013.Google Scholar
  32. [178]
    M. Krein. Topics in Differential and Integral Equations and Operator Theory, volume 7 of Operator Theory: Advances and Applications, pp. 1–98. Birkhäuser, Basel, 1983.Google Scholar
  33. [180]
    S. Lafortune and J. Lega. Spectral stability of local deformations of an elastic rod: Hamiltonian formalism. SIAM J. Math. Anal., 36(6): 1726–1741, 2005.MathSciNetzbMATHCrossRefGoogle Scholar
  34. [194]
    M. Lukas, D. Pelinovsky, and P. Kevrekidis. Lyapunov–Schmidt reduction algorithm for three-dimensional discrete vortices. Physica D, 212:339–350, 2008.MathSciNetCrossRefGoogle Scholar
  35. [196]
    R. MacKay. Stability of equilibria of Hamiltonian systems. In R. MacKay and J. Meiss, editors, Hamiltonian Dynamical Systems, pp. 137–153. Adam Hilger, Briston, UK, 1987.Google Scholar
  36. [197]
    R. MacKay. Movement of eigenvalues of Hamiltonian equilibria under non-Hamiltonian perturbation. Phys. Lett. A, 155:266–268, 1991.MathSciNetCrossRefGoogle Scholar
  37. [214]
    E. Ostrovskaya, Y. Kivshar, D. Skryabin, and W. Firth. Stability of multihump optical solitons. Phys. Rev. Lett., 83(2):296–299, 1999.zbMATHCrossRefGoogle Scholar
  38. [222]
    D. Pelinovsky. Inertia law for spectral stability of solitary waves in coupled nonlinear Schrödinger equations. Proc. Royal Soc. London A, 461: 783–812, 2005.MathSciNetzbMATHCrossRefGoogle Scholar
  39. [224]
    D. Pelinovsky. Spectral stability of nonlinear waves in KdV-type evolution equations. preprint, 2013.Google Scholar
  40. [226]
    D. Pelinovsky and P. Kevrekidis. Stability of discrete dark solitons in nonlinear Schrödinger lattices. J. Phys. A: Math. Gen., 41:185206, 2008b.MathSciNetCrossRefGoogle Scholar
  41. [227]
    D. Pelinovsky and Y. Kivshar. Stability criterion for multicomponent solitary waves. Phys. Rev. E, 62(6):8668–8676, 2000.MathSciNetCrossRefGoogle Scholar
  42. [228]
    D. Pelinovsky and A. Sakovich. Internal modes of discrete solitons near the anti-continuum limit of the dNLS equation. Physica D, 240:265–281, 2011.MathSciNetzbMATHCrossRefGoogle Scholar
  43. [231]
    D. Pelinovsky and J. Yang. Instabilities of multihump vector solitons in coupled nonlinear Schrödinger equations. Stud. Appl. Math., 115(1): 109–137, 2005.MathSciNetzbMATHCrossRefGoogle Scholar
  44. [232]
    D. Pelinovsky, P. Kevrekidis, and D. Frantzeskakis. Persistence and stability of discrete vortices in nonlinear Schrödinger lattices. Physica D, 212:20–53, 2005a.MathSciNetzbMATHCrossRefGoogle Scholar
  45. [233]
    D. Pelinovsky, P. Kevrekidis, and D. Frantzeskakis. Stability of discrete solitons in nonlinear Schrödinger lattices. Physica D, 212:1–19, 2005b.MathSciNetzbMATHCrossRefGoogle Scholar
  46. [264]
    D. Skryabin. Energy of internal modes of nonlinear waves and complex frequencies due to symmetry breaking. Phys. Rev. E, 64(055601(R)), 2001.Google Scholar
  47. [268]
    J. Soto-Crespo, N. Akhmediev, and V. Afanasjev. Stability of the pulselike solutions of the quintic complex Ginzburg–Landau equation. J. Opt. Soc. Am. B, 13(7): 1439–1449, 1996.CrossRefGoogle Scholar
  48. [275]
    V. Vougalter and D. Pelinovsky. Eigenvalues of zero energy in the linearized NLS problem. J. Math. Phys., 47:062701, 2006.MathSciNetCrossRefGoogle Scholar
  49. [285]
    A. Yew. Stability analysis of multipulses in nonlinearly coupled Schrödinger equations. Indiana U. Math. J., 49(3):1079–1124, 2000.MathSciNetzbMATHCrossRefGoogle Scholar
  50. [286]
    A. Yew, B. Sandstede, and C.K.R.T. Jones. Instability of multiple pulses in coupled nonlinear Schrödinger equations. Phys. Rev. E, 61(5):5886–5892, 2000.MathSciNetCrossRefGoogle 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

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