In its original formulation the Krein matrix was used to locate the spectrum of first-order star-even polynomial operators where both operator coefficients are nonsingular. Such operators naturally arise when considering first-order-in-time Hamiltonian PDEs. Herein the matrix is reformulated to allow for operator coefficients with nontrivial kernel. Moreover, it is extended to allow for the study of the spectral problem associated with quadratic star-even operators, which arise when considering the spectral problem associated with second-order-in-time Hamiltonian PDEs. In conjunction with the Hamiltonian-Krein index (HKI) the Krein matrix is used to study two problems: conditions leading to Hamiltonian-Hopf bifurcations for small spatially periodic waves, and the location and Krein signature of small eigenvalues associated with, e.g., $n$-pulse problems. For the first case we consider in detail a first-order-in-time fifth-order KdV-like equation. In the latter case we use a combination of Lin's method, the HKI, and the Krein matrix to study the spectrum associated with $n$-pulses for a second-order-in-time Hamiltonian system which is used to model the dynamics of a suspension bridge.


  1. Krein matrix
  2. multipulses
  3. Hamiltonian PDEs

MSC codes

  1. 35P30
  2. 47A55
  3. 47A56
  4. 70H14

Get full access to this article

View all available purchase options and get full access to this article.


J. Alexander, R. Gardner, and C. Jones, A topological invariant arising in the stability of travelling waves, J. Reine Angew. Math., 410 (1990), pp. 167--212.
J. Bronski, M. Johnson, and T. Kapitula, An instability index theory for quadratic pencils and applications, Comm. Math. Phys., 327 (2014), pp. 521--550.
C. Buzzi and J. Lamb, Reversible Hamiltonian Liapunov center theorem, Discrete Contin. Dyn. Syst. Ser. B, 5 (2005), pp. 51--66.
J. Chamard, J. Otta, and D. Lloyd, Computation of minimum energy paths for quasi-linear problems, J. Sci. Comput., 49 (2011), pp. 180--194.
A. Champneys and M. Groves, A global investigation of solitary-wave solutions to a two-parameter model for water waves, J. Fluid Mech., 342 (1997), pp. 199--229.
Y. Chen and P. McKenna, Traveling waves in a nonlinearly suspended beam: Theoretical results and numerical observations, J. Differential Equations, 136 (1997), pp. 325--355.
W. Coppel, Dichotomies in Stability Theory, Springer, Berlin, 1978.
B. Deconinck and T. Kapitula, On the spectral and orbital stability of spatially periodic stationary solutions of generalized Korteweg-de Vries equations, in Hamiltonian Partial Differential Equations and Applications, P. Guyenne, D. Nicholls, and C. Sulem, eds., Fields Inst. Commun. 75, Springer, New York, 2015, pp. 285--322.
B. Deconinck and O. Trichtchenko, High-frequency instabilities of small-amplitude solutions of Hamiltonian PDEs, Discrete Contin. Dyn. Syst., 37 (2015), pp. 1323--1358.
J. Evans, N. Fenichel, and J. Feroe, Double impulse solutions in nerve axon equations, SIAM J. Appl. Math., 42 (1982), pp. 219-234.
L. Evans, Partial Differential Equations, AMS, Providence, RI, 2010.
M. Grillakis, J. Shatah, and W. Strauss, Stability theory of solitary waves in the presence of symmetry, I, J. Funct. Anal., 74 (1987), pp. 160--197.
S. Hammarling, C. Munro, and F. Tisseur, An algorithm for the complete solution of quadratic eigenvalue problems, ACM Trans. Math. Software, 39 (2013), pp. 1--19.
M. Hǎrǎguş and G. Iooss, Local Bifurcations, Center Manifolds, and Normal Forms in Infinite-Dimensional Dynamical Systems, Springer, New York, 2011.
M. Hǎrǎguş and T. Kapitula, On the spectra of periodic waves for infinite-dimensional Hamiltonian systems, Phys. D, 237 (2008), pp. 2649--2671.
I. Ipsen and R. Rehman, Perturbation bounds for determinants and characteristic polynomials, SIAM J. Matrix Anal. Appl., 30 (2008), pp. 762--776.
T. Kapitula, The Krein signature, Krein eigenvalues, and the Krein oscillation theorem, Indiana U. Math. J., 59 (2010), pp. 1245--1276.
T. Kapitula, E. Hibma, H.-P. Kim, and J. Timkovich, Instability indices for matrix polynomials, Linear Algebra Appl., 439 (2013), pp. 3412--3434.
T. Kapitula, P. Kevrekidis, and B. Sandstede, Counting eigenvalues via the Krein signature in infinite-dimensional Hamiltonian systems, Phys. D, 195 (2004), pp. 263--282.
T. Kapitula, P. Kevrekidis, and B. Sandstede, Addendum: Counting eigenvalues via the Krein signature in infinite-dimensional Hamiltonian systems, Phys. D, 201 (2005), pp. 199--201.
T. Kapitula, P. Kevrekidis, and D. Yan, The Krein matrix: General theory and concrete applications in atomic Bose-Einstein condensates, SIAM J. Appl. Math., 73 (2013), pp. 1368--1395.
T. Kapitula and K. Promislow, Stability indices for constrained self-adjoint operators, Proc. Amer. Math. Soc., 140 (2012), pp. 865--880.
T. Kapitula and K. Promislow, Spectral and Dynamical Stability of Nonlinear Waves, Springer-Verlag, Berlin, 2013.
T. Kapitula and A. Stefanov, A Hamiltonian-Krein (instability) index theory for KdV-like eigenvalue problems, Stud. Appl. Math., 132 (2014), pp. 183--211.
T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin, 1980.
R. Kollár, B. Deconinck, and O. Trichtchenko, Direct characterization of spectral stability of small--amplitude periodic waves in scalar Hamiltonian problems via dispersion relation. SIAM J. Math. Anal., 51 (2019), pp. 3145--3169.
R. Kollár and P. Miller, Graphical Krein signature theory and Evans-Krein functions, SIAM Rev., 56 (2014), pp. 73--123.
Y. Li and K. Promislow, Structural stability of non-ground state traveling waves of coupled nonlinear Schrödinger equations, Phys. D, 124 (1998), pp. 137--165.
P. McKenna and W. Walter, Travelling waves in a suspension bridge, SIAM J. Appl. Math., 50 (1990), pp. 703--715.
K. Palmer, Exponential transversal dichotomies and homoclinic points J. Differential Equations, 55 (1984), pp. 225-256
D. Pelinovsky, Inertia law for spectral stability of solitary waves in coupled nonlinear Schrödinger equations, Proc. A, 461 (2005), pp. 783--812.
D. Pelinovsky, Spectral stability of nonlinear waves in KdV-type evolution equations, in Nonlinear Physical Systems: Spectral Analysis, Stability, and Bifurcations, O. Kirillov and D. Pelinovsky, eds., Wiley-ISTE, New York, 2014, pp. 377--400.
B. Sandstede, Verzweigungstheorie homokliner Verdopplungen, Ph.D. thesis, University of Stuttgart, 1993.
B. Sandstede, Instability of localized buckling modes in a one-dimensional strut model, Philos. Trans. A, 355 (1997), pp. 2083--2097.
B. Sandstede, Stability of multiple-pulse solutions, Trans. Amer. Math. Soc., 350 (1998), pp. 429--472.
B. Sandstede, Homoclinic flip bifurcations in conservative reversible systems, in Recent Trends in Dynamical Systems, Springer, 2013, pp. 107--124.
B. Sandstede and A. Scheel, Relative Morse indices, Fredholm indices, and group velocities, Discrete Contin. Dyn. Syst., 20 (2008), pp. 139--158.
D. Smets and J. van den Berg, Homoclinic solutions for Swift-Hohenberg and suspension bridge type equations, J. Differential Equations, 184 (2002), pp. 78--96.
O. Trichtchenko, B. Deconinck, and R. Kollár, Stability of periodic traveling waves solutions to the Kawahara equation, SIAM J. Appl. Dyn., 17 (2018), pp. 2761--2783.
J. van den Berg, M. Breden, J.-P. Lessard, and M. Murray, Continuation of homoclinic orbits in the suspension bridge equation: A computer-assisted proof, J. Differential Equations, 264 (2018), pp. 3086--3130.
V. Vougalter and D. Pelinovsky, Eigenvalues of zero energy in the linearized NLS problem, J. Math. Phys., 47 (2006), 062701.
A. Weinstein, Normal modes for nonlinear Hamiltonian systems, Invent. Math., 20 (1973), pp. 47--57.
K. Yosida, Functional Analysis, Springer-Verlag, Berlin, 1965.

Information & Authors


Published In

cover image SIAM Journal on Mathematical Analysis
SIAM Journal on Mathematical Analysis
Pages: 4705 - 4750
ISSN (online): 1095-7154


Submitted: 4 February 2019
Accepted: 29 June 2020
Published online: 30 September 2020


  1. Krein matrix
  2. multipulses
  3. Hamiltonian PDEs

MSC codes

  1. 35P30
  2. 47A55
  3. 47A56
  4. 70H14



Funding Information

National Science Foundation https://doi.org/10.13039/100000001 : DMS-1108783, DMS-1148284, DMS-1714429

Metrics & Citations



If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.

View Options

View options


View PDF







Copy the content Link

Share with email

Email a colleague

Share on social media