Geodesic
The Algebraic Multiplicity of Eigenvalues and the Evans Function Revisited
Y. Latushkin1 ; A. Sukhtayev1
1 Department of Mathematics, University of Missouri, Columbia, MO 65211, USA
Mathematical modelling of natural phenomena, Tome 5 (2010) no. 4, pp. 269-292

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This paper is related to the spectral stability of traveling wave solutions of partial differential equations. In the first part of the paper we use the Gohberg-Rouche Theorem to prove equality of the algebraic multiplicity of an isolated eigenvalue of an abstract operator on a Hilbert space, and the algebraic multiplicity of the eigenvalue of the corresponding Birman-Schwinger type operator pencil. In the second part of the paper we apply this result to discuss three particular classes of problems: the Schrödinger operator, the operator obtained by linearizing a degenerate system of reaction diffusion equations about a pulse, and a general high order differential operator. We study relations between the algebraic multiplicity of an isolated eigenvalue for the respective operators, and the order of the eigenvalue as the zero of the Evans function for the corresponding first order system.
DOI : 10.1051/mmnp/20105412

Y. Latushkin 1 ; A. Sukhtayev 1

1 Department of Mathematics, University of Missouri, Columbia, MO 65211, USA 
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Y. Latushkin; A. Sukhtayev. The Algebraic Multiplicity of Eigenvalues and the Evans Function Revisited. Mathematical modelling of natural phenomena, Tome 5 (2010) no. 4, pp. 269-292. doi: 10.1051/mmnp/20105412

[1] J. Alexander, R. Gardner, C. Jones A topological invariant arising in the stability analysis of traveling waves J. reineangew. Math. 1990 167 212

[2] M. S. Birman, M. Z. Solomyak.Spectral theory of self-adjoint operators in Hilbert space. Reidel, Dordrecht, 1987.

[3] C. Chicone, Y. Latushkin.Evolution semigroups in dynamical systems and differential equations. Amer. Math. Soc., Providence, RI, 1999.

[4] R.A. Gardner, C. K. R. T. Jones Traveling waves of a perturbed diffusion equation arising in a phase field model Indiana Univ. Math. J. 1989 1197 1222

[5] F. Gesztesy, Y. Latushkin, K. A. Makarov Evans functions, Jost functions, and Fredholm determinants Arch. Rat. Mech. Anal. 2007 361 421

[6] F. Gesztesy, Y. Latushkin, M. Mitrea, M. Zinchenko Non-self-adjoint operators, infinite determinants, and some applications Russ. J. Math. Phys. 2005 443 471

[7] F. Gesztesy, Y. Latushkin, K. Zumbrun Derivatives of (modified) Fredholm determinants and stability of standing and traveling waves J. Math. Pures Appl. 2008 160 200

[8] F. Gesztesy, K. A. Makarov (Modified ) Fredholm determinants for operators with matrix-valued semi-separable integral kernels revisited Integral Eq. Operator Theory 2003 457 497

[9] I. Gohberg, S. Goldberg, M. Kaashoek. Classes of linear operators. Vol. 1. Birkhäuser, 1990.

[10] K. F. Gurski, R. Kollar, R. L. Pego Slow damping of internal waves in a stably stratified fluid Proc. Royal Soc. Lond. Ser. A Math. Phys. Engrg. Sci. 2004 977 994

[11] K. F. Gurski, R. L. Pego Normal modes for a stratified viscous fluid layer Proc. Royal Soc. Edinburgh Sect. A 2002 611 625

[12] T. Kapitula, B. Sandstede Edge bifurcations for near integrable systems via Evans function techniques SIAM J. Math. Anal. 2002 1117 1143

[13] T. Kapitula, B. Sandstede Eigenvalues and resonances using the Evans function Discrete Contin. Dyn. Syst. 2004 857 869

[14] T. Kato Wave operators and similarity for some non-selfadjoint operators Math. Ann. 1966 258 279

[15] R. L. Pego, M. I. Weinstein Eigenvalues and instabilities of solitary waves Philos. Trans. Royal Soc. London Ser. A 1992 47 94

[16] M. Reed, B. Simon. Methods of modern mathematical physics. I: Functional analysis. Academic Press, New York, 1980.

[17] M. Reed, B. Simon.Methods of Modern Mathematical Physics. II: Fourier Analysis, Self-adjointness. Academic Press, New York, 1975.

[18] B. Sandstede. Stability of traveling waves. In: Handbook of dynamical systems. Vol. 2. B. Hasselblatt, A. Katok (eds.). North-Holland, Elsevier, Amsterdam, 2002, pp. 983–1055.

[19] B. Simon. Trace ideals and their applications. Cambridge University Press, Cambridge, 1979.

[20] K. Zumbrun. Multidimensional stability of planar viscous shock waves. In:Advances in the Theory of Shock Waves. T.-P. Liu, H. Freistühler, A. Szepessy (eds.). Progress Nonlin. Diff. Eqs. Appls.,47, Birkhäuser, Boston, 2001, pp. 307–516.

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