The word bifurcation refers to changes in the number and stability of equilibria supported by a governing system as its parameters are varied. The classical bifurcation problem begins with an analysis of the point spectrum of the linearized operator associated with the equilibria under investigation. In this chapter we investigate finite-rank bifurcations for which a finite number of point eigenvalues cross the imaginary axis, either transversely or more degenerately, as the system parameters are varied. In particular, we derive the perturbative motion of such point spectra. This analysis is most informative in those cases for which the associated linearized operator initially has purely imaginary eigenvalues, and a small change in parameters moves the eigenvalues decisively off the imaginary axis.
Imaginary Axis Point Spectrum Translational Symmetry Orbital Stability Simple Eigenvalue
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J. Angulo. Nonlinear stability of periodic travelling wave solutions to the Schrödinger and the modified Korteweg–de Vries equations. J. Diff. Eq., 235:1–30, 2007.zbMATHCrossRefGoogle Scholar
J. Angulo and J. Quintero. Existence and orbital stability of cnoidal waves for a 1D Boussinesq equation. Int. J. Math. Math. Sci., 2007:52020, 2007.MathSciNetCrossRefGoogle Scholar
J. Bronski, M. Johnson, and T. Kapitula. An index theorem for the stability of periodic traveling waves of KdV type. Proc. Roy. Soc. Edinburgh: Section A, 141(6):1141–1173, 2011.MathSciNetzbMATHCrossRefGoogle Scholar
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
B. Deconinck and M. Nivala. The stability analysis of the periodic traveling wave solutions of the mkdv equation. Stud. Appl. Math., 126:17–48, 2010.MathSciNetCrossRefGoogle Scholar
T. Gallay and M. Hǎrǎguş. Orbital stability of periodic waves for the nonlinear Schrödinger equation. J. Dyn. Diff. Eqns., 19:825–865, 2007a.zbMATHCrossRefGoogle Scholar
T. Gallay and M. Hǎrǎguş. Stability of small periodic waves for the nonlinear Schrödinger equation. J. Diff. Eq., 234:544–581, 2007b.zbMATHCrossRefGoogle Scholar
M. Johnson. Nonlinear stability of periodic traveling wave solutions of the generalized Korteweg–de Vries equation. SIAM J. Math. Anal., 41(5): 1921–1947, 2009.MathSciNetzbMATHCrossRefGoogle Scholar