Validated Spectral Stability via Conjugate Points

Margaret Beck, Jonathan Jaquette

Research output: Contribution to journalArticlepeer-review

Abstract

Classical results from Sturm-Liouville theory state that the number of unstable eigenvalues of a scalar, second-order linear operator is equal to the number of associated conjugate points. Recent work has extended these results to a much more general setting, thus allowing for spectral stability of nonlinear waves in a variety of contexts to be determined by counting conjugate points. However, in practice, it is not yet clear whether it is easier to compute conjugate points than to just directly count unstable eigenvalues. We address this issue by developing a framework for the computation of conjugate points using validated numerics. Moreover, we apply our method to a parameter-dependent system of bistable equations and show that there exist both stable and unstable standing fronts. This application can be seen as complementary to the classical result via Sturm-Liouville theory that in scalar reaction-diffusion equations pulses are unstable whereas fronts are stable and to the more recent result of Beck et al. [Philos. Trans. Roy. Soc. A, 376 (2018), 20170187] that symmetric pulses in reaction-diffusion systems with gradient nonlinearity are also necessarily unstable.

Original languageEnglish (US)
Pages (from-to)366-404
Number of pages39
JournalSIAM Journal on Applied Dynamical Systems
Volume21
Issue number1
DOIs
StatePublished - 2022
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Analysis
  • Modeling and Simulation

Keywords

  • computer assisted proof
  • conjugate points
  • Maslov index
  • reaction-diffusion systems
  • stability

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