Stability bounds for nonlinear dispersive Hamiltonian partial differential equations

Existing approaches for analyzing the stability of periodic traveling wave solutions to dispersive partial differential equations (PDEs) often rely on numerical methods. However, due to computational limitations, these methods require an implicit assumption that all instabilities lie within a bounded region containing the origin. Recent research has revealed that this assumption does not always hold true, as in certain cases, the spectrum extends to infinity along curves having non-zero real part. Numerical methods are also susceptible to missing spectrum with small real part located further up the imaginary axis. To address these concerns, explicit bounds indicating the region within which all instabilities must be contained would be valuable. This dissertation focuses on providing such bounds for a specific subset of these problems, namely, dispersive, Hamiltonian PDEs. We provide explicit bounds for the region within which all instabilities must lie, along with giving an upper bound on the number of instabilities. For cases where the dispersion relation exhibits at least cubic growth rate these bounds are obtained via a Gershgorin disc theorem type argument coupled with application of the fact that the spectrum of Hamiltonian operators is symmetric with respect to reflection across both the real and imaginary axes. In instances where the growth rate of the dispersion relation is only quadratic we provide an alternative argument following the form of a second-order perturbation calculation to bound the desired region. We provide these results in a general form readily available for applications along with discussing several specific examples in detail including generalized Korteweg-de Vries, Benjamin-Bona-Mahony, Kawahara, and generalized Benjamin-Ono.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2023

Author dc:creator

• Simpson, Sarah E

Contributors dc:contributor

• Bronski, Jared
• DeVille, Lee
• Hur, Vera
• Rapti, Zoi

• Partial Differential Equations
• Dispersive Equations
• Hamiltonian Equations
• Gershgorin Disc Theorem
• Spectral Stability

Statement dc:rights

• Copyright 2023 Sarah E. Simpson

Language dc:language

en, eng

Handle dc:identifier

https://hdl.handle.net/2142/122014

OAI record oai:identifier

https://www.ideals.illinois.edu/oai-pmh?verb=GetRecord&metadataPrefix=etdms&identifier=oai%3Awww.ideals.illinois.edu%3A2142%2F122014