Bounds on unstable spectrum for dispersive Hamiltonian PDEs
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916453868896256 |
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| author | Bronski, Jared C Hur, Ver Mikyoung Simpson, Sarah E |
| author_facet | Bronski, Jared C Hur, Ver Mikyoung Simpson, Sarah E |
| contents | We study quasi-periodic eigenvalue problems that arise in the stability analysis of periodic traveling wave solutions to Hamiltonian PDEs. We establish bounds on regions in the complex plane when the eigenvalues may deviate from the imaginary axis, and estimates for the number of such off-axis eigenvalues. These relations hold when the dispersion relation grows sufficiently rapidly in the wavenumber. The proofs involve a Gershgorin disk argument together with the Hamiltonian symmetry of the spectrum. The results are applicable to a broad class of nonlinear dispersive equations including the generalized Korteweg--de Vries, Benjamin--Bona--Mahoney, and Kawanhara equations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_19113 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Bounds on unstable spectrum for dispersive Hamiltonian PDEs Bronski, Jared C Hur, Ver Mikyoung Simpson, Sarah E Analysis of PDEs Pattern Formation and Solitons 35B35, 37K45 We study quasi-periodic eigenvalue problems that arise in the stability analysis of periodic traveling wave solutions to Hamiltonian PDEs. We establish bounds on regions in the complex plane when the eigenvalues may deviate from the imaginary axis, and estimates for the number of such off-axis eigenvalues. These relations hold when the dispersion relation grows sufficiently rapidly in the wavenumber. The proofs involve a Gershgorin disk argument together with the Hamiltonian symmetry of the spectrum. The results are applicable to a broad class of nonlinear dispersive equations including the generalized Korteweg--de Vries, Benjamin--Bona--Mahoney, and Kawanhara equations. |
| title | Bounds on unstable spectrum for dispersive Hamiltonian PDEs |
| topic | Analysis of PDEs Pattern Formation and Solitons 35B35, 37K45 |
| url | https://arxiv.org/abs/2410.19113 |