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Bounds on unstable spectrum for dispersive Hamiltonian PDEs

Jared C Bronski, Ver Mikyoung Hur, Sarah E Simpson

Abstract

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.

Bounds on unstable spectrum for dispersive Hamiltonian PDEs

Abstract

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.

Paper Structure

This paper contains 7 sections, 6 theorems, 61 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The main result
  3. Examples
  4. The gKdV equation
  5. The Kawahara equation
  6. The BBM equation
  7. Concluding remarks

Key Result

Theorem 2.1

For each $\mu\in(-\frac{1}{2},\frac{1}{2}]$, a nonzero eigenvalue of eqn: General eigenvalue problem-eqn: Floquet quasiperiodic BCs must be contained in ${ \bigcup_{k\in\mathbb{Z}} D_k(\mu)}$, where the $k$-th Gershgorin (topological) disk is defined as Furthermore, any connected component consisting of $n$ intersecting Gershgorin disks must contain precisely $n$ eigenvalues. In particular, if a

Figures (3)

  • Figure 1: The Gershgorin disks and the eigenvalues for \ref{['eqn:mKdVStable']} and \ref{['eqn: Floquet quasiperiodic BCs']} for several values of $\mu$, where $\phi$ is in \ref{['eqn:phi(gKdV)']} for $A=1$ and $m=\frac{1}{2}$. The Gershgorin disks are shown in blue, and the eigenvalues in red. The insets show the essential spectrum (magenta) as well as the eigenvalues. The largest connected component of Gershgorin disks consists of four or five overlapping disks, depending on the value of $\mu$, leading to two or four eigenvalues off the imaginary axis.
  • Figure 2: The Gershgorin disks (blue), the eigenvalues (red), and the essential spectrum for the linearized Kawahara equation about a stationary periodic solution, for $\mu=0$, $0.1$, $0.2$, $0.3$, $0.4$, and $0.5$. The insets highlight the essential spectrum (magenta) and eigenvalues in the vicinity of $0$, where $\operatorname{Re}(\lambda)\in(-0.025,0.025)$ and $\operatorname{Im}(\lambda) \in (-0.2,0.2)$.
  • Figure 3: The Gershgorin disks and the spectrum for \ref{['eqn:BBM']}, for $\mu=0,0.1,0.2,0.3,0.4,0.5$. The disks outlined in red ($|k|>9$) are disjoint from all other disks for all values of $\mu$.

Theorems & Definitions (14)

  • Theorem 2.1
  • proof
  • Theorem 2.2
  • proof
  • Remark
  • Corollary 2.3
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • ...and 4 more