Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Spectral properties of the Frechet derivatives of stratified steady Stokes waves

Vladimir Kozlov

TL;DR

The paper analyzes local uniqueness of two-dimensional stratified steady Stokes waves by examining the spectral properties of the Frechet derivative of the water-wave operator. Using partial hodograph and boundary-flattening transforms, it reduces the free-boundary problem to fixed-domain spectral problems and derives detailed eigenvalue estimates. The key findings show that the first eigenvalue is always negative and, under a positive second eigenvalue assumption, there are no subharmonic or multi-periodic waves near a Stokes wave; a simple zero eigenvalue with an odd eigenfunction is identified for a Floquet parameter, enabling precise uniqueness conclusions. These results contribute to the understanding of local wave-structure uniqueness in stratified, rotational flows and connect to broader conjectures on wave parameter estimates in the Benjamin-Lighthill framework.

Abstract

We consider stratified steady water waves in a two dimensional channel. Our main subject is spectral properties of the Frechet derivatives of steady water Stokes waves. One of main results is the absence of subharmonic water waves in a neighborhood of a Stokes wave. The main assumption is formulated in terms of the eigenvalues of the Frechet derivative evaluated at this wave and considered in the class of periodic solutions of the same period. The first eigenvalue is always negative. We show that if the second eigenvalue is positive then there are no waves with multiple periods in a neighborhood of the Stokes wave.

Spectral properties of the Frechet derivatives of stratified steady Stokes waves

TL;DR

The paper analyzes local uniqueness of two-dimensional stratified steady Stokes waves by examining the spectral properties of the Frechet derivative of the water-wave operator. Using partial hodograph and boundary-flattening transforms, it reduces the free-boundary problem to fixed-domain spectral problems and derives detailed eigenvalue estimates. The key findings show that the first eigenvalue is always negative and, under a positive second eigenvalue assumption, there are no subharmonic or multi-periodic waves near a Stokes wave; a simple zero eigenvalue with an odd eigenfunction is identified for a Floquet parameter, enabling precise uniqueness conclusions. These results contribute to the understanding of local wave-structure uniqueness in stratified, rotational flows and connect to broader conjectures on wave parameter estimates in the Benjamin-Lighthill framework.

Abstract

We consider stratified steady water waves in a two dimensional channel. Our main subject is spectral properties of the Frechet derivatives of steady water Stokes waves. One of main results is the absence of subharmonic water waves in a neighborhood of a Stokes wave. The main assumption is formulated in terms of the eigenvalues of the Frechet derivative evaluated at this wave and considered in the class of periodic solutions of the same period. The first eigenvalue is always negative. We show that if the second eigenvalue is positive then there are no waves with multiple periods in a neighborhood of the Stokes wave.

Paper Structure

This paper contains 14 sections, 8 theorems, 152 equations.

Table of Contents

  1. Introduction
  2. Statement of the main result
  3. The Frechet derivative
  4. Results
  5. Proofs
  6. Partial hodograph transformation
  7. The Frechet derivative (\ref{['J25ab']}) in $(x,y)$ variables
  8. Flattening the boundary in the second case
  9. Proofs of Proposition \ref{['Pm23']} and the negativity of $\mu_1$
  10. Estimates of eigenvalues
  11. Proof of Theorem \ref{['TOkt13a']}
  12. Proof of Corollary \ref{['CorOkt22']}
  13. Generalized eigenfunctions for the eigenvalue $z=0$
  14. References

Key Result

Proposition 1.2

Assume that (Okt5ba) is valid. Let $a$ and $b$ be continuous, even, positive and $\Lambda$-periodic functions defined in $\overline{D}$ and in $\Bbb R$ respectively. Consider the spectral problems and where $\mu$ and $\theta$ are spectral parameters and $u$ is an even, $\Lambda$--periodic function. Then these problems have the same number of negative eigenvalues (accounting their multiplicities)

Theorems & Definitions (17)

  • Remark 1.1
  • Proposition 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Lemma 2.1
  • proof
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Corollary 2.4
  • ...and 7 more