Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A note on the periodic Hilbert Transform on a strip

Javier Gómez-Serrano, Sieon Kim

Abstract

In this note we prove a conjecture by Constantin--Strauss--Vărvărucă related to the finite depth water wave problem, tightening their results. The proof uses identities related to Jacobi Theta functions. We also discuss potential implications of the improvement.

A note on the periodic Hilbert Transform on a strip

Abstract

In this note we prove a conjecture by Constantin--Strauss--Vărvărucă related to the finite depth water wave problem, tightening their results. The proof uses identities related to Jacobi Theta functions. We also discuss potential implications of the improvement.

Paper Structure

This paper contains 2 sections, 4 theorems, 27 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['main_thm']}

Key Result

Theorem 1.2

Conjecture main_conjecture is true.

Figures (1)

  • Figure 1: The function $\beta_d\left(\frac{\pi}{2}\right)$ as a function of $x$.

Theorems & Definitions (8)

  • Conjecture 1.1: Constantin-Strauss-Varvaruca:large-amplitude-steady-downstream-water-waves Lemma 1, Remark, p.252, abridged, see also Haziot-Strauss:amplitude-bounds-steady-rotational-water-waves Lemma 2)
  • Theorem 1.2
  • Remark 1.3
  • Corollary 1.4: Strengthening of Theorem 4 of Constantin-Strauss-Varvaruca:large-amplitude-steady-downstream-water-waves
  • Corollary 1.5: Strengthening of Theorem 1.3 of Haziot-Strauss:amplitude-bounds-steady-rotational-water-waves
  • proof
  • Lemma 2.1
  • proof