Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Large-Amplitude Steady Solitary Water Waves with General Vorticity

Jifeng Chu, Zihao Wang, Yong Zhang

Abstract

In this paper, we study two-dimensional steady solitary gravity waves propagating along the surface of a fluid of finite depth. In particular, we can deal with general vorticity distributions and overhanging wave profiles. By conformal mappings, we reformulate the problem into an overdetermined elliptic system coupled with an elliptic boundary value problem in a fixed strip domain. To avoid imposing extra constraints on vorticity function, we further reformulate the problem into the form of an abstract operator. Based on the formulations, the existence of small-amplitude solitary waves is proved by the center manifold reduction method, while the large-amplitude waves are obtained based on the analytic global bifurcation theorem.

Large-Amplitude Steady Solitary Water Waves with General Vorticity

Abstract

In this paper, we study two-dimensional steady solitary gravity waves propagating along the surface of a fluid of finite depth. In particular, we can deal with general vorticity distributions and overhanging wave profiles. By conformal mappings, we reformulate the problem into an overdetermined elliptic system coupled with an elliptic boundary value problem in a fixed strip domain. To avoid imposing extra constraints on vorticity function, we further reformulate the problem into the form of an abstract operator. Based on the formulations, the existence of small-amplitude solitary waves is proved by the center manifold reduction method, while the large-amplitude waves are obtained based on the analytic global bifurcation theorem.
Paper Structure (14 sections, 35 theorems, 262 equations, 1 figure)

This paper contains 14 sections, 35 theorems, 262 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Governing equations and reformulations
  3. Governing equations
  4. The laminar solution
  5. Conformal mapping
  6. Equivalent systems
  7. Linearization and Functional Analytic Formulation
  8. Existence of Small-amplitude Waves
  9. Velocity Fields and Nodal Properties
  10. Flow Forces and Compactness
  11. Global Continuum and Main Result
  12. Uniform Regularity
  13. Maximum Principles
  14. Schauder Estimates for Elliptic Problems in Infinite Strips

Key Result

Lemma 3.1

For all $\theta\in\tilde{\mathcal{X}},$ it holds that

Figures (1)

  • Figure 1: The phase portrait of equation \ref{['reducedq']} when $\varepsilon\to0^+$ and $M_0>0$.

Theorems & Definitions (64)

  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Example 3.4
  • Example 3.5
  • Lemma 3.6
  • proof
  • ...and 54 more