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Existence of pure capillary solitary waves in constant vorticity flows

Ting-Yang Hsiao, Zhengjun Liang, Giang To, Ye Zhang

TL;DR

This work establishes the existence of finite-depth pure-capillary solitary waves for the 2D Euler equations with constant vorticity. It develops a spatial-dynamics Hamiltonian framework, uses a nonlinear change of variables to flatten the free surface and puts the symplectic form in Darboux coordinates, and applies a center-manifold reduction to obtain a finite-dimensional reversible Hamiltonian system. Through a KdV-type normal form and long-wave scaling, it constructs a reversible homoclinic orbit whose persistence yields solitary waves in the full Euler dynamics. The results isolate constant vorticity as the mechanism enabling solitary waves in the pure-capillary regime and connect the full dynamics to a KdV-type profile in the small-parameter limit, providing explicit leading-order asymptotics for the wave shape.

Abstract

We prove the existence of pure capillary solitary waves for the 2D finite-depth Euler equations with nonzero constant vorticity. In the irrotational case, nonexistence of solitary waves was established by Ifrim--Pineau--Tataru--Taylor, so our theorem isolates constant vorticity as a mechanism that enables solitary waves in the pure-capillary regime. The proof uses a spatial-dynamics Hamiltonian formulation of the travelling-wave equations and a nonlinear change of variables that flattens the free surface while putting the symplectic form into Darboux coordinates. Near a distinguished curve in the vorticity--capillarity parameter space, the linearization has a two-dimensional center subspace; a parameter-dependent center-manifold reduction yields a canonical planar Hamiltonian system. A cubic normal-form expansion and long-wave scaling produce a KdV-type profile equation with a reversible homoclinic orbit, which persists under the full dynamics and generates the solitary-wave solutions.

Existence of pure capillary solitary waves in constant vorticity flows

TL;DR

This work establishes the existence of finite-depth pure-capillary solitary waves for the 2D Euler equations with constant vorticity. It develops a spatial-dynamics Hamiltonian framework, uses a nonlinear change of variables to flatten the free surface and puts the symplectic form in Darboux coordinates, and applies a center-manifold reduction to obtain a finite-dimensional reversible Hamiltonian system. Through a KdV-type normal form and long-wave scaling, it constructs a reversible homoclinic orbit whose persistence yields solitary waves in the full Euler dynamics. The results isolate constant vorticity as the mechanism enabling solitary waves in the pure-capillary regime and connect the full dynamics to a KdV-type profile in the small-parameter limit, providing explicit leading-order asymptotics for the wave shape.

Abstract

We prove the existence of pure capillary solitary waves for the 2D finite-depth Euler equations with nonzero constant vorticity. In the irrotational case, nonexistence of solitary waves was established by Ifrim--Pineau--Tataru--Taylor, so our theorem isolates constant vorticity as a mechanism that enables solitary waves in the pure-capillary regime. The proof uses a spatial-dynamics Hamiltonian formulation of the travelling-wave equations and a nonlinear change of variables that flattens the free surface while putting the symplectic form into Darboux coordinates. Near a distinguished curve in the vorticity--capillarity parameter space, the linearization has a two-dimensional center subspace; a parameter-dependent center-manifold reduction yields a canonical planar Hamiltonian system. A cubic normal-form expansion and long-wave scaling produce a KdV-type profile equation with a reversible homoclinic orbit, which persists under the full dynamics and generates the solitary-wave solutions.
Paper Structure (18 sections, 14 theorems, 238 equations)

This paper contains 18 sections, 14 theorems, 238 equations.

Table of Contents

  1. Introduction
  2. Free-boundary Euler equations and capillary water waves with constant vorticity
  3. Previous work on solitary waves and our the main result
  4. Main ideas and comparison with related work
  5. Organization of the paper
  6. Notations and conventions
  7. The derivation of Hamiltonian formulation
  8. Infinite-dimensional dynamical system
  9. Hamiltonian reformulation theorem
  10. Center Manifold: The Reduced Space
  11. Linearized boundary conditions
  12. Linearized operator $\mathbf{L}$
  13. Mielke’s center-manifold theorem
  14. The Korteweg-de Vries limit
  15. Resolvent estimate
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $c \neq 0$. Suppose ${\omega d}/{c}=1+\varepsilon$ for a sufficiently small absolute constant $\varepsilon > 0$ and ${\sigma}/{(c^{2}d)}>\frac{1}{3}$, the system (main second) admits a $C^2$ solitary wave solution with velocity $c$. Moreover, the solitary wave profile $\eta$ has the asymptotic e where $Q(\bar{x})=-3\mathrm{sech}^2(\bar{x}/2)$ solves the steady the KdV equation

Theorems & Definitions (25)

  • Theorem 1.1
  • Lemma 3.1
  • proof
  • Theorem 3.1
  • proof
  • Lemma 4.1
  • Lemma 4.2
  • proof
  • Lemma 4.3
  • proof
  • ...and 15 more