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On non-uniqueness of solitary waves on two-dimensional rotational flow

Vladimir Kozlov

TL;DR

The paper addresses non-uniqueness of solitary waves on a two-dimensional rotational flow by proving that, near the first bifurcation point on the solitary-wave branch, infinitely many solitary-wave pairs share the same Bernoulli constant $R$. It develops a new bifurcation-based approach using a partial hodograph transform to reformulate the problem into an analytic, Fredholm framework, enabling both local and global bifurcation analyses and the construction of multiple bifurcating branches. A key contribution is showing that, as the branch approaches the extreme wave, the multiplicity persists, extending previous irrotational results to rotational flows. The results rely on a rigorous spectral analysis of the linearized operator, a priori estimates, and an abstract global bifurcation theory, providing a detailed description of the global structure of the solution set and the mechanism for non-uniqueness with the same $R$. This work has implications for understanding the multiplicity and structure of water-wave solutions in vortical regimes and introduces techniques applicable to related free-surface flow problems.

Abstract

We consider solitary water waves on a rotational, unidirectional flow in a two-dimensional channel of finite depth. Ovsyannikov has conjectured in 1983 that the solitary wave is uniquely determined by the Bernoulli constant, mass flux and by the flow force. This conjecture was disproved by Plotnikov in 1992 for the ir-rotational flow. In this paper we show that this conjecture is wrong also for rotational flows. Moreover we prove that in any neighborhood of the first bifurcation point on the branch of solitary waves, approaching the extreme wave, there are infinitely many pairs of solitary waves corresponding to the same Bernoulli constant. We give a description of the structure of this set of pairs. The proof is based on a bifurcation analysis of the global branch of solitary waves which is of independent interest.

On non-uniqueness of solitary waves on two-dimensional rotational flow

TL;DR

The paper addresses non-uniqueness of solitary waves on a two-dimensional rotational flow by proving that, near the first bifurcation point on the solitary-wave branch, infinitely many solitary-wave pairs share the same Bernoulli constant . It develops a new bifurcation-based approach using a partial hodograph transform to reformulate the problem into an analytic, Fredholm framework, enabling both local and global bifurcation analyses and the construction of multiple bifurcating branches. A key contribution is showing that, as the branch approaches the extreme wave, the multiplicity persists, extending previous irrotational results to rotational flows. The results rely on a rigorous spectral analysis of the linearized operator, a priori estimates, and an abstract global bifurcation theory, providing a detailed description of the global structure of the solution set and the mechanism for non-uniqueness with the same . This work has implications for understanding the multiplicity and structure of water-wave solutions in vortical regimes and introduces techniques applicable to related free-surface flow problems.

Abstract

We consider solitary water waves on a rotational, unidirectional flow in a two-dimensional channel of finite depth. Ovsyannikov has conjectured in 1983 that the solitary wave is uniquely determined by the Bernoulli constant, mass flux and by the flow force. This conjecture was disproved by Plotnikov in 1992 for the ir-rotational flow. In this paper we show that this conjecture is wrong also for rotational flows. Moreover we prove that in any neighborhood of the first bifurcation point on the branch of solitary waves, approaching the extreme wave, there are infinitely many pairs of solitary waves corresponding to the same Bernoulli constant. We give a description of the structure of this set of pairs. The proof is based on a bifurcation analysis of the global branch of solitary waves which is of independent interest.
Paper Structure (17 sections, 26 theorems, 200 equations)

This paper contains 17 sections, 26 theorems, 200 equations.

Table of Contents

  1. Introduction
  2. Statement of the main result
  3. Main steps of the proof. Global branch of solitary waves and its first bifurcation.
  4. Uniform stream solutions and the dispersion equation
  5. Partial hodograph transform
  6. Some properties of solitary waves
  7. Uniform boundedness of solitary waves in ${\mathcal{U}}_\delta$
  8. Uniform exponential decay of functions in ${\mathcal{U}}_\delta$
  9. Abstract bifurcation analysis
  10. Small bifurcation curves
  11. Global bifurcation, continuation of local bifurcation curve.
  12. The first bifurcation on the branch of solitary waves
  13. Bifurcation equation
  14. Lemmas
  15. Proofs of Theorems \ref{['TrF18b']} and \ref{['ThF18']}
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $\omega$ satisfy (Bov20a). Then in any neighborhood of the first bifurcation point on the branch of solitary wave solutions to the problem (K2a), approaching the extreme wave, there are infinitely many pairs of solitary waves corresponding to the same Bernoulli constant.

Theorems & Definitions (39)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Proposition 1.4
  • Remark 1.5
  • Proposition 1.6
  • Theorem 1.7
  • Proposition 1.8
  • Theorem 1.9
  • Theorem 1.10
  • ...and 29 more