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On the first bifurcation of Stokes waves

Vladimir Kozlov

Abstract

We consider Stokes water waves on the vorticity flow in a two-dimensional channel of finite depth. In the paper "V.Kozlov, On first subharmonic bifurcations in a branch of Stokes waves, JDE, 2024," it was proved existence of subharmonic bifurcations on a branch of Stokes waves. Such bifurcations occur near the first bifurcation in the set of Stokes waves. Moreover it is shown in that paper that the bifurcating solutions build a connected continuum containing large amplitude waves. This fact was proved under a certain assumption concerning the second eigenvalue of the Frechet derivative. In this paper we investigate this assumption and present explicit conditions when it is satisfied.

On the first bifurcation of Stokes waves

Abstract

We consider Stokes water waves on the vorticity flow in a two-dimensional channel of finite depth. In the paper "V.Kozlov, On first subharmonic bifurcations in a branch of Stokes waves, JDE, 2024," it was proved existence of subharmonic bifurcations on a branch of Stokes waves. Such bifurcations occur near the first bifurcation in the set of Stokes waves. Moreover it is shown in that paper that the bifurcating solutions build a connected continuum containing large amplitude waves. This fact was proved under a certain assumption concerning the second eigenvalue of the Frechet derivative. In this paper we investigate this assumption and present explicit conditions when it is satisfied.
Paper Structure (13 sections, 2 theorems, 158 equations)

This paper contains 13 sections, 2 theorems, 158 equations.

Table of Contents

  1. Formulation of the problem
  2. Uniform stream solution, dispersion equation
  3. A connection between the functions $\mu(t)$ and $\Lambda(t)$ for small $t$
  4. Partial hodograph transform
  5. Bifurcation equation
  6. Stokes waves for small $t$
  7. Formula for $\lambda_2$ and the proof of the relation (\ref{['J3aa']})
  8. The coefficient $\lambda_2$ for the irrotational flow
  9. Asymptotic analysis
  10. Sign of $\lambda_2$
  11. Upper estimates of the Froude number and the validity of Assumption
  12. Acknowledgments
  13. References

Key Result

Theorem 1.1

Let Assumption be fulfilled. Then there exists an integer $M_0$ and pairs $(t_M,M)$, where $M$ is integer, $M>M_0$ and $t_M>t_0$, satisfying such that $t_M$ is $M$- subharmonic bifurcation point. There are no subharmonic bifurcations for $t<t_0$.

Theorems & Definitions (2)

  • Theorem 1.1
  • Proposition 1.2