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Asymmetric travelling wave solutions of the capillary-gravity Whitham Equation

Ola Mæhlen, Douglas Svensson Seth

Abstract

By a bifurcation argument we prove that the capillary-gravity Whitham equation features asymmetrical periodic travelling wave solution of arbitrarily small amplitude. Such waves exist only in the weak surface tension regime $0<T<\frac{1}{3}$ and are necessarily bimodal; they are located at double bifurcation points satisfying a certain symmetry breaking condition. Our bifurcation argument is an extension of the one applied by Ehrnström et al to find symmetric waves: Here, two additional scalar equations arise. Combining the variational structure of our problem with its translation symmetry, we show that these two additional equations are in fact linearly dependent, and can (at 'symmetry breaking' bifurcation points) be solved by incorporating the surface tension as a bifurcation parameter. Contrary to the symmetric case, only very specific modal pairs $(k_1,k_2)$ give rise to (small) asymmetrical periodic waves and we here provide a partial characterization of such pairs.

Asymmetric travelling wave solutions of the capillary-gravity Whitham Equation

Abstract

By a bifurcation argument we prove that the capillary-gravity Whitham equation features asymmetrical periodic travelling wave solution of arbitrarily small amplitude. Such waves exist only in the weak surface tension regime and are necessarily bimodal; they are located at double bifurcation points satisfying a certain symmetry breaking condition. Our bifurcation argument is an extension of the one applied by Ehrnström et al to find symmetric waves: Here, two additional scalar equations arise. Combining the variational structure of our problem with its translation symmetry, we show that these two additional equations are in fact linearly dependent, and can (at 'symmetry breaking' bifurcation points) be solved by incorporating the surface tension as a bifurcation parameter. Contrary to the symmetric case, only very specific modal pairs give rise to (small) asymmetrical periodic waves and we here provide a partial characterization of such pairs.
Paper Structure (20 sections, 27 theorems, 122 equations, 3 figures)

This paper contains 20 sections, 27 theorems, 122 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Setup and main results
  3. Formulating the problem
  4. Characterizing the bifurcation points
  5. Lyapunov--Schmidt reduction
  6. The symmetry breaking criterion
  7. Main results
  8. Preliminaries
  9. Zygmund spaces
  10. Smoothing effects and Fredholm property of the linear part
  11. Properties of the Fourier symbol
  12. Asymmetry of sums of sinusoids
  13. Symmetry-preservation of Lyapunov--Schmidt reduction
  14. Proof of the Existence Theorem
  15. Lyapunov--Schmidt reduction
  16. ...and 5 more sections

Key Result

Theorem 1.2

There exist asymmetric periodic travelling wave solutions to the Whitham equation with arbitrarily small amplitude. These are found only near double bifurcation points that satisfy a symmetry breaking condition. The wave number pair $(k_1,k_2)=(2,5)$ is the smallest one that admits a corresponding '

Figures (3)

  • Figure 1: Respective plots of the domains $[0,\infty)^2$ and $\Theta=\left(\mathbb{R}/{\pi}\mathbb{Z}\right)\times \left(\mathbb{R}/\tfrac{2\pi}{5}\mathbb{Z}\right)$. Blue regions denote points where $r_1r_2=0$ or $\theta_1-\theta_2\in \frac{\pi}{10}\mathbb{Z}$; the latter set can be split in $\theta_1-\theta_2\in \frac{\pi}{10}(2\mathbb{Z})$ and $\theta_1-\theta_2\in \frac{\pi}{10}(\mathbb{Z}\setminus2\mathbb{Z})$ marked by solid and dotted lines respectively.
  • Figure 2: Plot of the function $T\mapsto\phi(T;2,5)$ on the interval $(0,1/3)$. Here $\phi(T_0;2,5)=0$ for a value $T_0\approx 0.1215$.
  • Figure 3: The dots mark coprime pairs $1\leq k_1<k_2$ for which we obtained different signs in the limits \ref{['eq:phiLims']} by a numerical computation. Thus all plotted wave number pairs admit symmetry breaking.

Theorems & Definitions (65)

  • Definition 1.1: Symmetric and asymmetric functions
  • Theorem 1.2: Summary of main result
  • Definition 2.1: Simple and double bifurcation points
  • Remark 2.2
  • Definition 2.3: Symmetry breaking criterion
  • Remark 2.4
  • Remark 2.5
  • Theorem 2.6
  • Remark 2.7
  • Theorem 2.8
  • ...and 55 more