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Existence of All Wilton Ripples of the Kawahara Equation

Ryan P. Creedon

Abstract

We investigate the existence of Wilton ripple solutions of the Kawahara equation. Without loss of generality, these are $2π$-periodic, traveling-wave solutions whose profiles at zero amplitude have a codimension-1 bifurcation from a linear combination of $\cos(x)$ and $\cos(Kx)$ for $K \in \mathbb{N} \setminus \{1\}$. Using a Lyapunov-Schmidt reduction, we prove the existence of these solutions for all $K$, in contrast to previous work demonstrating existence only for $K = 2$. Although the proof holds only for the Kawahara equation, many ideas introduced in the proof can be applied to more general contexts, including Wilton ripples of the gravity-capillary water wave equations.

Existence of All Wilton Ripples of the Kawahara Equation

Abstract

We investigate the existence of Wilton ripple solutions of the Kawahara equation. Without loss of generality, these are -periodic, traveling-wave solutions whose profiles at zero amplitude have a codimension-1 bifurcation from a linear combination of and for . Using a Lyapunov-Schmidt reduction, we prove the existence of these solutions for all , in contrast to previous work demonstrating existence only for . Although the proof holds only for the Kawahara equation, many ideas introduced in the proof can be applied to more general contexts, including Wilton ripples of the gravity-capillary water wave equations.

Paper Structure

This paper contains 11 sections, 2 theorems, 93 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Proof of Theorem 1
  3. Notation and Set-Up
  4. Solving the Auxiliary Equation
  5. Solving the Bifurcation Equation
  6. Case 1: $K = 2$
  7. Case 2: $K = 3$
  8. Case 3: $K \geq 4$
  9. Appendix: The Taylor Coefficients $u_r^{(i,j,k)}(x)$
  10. Case 1: $K = 2$
  11. Case 2: $K \geq 3$

Key Result

Theorem 1

For each $K \in \mathbb{N} \setminus \{1\}$, the Kawahara equation has the following Wilton ripple solutions:

Figures (1)

  • Figure 1: (a) A Stokes wave of the Kawahara equation for $\beta = 1/2$, (b) A Wilton ripple of the Kawahara equation for $\beta = 1/5$, and (c) A Wilton ripple of the Kawahara equation for $\beta = 1/10$. In all plots, the amplitude parameter $a = 0.01$. The blue dots are numerically computed wave profiles using the continuation method presented in trichtchenko2018stability. The red curves are the leading-order asymptotic behavior of the wave profiles according to Theorem 1 below. The wave profiles are normalized by $a$ for ease of comparisons.

Theorems & Definitions (3)

  • Theorem 1
  • Lemma 1
  • proof