Existence of All Wilton Ripples of the Kawahara Equation

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.

💡 Research Summary

The paper addresses the long‑standing open problem of proving the existence of Wilton ripple solutions—periodic traveling waves with a two‑mode resonance 1:K—for the Kawahara equation, a fifth‑order dispersive PDE that models shallow water gravity‑capillary waves near a critical Bond number. After a Galilean shift and nondimensional scaling, the authors reduce the problem to the normalized equation

  c u + u_{xx} + β u_{xxxx} + u² = 0,

with periodic, even solutions of period 2π. Linearizing about the trivial state yields the operator L = ∂{xx} + β∂{xxxx}. Choosing the wave speed c₀ = 1 − β makes the kernel of (c₀ + L) two‑dimensional, spanned by cos x and cos Kx for any integer K ≥ 2. This is precisely the codimension‑one bifurcation that defines a Wilton ripple.

The authors perform a Lyapunov–Schmidt reduction. They decompose a candidate solution as

  u(x) = a cos x + b cos Kx + u_r(x),

where (a,b) are small amplitudes and u_r lies in the orthogonal complement of the kernel. Introducing projection operators P (onto the kernel) and Q (onto the range), the original equation F(u,c)=0 is split into an auxiliary equation Q F=0 and a bifurcation equation P F=0. The auxiliary equation is solved via the analytic Implicit Function Theorem, yielding a real‑analytic map

  u_r = u_r(x; a, b, c_r)

with c = c₀ + c_r. The map is expanded in a multivariate power series in (a,b,c_r).

The core of the analysis is the reduced bifurcation equations obtained by applying P. These give two scalar equations for a, b, and c_r. The authors show that b can be expressed as a real‑analytic function b(a) whose leading non‑zero term appears at order a^{K‑2}. To establish that the coefficient is non‑zero for every K, they carry out high‑order formal expansions (previously derived in the literature) and then rigorously justify each term using the analyticity of u_r. This step overcomes the main obstacle that prevented earlier works from treating K > 2.

The main result, Theorem 1, is presented in three parts:

1. K = 2 (β = 1/5): Two symmetric families of solutions u⁺ and u⁻ exist for small a. Their speeds are c_{±}=4/5 ± a√2 + O(a²). The wave profiles contain a leading cos x term, a cos 2x term with amplitude ±(1/√2) b_{±}(a) (b_{±}(0)=±1), and higher‑order corrections that are analytic in (a, b, c_r).

2. K = 3 (β = 1/10): Three distinct families u^{(σ)} (σ=1,2,3) arise. The speed expands as c_{σ}=9/10 + a² c̃_{σ}(a) with distinct leading constants (≈4.27863, 1.48289, 0.37396). The cos 3x amplitude b_{σ}(a) also has distinct non‑zero limits at a=0 (≈−1.78374, −0.54488, −0.59468). All higher‑order terms are analytic.

3. K ≥ 4: A single family exists. The speed is c = K²/(K²+1) + a² c̃(a) with c̃(0)= (K²+1)(5K²−24) /