arXiv subjects

Ryan P. Creedon

Publications and source records attributed to Ryan P. Creedon.

4 recordsLinked to original sources

arXiv2026

Sheets of Spectral Data of Stokes Waves in Weakly Nonlinear Models

We study the spectral stability of small-amplitude Stokes waves in a family of weakly nonlinear, unidirectional models of the form $u_t + L u + (u^2)_x = 0$. We introduce a perturbation method to expand the spectral data in wave amplitude near flat-state eigenvalue collisions, with the ratio of the colliding modes as a free parameter. This yields sheets of spectral data whose slices at fixed amplitude give isolas of instability. The same perturbation framework treats both high-frequency and Benjamin--Feir instabilities, extends to discontinuous dispersion relations (including the Akers--Milewski equation), and, for the first time, provides an analytic approximation of the Benjamin--Feir spectrum for this model and a direct comparison of high-frequency and Benjamin--Feir growth rates across the full family of models. Asymptotic predictions are validated against numerical spectra computed by Floquet--Fourier--Hill and quasi-Newton methods.

math.AP↗

arXiv2026

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.

math.AP↗

arXiv2026

Transverse Instability of Stokes Waves at Finite Depth

A Stokes wave is a traveling free-surface periodic water wave that is constant in the direction transverse to the direction of propagation. In 1981 McLean discovered via numerical methods that Stokes waves are unstable with respect to transverse perturbations. In \cite{CreNguStr} for the case of infinite depth we proved rigorously that the spectrum of the water wave system linearized at small Stokes waves, with respect to transverse perturbations, contains unstable eigenvalues lying approximately on an ellipse. In this paper we consider the case of finite depth and prove that the same spectral instability result holds for all but finitely many values of the depth. The computations and some aspects of the theory are considerably more complicated in the finite depth case.

math.AP↗

arXiv2023

Proof of the transverse instability of Stokes waves

A Stokes wave is a traveling free-surface periodic water wave that is constant in the direction transverse to the direction of propagation. In 1981 McLean discovered via numerical methods that Stokes waves at infinite depth are unstable with respect to transverse perturbations of the initial data. Even for a Stokes wave that has very small amplitude $\varepsilon$, we prove rigorously that transverse perturbations, after linearization, will lead to exponential growth in time. To observe this instability, extensive calculations are required all the way up to order $O(\varepsilon^3)$. All previous rigorous results of this type were merely two-dimensional, in the sense that they only treated long-wave perturbations in the longitudinal direction. This is the first rigorous proof of three-dimensional instabilities of Stokes waves.

math.AP↗