Special Session 95: The Euler Water Wave Problem

Rational approximation and branch cuts for standing water waves

Jon A Wilkening UC Berkeley USA Co-Author(s):    Ahmad Abassi

Abstract: We compute arbitrary-order asymptotic expansions of standing water waves, generalizing the conformal mapping framework of Schwartz and Whitney (1981) to handle finite-depth domains and/or surface tension effects. The calculations are performed in arbitrary precision with up to 3000 bits using MPFR and parallelized to run on a supercomputer. We explore Pade approximations of the expansions and observe branch cuts that leave gaps in the families of solutions. We also explore analytic continuation of standing waves into the upper half-plane and observe branch cuts there as well.

Recovery of finite-genus KP solutions from data

Levent Batakci University of Washington USA Co-Author(s):    Daniele Agostini, Turku Ozlum Celik, Bernard Deconinck, and Charles Wang

Abstract: The Kadomtsev-Petviashvili (KP) equation describes 3-dimensional waves in shallow water with weak dependence on the second spatial variable. Solutions of the KP equation with finitely many interacting phases can be expressed in terms of the Riemann theta function, and complex algebraic curves can be used to determine the parameters (e.g. wave vectors) therein. Particularly, a KP solution corresponding to a curve on a genus-$g$ surface will have $g$ phases present. We develop a new framework for the recovery problem: given a KP solution as a set of data values, we determine the corresponding genus and parameters. Our framework is rooted in Fourier analysis alongside a non-linear optimization step, which we recast as two linear least-squares problems. Whereas previous works were limited to recovery of genus 2 solutions, our framework is applicable to higher genus.

Amplification of interacting solitons in Kadomtsev-Petviashvili and bi-directional water-wave equations

Onno Bokhove University of Leeds England Co-Author(s):    Onno Bokhove and Shoaib Mohammed

Abstract: Extreme water-wave motion is investigated by considering soliton interactions on a horizontal plane. We determine numerically that soliton solutions of the unidirectional Kadomtsev-Petviashvili equation (KPE), with equal far-field individual amplitudes, survive well in the bidirectional and higher-order Benney-Luke and potential-flow equations (BLE and PFE). An exact two-soliton solution of the KPE on the infinite horizontal plane is used to seed BLE and PFE at an initial time, verifying that the KPE-fourfold amplification approximately persists. More extremely, a known three-soliton solution of the KPE is analysed, in a combined geometric-analytical approach proving its ninefold amplification. This three-soliton solution leads to an extreme splash at one location in space and time. Subsequently, we seed BLE and PFE with that three-soliton solution at a suitable initial time prior to and establish its maximum numerical amplification: it is at least 7.6 to 9 for an exact KPE amplification of 9 (depending on the choice of small parameters). In our simulations, computational domain and solutions are truncated approximately to a fully periodic or half-periodic channel geometry of sufficient size, essentially leading to (``time-periodic") cnoidal-wave solutions. Special geometric (finite-element) variational integrators in space and time have been used to avoid artificial numerical damping of wave amplitude. A larger goal, in progress, is to use these simulations for designs of suitable wave-tank experiments. Finally, we investigate whether an analytical solution of the KPE with four line-solitons of far-field amplitude A can yield amplitude 16A at the origin in space-time.

Spectral Stability of High Amplitude Waves

Eleanor D Byrnes University of Washington USA Co-Author(s):

Abstract: The spectral stability of the Euler water wave problem has been well studied both numerically and analytically for low amplitude waves. High amplitude waves in infinite depth have also been thoroughly studied numerically, with asymptotic results regarding a number of critical eigenvalue bifurcations emerging recently. I will present numerical and asymptotic work on the stability of high amplitude waves in finite depth.

Stability of Near-Extreme Solutions of the Whitham Equation

John Carter Seattle University USA Co-Author(s):

Abstract: The Whitham equation is a model for the evolution of small-amplitude, unidirectional waves of all wavelengths on shallow water. It has been shown to accurately model the evolution of waves in laboratory experiments. We compute $2\pi$-periodic traveling-wave solutions of the Whitham equation and numerically study their stability with a focus on solutions with large steepness. We show that the Hamiltonian oscillates as a function of wave steepness when the solutions are sufficiently steep. We show that a superharmonic instability is created at each extremum of the Hamiltonian and that between each extremum the stability spectra undergo similar bifurcations. Finally, we compare these results with those from the Euler equations.

The Stokes waves on ideal fluid: modulational instability and wave breaking

Sergey Dyachenko State University of New York at Buffalo USA Co-Author(s):    Bernard Deconinck, Elleanor Byrnes, Anastassiya Semenova, Pavel Lushnikov

Abstract: The long-standing problem of stability of surface waves on 2D fluid is solved in conformal variables for Stokes up to nearly extreme steepness. The stability spectrum of Stokes waves exhibits recurrent transitions, multiple modulation, or Benjamin-Feir instability branches. We show that all Stokes waves are, in fact, unstable, but the nature of these instabilities varies -- in some cases it leads directly to wave-breaking, and, in others, to modulational disturbance and appearance of rogue waves in the ocean swell. We discuss the profound change in the numerical approach that allowed us to consider nearly extreme Stokes waves.

Free surface fluid dynamics on Riemann sheets

Pavel M Lushnikov University of New Mexico USA Co-Author(s):

Abstract: A fully nonlinear dynamics for potential flow of ideal incompressible fluid with a free surface is considered in two dimensions. The exact solutions are found which are characterized by motion of infinite number of complex singularities in infinite number of sheets of Riemann surface for the analytical continuation of surface dynamics to outside of fluid. it allows an efficient characterization of fluid motion inside fluid.

Wave resonances in a water world

Paul Milewski Penn State USA Co-Author(s):    Matt Durey, Emilian Parau

Abstract: We consider nonlinear waves on a self-gravitating planet and discuss wave resonances.

Transverse Instability of Small-Amplitude Finite-Depth Stokes Waves

Luis Miguel Rodrigues Univ Rennes France Co-Author(s):    Ziang Jiao, Zhao Yang (Beijing, China), Changzhen Sun (Besan\c{c}on, France)

Abstract: We report on a recent proof that all irrotational planar periodic traveling waves of sufficiently small-amplitude are spectrally unstable as solutions to three-dimensional inviscid finite-depth gravity water-waves equations. The associated temporal growth scales sharply with respect to the amplitude of the wave.

Two-Crested Stokes Waves

Anastassiya Semenova Rochester Institute of Technology USA Co-Author(s):

Abstract: The study of surface gravity waves is crucial for understanding the formation of rogue waves and whitecaps in ocean swell. Waves originating from the epicenter of a storm can often be approximated as unidirectional. In this presentation, we explore periodic traveling waves on the free surface of an ideal two-dimensional fluid. We focus on surface waves of permanent shape, and present new families of Stokes waves and discuss their stability.

Well-posedness of vortex sheets in 3D flow with unequal density fluids

Michael Siegel New Jersey Institute of Technology USA Co-Author(s):

Abstract: We prove local well-posedness for a 2D vortex sheet separating two fluids of different densities in a 3D flow. The sheet evolves in a potential flow under the combined effects of gravity and surface tension. Our approach follows the method of Ambrose and Masmoudi, who treated the density-matched case. The main challenge in the unequal-density setting is a more complicated equation governing the vortex sheet strength, which is determined by the jump in velocity potential across the interface. The velocity potential can be represented as a boundary integral of dipoles. In the density-matched case, the dipole strength $\mu$ satisfies an explicit time-evolution equation. In contrast, when the densities differ, $\mu_t$ appears implicitly in a singular integral equation, which must first be inverted to find the evolution equation for $\mu$. The regularity of the vortex-sheet strength is established by combining new estimates with a bootstrap argument. This is joint work with Joseph Cavatchel (NJIT) and David Ambrose (Drexel).

Instability Mechanisms in 2D Fluids

Paolo Ventura EPFL Switzerland Co-Author(s):    Gonzalo Cao-Labora, Maria Colombo, Michele Dolce, Luca Franzoi, Riccardo Montalto

Abstract: In this talk I will discuss some instability results for two-dimensional fluid equations. I will first present results on the long-wave instability of periodic shear flows for the 2D Navier--Stokes equations (joint work with M. Colombo, M. Dolce, and R. Montalto), as well as results on the instability of a family of equilibria for the two-dimensional Euler equations (joint work with G. Cao Labora, M. Colombo, and M. Dolce). I will also describe work in progress with A. Franzoi and R. Montalto on the existence of infinitely many unstable eigenvalues for the 2D Euler equations linearized around a traveling wave.

Stability and instability of gravity-capillary waves

Erik Wahl\`en Lund University Sweden Co-Author(s):    Mariana Haragus, Wei Lian, Changzhen Sun, Tien Truong

Abstract: I will report on recent progress on the spectral stability and instability of periodic travelling gravity-capillary waves for the irrotational water wave problem in two space dimensions (one-dimensional surface). This includes both co-periodic, transverse and localised perturbations. I will also discuss some implications for the stability of generalised solitary waves, consisting of a localised hump with periodic ripples in the far field.