Small divisor problem in the theory of three-dimensional water gravity waves

by GГ©rard Iooss

Publisher: American Mathematical Society in Providence, R.I

Written in English
Published: Downloads: 101
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Subjects:

  • Water waves,
  • Gravity waves,
  • Multiphase flow -- Mathematical models,
  • Pseudodifferential operators,
  • Boundary value problems,
  • Bifurcation theory,
  • Small divisors
  • Edition Notes

    StatementGérard Iooss and Pavel I. Plotnikov.
    SeriesMemoirs of the American Mathematical Society -- no. 940
    ContributionsPlotnikov, Pavel I.
    Classifications
    LC ClassificationsQA922 .L66 2009
    The Physical Object
    Paginationp. cm.
    ID Numbers
    Open LibraryOL23175256M
    ISBN 109780821843826
    LC Control Number2009008894

three-dimensional standing waves are found in both shallow and deep regimes, and their physical characteristics are examined and compared to previously known two-dimensional solutions. Keywords: optimization, multigrid methods, water waves 1. Introduction Gravity-driven water waves have been studied for well over a century and have a rich mathemat-File Size: 5MB.   In this article we show how Kirchgässner’s spatial dynamics approach can be used to construct doubly periodic travelling gravity-capillary surface waves on water of infinite depth. The hydrodynamic problem is formulated as a reversible Hamiltonian system in which an arbitrary horizontal spatial direction is the time-like variable and the infinite-dimensional function space consists of wave Cited by: 2.   This paper establishes the existence of quasipatterns solutions of the Swift–Hohenberg PDE. In a former approach we avoided the use of Nash–Moser scheme, but our proof contains a gap. The present proof of existence is based on the works by Berti et al related to the Nash–Moser scheme. For solving the small divisor problem, we need to introduce a new free parameter related to the . In this paper we prove the rst existence result of small amplitude time quasi-periodic standing waves solutions of the pure gravity water waves equations (), for most values of the depth h, see Theorem The existence of standing water waves is a small divisor problem, which is .

Part A will cover the linear theory of regular gravity waves on the surface of a fluid, in our case, the surface of water. For gravity waves, gravitation constitutes the restoration force, that is the force that keep the waves going. This applies to waves with wavelengths larger than a few Size: KB. Here previous work on the two-dimensional problem is extended to give motion trapping structures in the three-dimensional water-wave problem that have a vertical axis of symmetry. 1 Introduction Within the linearized theory of water waves, certain structures when held fixed can support a trapped mode of a particular frequency [1]. Justification of the nonlinear Schödinger equation for two-dimensional gravity driven water waves In V.E. Zakharov derived the Nonlinear Schödinger equation for the 2D water wave problem in the absence of surface tension in order to describe slow temporal and spatial modulations of a spatially and temporarily oscillating wave packet. ‘The book is a valuable source of information on the mathematic theory of water gravity waves and I am very pleased to have it on my shelf. I would recommend it to anyone who deals or is going to deal with this subject, but first for all to mathematically inclined readers.’.

For a complete bibliography on the numerical computation of three-dimensional water waves, one can refer to the paper [44]. Our review is more in the spirit of the section entitled “Existence theorems” in Wehausen and Laitone's contribution to the Encyclopedia of Physics []. Since the water-wave problem is a difficult nonlinear problem. We consider a Boussinesq system which describes three-dimensional water waves in uid layers with a depth small with respect to the wave (the depth is small with respect to wave length) gravity waves of an ideal, incompressible liquid. Here the horizontal coordinate x and due to a small divisor problem .   The objective of this book is to introduce new researchers to the rich dynamical system of water waves, and to show how (some) abstract mathematical concepts can be applied fruitfully in a practical physical problem and to make the connection between theory and experiment. • Craig & Nicholls’ method fails for pure gravity waves (no surface tension) because of small divisors. • Iooss & Plotnikov () prove existence of three-dimensional wave patterns of permanent form on deep water, for pure gravity waves (no surface tension). (“Small divisor problem in the theory .

Small divisor problem in the theory of three-dimensional water gravity waves by GГ©rard Iooss Download PDF EPUB FB2

For non-resonant cases, we first give a large family of formal three-dimensional gravity travelling waves, in the form of an expansion in powers of the amplitudes of two basic travelling waves. “Diamond waves” are a particular case of such waves, when they are.

For non-resonant cases, we first give a large family of formal three-dimensional gravity travelling waves, in the form of an expansion in powers of the amplitudes of two basic travelling waves. ”Diamond waves ” are a particular case of such waves, when they are symmetric with respect to the direction of propagation.

For non-resonant cases, we first give a large family of formal three-dimensional gravity travelling waves, in the form of an expansion in powers of the amplitudes of two basic travelling by: Small divisor problem in the theory of three-dimensional water gravity waves Small divisor problem in the theory of three-dimensional water gravity waves by we first give a large family of formal three-dimensional gravity travelling waves, in the form of an expansion in powers of the amplitudes of two basic travelling waves.

The existence of standing water waves is a small divisor problem, which is particularly difficult because () is a system of quasi-linear PDEs. Small divisor problem in the theory of three-dimensional water gravity waves.

"Diamond waves" are a particular case of such waves, when they are symmetric with respect to the direction of propagation.\newline \emph{The main object of the paper is the proof of existence} of such symmetric waves having the above mentioned asymptotic.

Small Divisor Problem in the Theory of Three-dimensional Water Gravity Waves (Memoirs of the American Mathematical Society) free ebook download. Title: Small divisor problem in the theory of three-dimensional water gravity waves Authors: Gérard Iooss (INLN), Pavel I. Plotnikov (Submitted on 23 Jan )Author: Gérard Iooss, Pavel I.

Plotnikov. Small divisor problem in the theory of three-dimensional water gravity waves. For non-resonant cases, we first give a large family of formal three-dimensional gravity travelling waves, in the form of an expansion in powers of the amplitudes of two basic travelling waves.

" Diamond waves " are a particular case of such waves, when they are Author: Gérard Iooss and Pavel Plotnikov. concentrate on cases where the small divisor problem arises in various ways for steady °ows (no dynamics here).

In particular, we present two water wave problems where this di–culty happens in an a priori unexpected way. The flrst example is 3D Travelling gravity waves, with a 2D periodic horizontal pattern on the free surface.

Small divisor problem in the theory of three-dimensional water gravity waves. For non-resonant cases, we first give a large family of formal three-dimensional gravity travelling waves, in the form of an expansion in powers of the amplitudes of two basic travelling waves.

”Diamond waves ” are a particular case of such waves, when they. Asymmetrical three-dimensional travelling gravity waves Abstract We consider periodic travelling gravity waves at the surface of an infinitely deep perfect fluid.

The pattern is non symmetric with respect to the propagation direction of the waves and we consider a general non resonant situation. Small divisor problem in the theory of three-dimensional water gravity waves.

For non-resonant cases, we first give a large family of formal three-dimensional gravity travelling waves, in the form of an expansion in powers of the amplitudes of two basic travelling waves.

"Diamond waves" are a particular case of such waves, when they are. Small divisor problem in the theory of three-dimensional water gravity waves. For non-resonant cases, we first give a large family of formal three-dimensional gravity travelling waves, in the form of an expansion in powers of the amplitudes of two basic travelling waves.

" Diamond waves " are a particular case of such waves, when they are. Small divisor problem in the theory of three-dimensional water gravity waves About this Title.

Gérard Iooss and Pavel I. Plotnikov. Publication: Memoirs of the American Mathematical Society Publication Year: ; VolumeNumber ISBNs: (print); (online) Cited by: Solutions of the water waves equations () satisfying () and () are called gravity standing water waves. In this paper we prove the first existence result of small amplitude time quasi-periodic standing waves solutions of the pure gravity water waves equations (), for most values of the depth h, see Theorem Enter the password to open this PDF file: Cancel OK.

File name:. Non-existence of three-dimensional travelling water waves 3 in. Here Pis the pressure. Furthermore, the velocity eld satis es the kinematic bound-ary condition w= 0 () on the bottom fz= 0gand the kinematic and dynamic boundary conditions w u x v y= 0; () P= const.

() on the surface fz= (x;y)g. By a solution of the water wave problem. Gravity waves on water Waves on the surface of water can arise from the restoring force of gravity or of surface tension, or a combination.

For wavelengths longer than a couple of centimeters surface tension can be neglected, and the waves are called gravity waves. Short wavelength surface waves are called capillary Size: KB. Small divisor problem in the theory of three-dimensional water gravity waves - Gérard Iooss and Pavel I.

Plotnikov MEMO/ The minimal polynomials of unipotent elements in irreducible representations of the classical groups in odd characteristic - I. Suprunenko. Traveling gravity water waves in two and three dimensions.

We present a perturbation analysis of the resulting bifurcation surfaces for the three-dimensional problem, some analytic results for these bifurcation problems, and numerical solutions of the surface water waves problem, based on a numerical continuation method which uses the Cited by: Get this from a library.

Small divisor problem in the theory of three-dimensional water gravity waves. [Gérard Iooss; Pavel I Plotnikov]. Traveling gravity water waves in two and three dimensions to construct the basic building blocks of a practical theory for inviscid, three-dimensional, fully nonlinear water waves in arbitrary.

Small divisor problem in the theory of three-dimensional water gravity waves / Material Type: Document, Internet resource: Document Type: Internet Resource, Computer File: All Authors / Contributors: Gérard Iooss; Pavel I Plotnikov.

The main results of this paper are existence theorems for traveling gravity and capillary gravity water waves in two dimensions, and capillary gravity water waves in three dimensions, for any periodic fundamental domain. This is a problem in bifurcation theory, yielding curves in the two dimensional case and bifurcation surfaces in the three dimensional by: Solutions of the water waves equations () satisfying () and () are called gravity standing water waves.

The existence of standing water waves is a small divisor problem, which is particularly difficult because () is a quasi-linear system of PDEs. The main results of this paper are existence theorems for traveling gravity and capillary gravity water waves in two dimensions, and capillary gravity water waves in three dimensions, for any periodic fundamental by: Global solutions for the gravity water waves equation in dimension 3 By P.

Germain, N. Masmoudi, and J. Shatah Abstract We show existence of global solutions for the gravity water waves equa-tion in dimension 3, in the case of small data. The proof combines energy estimates, which yield control of L2 related norms, with dispersive esti. Groves, M.

Haragus, A bifurcation theory for three-dimensional oblique travelling gravity-capillary water waves, J. Nonlinear Sci. 13 () – [8] G. Iooss, P. Plotnikov, Small divisor problem in the theory of three-dimensional water gravity waves, Mem. Amer. Math. Soc. () [9]Cited by: 1. the three-dimensional problem, some analytic results for these bifurcation problems, and numerical solutions of the surface water waves problem, based on a numerical continuation method which uses the spectral formulation of the problem in surface variables.

—, Local well posedness for the gravity water waves system, in preparation. T. Alazard and G. Métivier, Paralinearization of the Dirichlet to Neumann operator, and regularity of three-dimensional water waves, Comm.

Partial Differential Equations 34 (), –Cited by: A Hamiltonian formulation for three-dimensional nonlinear flexural-gravity waves propagating at the surface of an ideal fluid covered by a thin ice sheet is presented.

This is accomplished by introducing the Dirichlet–Neumann operator which reduces the original Laplace problem to a lower-dimensional system involving quantities evaluated at.Global solutions for the gravity water waves equation in dimension 3 Pages from Volume (), Issue 2 by Pierre Germain, Nader Masmoudi, Jalal Shatah AbstractCited by: