# Small Divisor Problem in the Theory of Three-Dimensional Water Gravity Waves

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*Gérard Iooss; Pavel I. Plotnikov*

The authors consider doubly-periodic
travelling waves at the surface of an infinitely deep perfect fluid,
only subjected to gravity \(g\) and resulting from the
nonlinear interaction of two simply periodic travelling waves making
an angle \(2\theta\) between them.

Denoting by \(\mu =gL/c^{2}\) the dimensionless bifurcation
parameter ( \(L\) is the wave length along the direction of the
travelling wave and \(c\) is the velocity of the wave),
bifurcation occurs for \(\mu = \cos \theta\). 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.

The main object of the paper is the proof of existence of
such symmetric waves having the above mentioned asymptotic
expansion. Due to the occurence of small divisors, the main
difficulty is the inversion of the linearized operator at a non
trivial point, for applying the Nash Moser theorem. This operator is
the sum of a second order differentiation along a certain direction,
and an integro-differential operator of first order, both depending
periodically of coordinates. It is shown that for almost all angles
\(\theta\), the 3-dimensional travelling waves bifurcate for a
set of “good” values of the bifurcation parameter having
asymptotically a full measure near the bifurcation curve in the
parameter plane \((\theta ,\mu ).\)

#### Table of Contents

# Table of Contents

## Small Divisor Problem in the Theory of Three-Dimensional Water Gravity Waves

- Contents v6 free
- Chapter 1. Introduction 110 free
- Chapter 2. Formal Solutions 1524
- Chapter 3. Linearized Operator 2332
- Chapter 4. Small Divisors. Estimate of L – Resolvent 3544
- Chapter 5. Descent Method-Inversion of the Linearized Operator 5160
- Chapter 6. Nonlinear Problem. Proof of Theorem 1.3 7180
- Appendix A. Analytical study of G[sub(n)] 7584
- Appendix B. Formal computation of 3-dimensional waves 7988
- Appendix C. Proof of Lemma 3.6 8796
- Appendix D. Proofs of Lemmas 3.7 and 3.8 8998
- Appendix E. Distribution of Numbers {ω[sub(0)]n[sup(2)]} 93102
- Appendix F. Pseudodifferential Operators 99108
- Appendix G. Dirichlet-Neumann Operator 107116
- Appendix H. Proof of Lemma 5.8 119128
- Appendix I. Fluid particles dynamics 123132
- Bibliography 127136