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Small Divisor Problem in the Theory of Three-Dimensional Water Gravity Waves
Gérard Iooss, Université de Nice, France, and Pavel I. Plotnikov, Lavrentyev Institute of Hydrodynamics RAS, Novosibirsk, Russia

Memoirs of the American Mathematical Society
2009; 128 pp; softcover
Volume: 200
ISBN-10: 0-8218-4382-6
ISBN-13: 978-0-8218-4382-6
List Price: US$67
Individual Members: US$40.20
Institutional Members: US$53.60
Order Code: MEMO/200/940
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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

  • Introduction
  • Formal solutions
  • Linearized operator
  • Small divisors. Estimate of \(\mathfrak{L}-\) resolvent
  • Descent method-inversion of the linearized operator
  • Nonlinear Problem. Proof of Theorem 1.3
  • Appendix A. Analytical study of \(\mathcal{G}_\eta\)
  • Appendix B. Formal computation of 3-dimensional waves
  • Appendix C. Proof of Lemma 3.6
  • Appendix D. Proofs of Lemmas 3.7 and 3.8
  • Appendix E. Distribution of numbers \(\{\omega_{0}n^{2}\}\)
  • Appendix F. Pseudodifferential operators
  • Appendix G. Dirichlet-Neumann operator
  • Appendix H. Proof of Lemma 5.8
  • Appendix I. Fluid particles dynamics
  • Bibliography
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