eBook ISBN:  9781470405540 
Product Code:  MEMO/200/940.E 
List Price:  $71.00 
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AMS Member Price:  $42.60 
eBook ISBN:  9781470405540 
Product Code:  MEMO/200/940.E 
List Price:  $71.00 
MAA Member Price:  $63.90 
AMS Member Price:  $42.60 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 200; 2009; 128 ppMSC: Primary 76; 47; 35
The authors consider doublyperiodic 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 nonresonant cases, we first give a large family of formal threedimensional 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 integrodifferential operator of first order, both depending periodically of coordinates. It is shown that for almost all angles \(\theta\), the 3dimensional 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

Chapters

Chapter 1. Introduction

Chapter 2. Formal solutions

Chapter 3. Linearized operator

Chapter 4. Small divisors. Estimate of $\mathfrak {L}$resolvent

Chapter 5. Descent methodinversion of the linearized operator

Chapter 6. Nonlinear problem. Proof of Theorem 1.3

Appendix A. Analytical study of $G_n$

Appendix B. Formal computation of 3dimensional 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. DirichletNeumann operator

Appendix H. Proof of Lemma 5.8

Appendix I. Fluid particles dynamics


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The authors consider doublyperiodic 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 nonresonant cases, we first give a large family of formal threedimensional 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 integrodifferential operator of first order, both depending periodically of coordinates. It is shown that for almost all angles \(\theta\), the 3dimensional 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 \((\theta ,\mu ).\)

Chapters

Chapter 1. Introduction

Chapter 2. Formal solutions

Chapter 3. Linearized operator

Chapter 4. Small divisors. Estimate of $\mathfrak {L}$resolvent

Chapter 5. Descent methodinversion of the linearized operator

Chapter 6. Nonlinear problem. Proof of Theorem 1.3

Appendix A. Analytical study of $G_n$

Appendix B. Formal computation of 3dimensional 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. DirichletNeumann operator

Appendix H. Proof of Lemma 5.8

Appendix I. Fluid particles dynamics