Contents
Chapter 1. Introduction 1
1.1. Presentation and History of the Problem 1
1.2. Formulation of the Problem 4
1.3. Results 7
1.4. Mathematical background 9
1.5. Structure of the paper 9
Chapter 2. Formal Solutions 15
2.1. Differential of 15
2.2. Linearized equations at the origin and dispersion relation 15
2.3. Formal computation of 3-dimensional waves 17
2.4. Geometric pattern of diamond waves 20
Chapter 3. Linearized Operator 23
3.1. Linearized system in (ψ, η) = 0 23
3.2. Pseudodifferential operators and diffeomorphism of the torus 25
3.3. Main orders of the diffeomorphism and coefficient ν 33
Chapter 4. Small Divisors. Estimate of L− Resolvent 35
4.1. Proof of Theorem 4.10 41
Chapter 5. Descent Method-Inversion of the Linearized Operator 51
5.1. Descent method 52
5.2. Proof of Theorem 5.1 61
5.3. Verification of assumptions of Theorem 5.1 66
5.4. Inversion of L 68
Chapter 6. Nonlinear Problem. Proof of Theorem 1.3 71
Appendix A. Analytical study of 75
A.1. Computation of the differential of 75
A.2. Second order Taylor expansion of in η = 0 77
Appendix B. Formal computation of 3-dimensional waves 79
B.1. Formal Fredholm alternative 79
B.2. Bifurcation equation 81
Appendix C. Proof of Lemma 3.6 87
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