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 Gη 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 Gη 75

A.1. Computation of the differential of Gη 75

A.2. Second order Taylor expansion of Gη 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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