Intro; Preface; Contents; 1 Introduction; 1.1 Main Theorem; 1.2 Introduction to the Proof; 1.3 Paradifferential Formulation and Good Unknown; 1.4 Reduction to Constant Coefficients; Step 1: Diagonalization of the System; Step 2: Reduction to Constant Coefficients at Principal Order; Step 3: Reduction to Constant Coefficients at Arbitrary Order; 1.5 Normal Forms; 2 Main Result; 2.1 The Periodic Capillarity-Gravity Equations; 2.2 Statement of the Main Theorem; 3 Paradifferential Calculus; 3.1 Classes of Symbols; 3.2 Quantization of Symbols; 3.3 Symbolic Calculus; 3.4 Composition Theorems

3.5 Paracomposition4 Complex Formulation of the Equation and Diagonalization of the Matrix Symbol; 4.1 Reality, Parity and Reversibility Properties; 4.2 Complex Formulation of the Capillary-Gravity Water Waves Equations; 4.3 Diagonalization of the System; 5 Reduction to a Constant Coefficients Operator and Proof of the Main Theorem; 5.1 Reduction to Constant Coefficients of the Highest Order Part; 5.2 Reduction to Constant Coefficient Symbols; 5.3 Normal Forms; 5.4 Proof of Theorem 4.7; 6 The Dirichlet-Neumann Paradifferential Problem; 6.1 Paradifferential and Para-Poisson Operators

6.2 Parametrix of Dirichlet-Neumann Problem6.3 Solving the Dirichlet-Neumann Problem; 7 Dirichlet-Neumann Operator and the Good Unknown; 7.1 The Good Unknown; 7.2 Paralinearization of the Water Waves System; 7.3 The Capillarity-Gravity Water Waves Equations in Complex Coordinates; 8 Proof of Some Auxiliary Results; 8.1 Non-Resonance Condition; 8.2 Precise Structure of the Dirichlet-Neumann Operator; References; Index