Felipe Linares

Introduction to nonlinear dispersive equations

1. The Fourier Transform.- 2. Interpolation of Operators.- 3. Sobolev Spaces and Pseudo-Differential Operators.- 4. The Linear Schrodinger Equation.- 5. The Non-Linear Schrodinger Equation.- 6. Asymptotic Behavior for NLS Equation.- 7. Korteweg-de Vries Equation.- 8. Asymptotic Behavior for k-gKdV Equations.- 9. Other Nonlinear Dispersive Models.- 10. General Quasilinear Schrodinger Equation.- Proof of Theorem 2.8.- Proof of Lemma 4.2.- References.- Index.

On the Davey-Stewartson systems

Abstract We study the initial value problem for the Davey-Stewartson systems. This model arises generically in both physics and mathematics. Using the classification in [15] we consider the elliptic-hyperbolic and hyperbolic-hyperbolic cases. Under smallness assumption on the data it is shown that the IVP is locally wellposed in weighted Sobolev spaces.

Well-Posedness for the Two-Dimensional Modified Zakharov–Kuznetsov Equation

We prove that the initial value problem for the two-dimensional modified Zakharov–Kuznetsov equation is locally well-posed for data in

Well-Posedness for the ZK Equation in a Cylinder and on the Background of a KdV Soliton

H^s(\mathbb{R}^2)

The Cauchy Problem for the Euler–Poisson System and Derivation of the Zakharov–Kuznetsov Equation

,

Global existence for the critical generalized KdV equation

s>3/4

The IVP for the dispersion generalized Benjamin–Ono equation in weighted Sobolev spaces

. Even though the critical space for this equation is

Well-Posedness of Strongly Dispersive Two-Dimensional Surface Wave Boussinesq Systems

L^2(\mathbb{R}^2)

On Decay Properties of Solutions of the k-Generalized KdV Equation

, we prove that well-posedness is not possible in such space. Global well-posedness and a sharp maximal function estimate are also established.

Stability and symmetry of solitary-wave solutions to systems modeling interactions of long waves

We establish the well-posedness of two Cauchy problems for the two-dimensional Zakharov–Kuznetsov equation motivated by the study of transverse stability properties of the N-soliton φ N of the Korteweg–de Vries equation. They differ by the functional setting: Sobolev spaces H s (ℝ × 𝕋), for the first one, φ N + H 1(ℝ2) for the second one. In the latter case, the solution is shown to be global.