Ademir Pastor

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)

Ill-posedness results for the (generalized) Benjamin-Ono-Zakharov-Kuznetsov equation

,

On the periodic Schrödinger–Boussinesq system

s>3/4

The Fourth-Order Dispersive Nonlinear Schrodinger Equation: Orbital Stability of a Standing Wave ∗

. Even though the critical space for this equation is

Large data scattering for the defocusing supercritical generalized KdV equation

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

Weak concentration and wave operator for a 3D coupled nonlinear Schrödinger system

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

Two dimensional solitary waves in shear flows

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.

Local and global well-posedness for the 2D generalized Zakharov–Kuznetsov equation

Here we consider results concerning ill-posedness for the Cauchy problem associated with the Benjamin-Ono-Zakharov-Kuznetsov equation, namely, (IVP) {u t -Hu xx +u xyy +u k u x =0, (x, y) ∈ ℝ 2 , t ∈ ℝ + , {u(x, y, 0)=φ(x, y). For k = 1, (IVP) is shown to be ill-posed in the class of anisotropic Sobolev spaces H s 1 ,s 2 (ℝ 2 ), s 1 , s 2 ∈ R, while for k ≥ 2 ill-posedness is shown to hold in H s 1 ,s 2 (ℝ 2 ), 2s 1 , + s 2 < 3/2 - 2/k. Furthermore, for k = 2,3, and some particular values of s 1 , s 2 , a stronger result is also established.

A NOTE ON THE 2D GENERALIZED ZAKHAROV-KUZNETSOV EQUATION: LOCAL, GLOBAL, AND SCATTERING RESULTS

Abstract We study the local and global well-posedness of the periodic boundary value problem for the nonlinear Schrodinger–Boussinesq system. The existence of periodic traveling-wave solutions as well as the orbital stability of such solutions are also considered.