Xavier Carvajal

Higher-Order Hamiltonian Model for Unidirectional Water Waves

Formally second-order correct, mathematical descriptions of long-crested water waves propagating mainly in one direction are derived. These equations are analogous to the first-order approximations of KdV- or BBM-type. The advantage of these more complex equations is that their solutions corresponding to physically relevant initial perturbations of the rest state may be accurate on a much longer timescale. The initial value problem for the class of equations that emerges from our derivation is then considered. A local well-posedness theory is straightforwardly established by a contraction mapping argument. A subclass of these equations possess a special Hamiltonian structure that implies the local theory can be continued indefinitely.

A Note on Local Well-Posedness of Generalized KdV Type Equations with Dissipative Perturbations

In this note we report local well-posedness results for the Cauchy problems associated to generalized KdV type equations with dissipative perturbation for given data in the low regularity

On well-posedness of the third-order nonlinear Schrödinger equation with time-dependent coefficients

L^2

On the ill-posedness for a nonlinear Schrödinger-Airy equation

-based Sobolev spaces. The method of proof is based on the {\em contraction mapping principle} employed in some appropriate time weighted spaces.

Operators That Achieve the Norm

We consider the Cauchy problem associated to the third-order nonlinear Schrodinger equation with time-dependent coefficients. Depending on the nature of the coefficients, we prove local as well as global well-posedness results for given data in L2-based Sobolev spaces. We also address the scaling limit to fast dispersion management and prove that it converges in H1 to the solution of the averaged equation.

On the critical KdV equation with time-oscillating nonlinearity

Abstract. Using ideas of Kenig Ponce and Vega and an explicit solution with two parameters we prove that the solution map of the initial value problem for a particular nonlinear Schrodinger-Airy equation ∂tu + ia ∂ xu + b ∂ 3 xu + ic |u|u + d |u|∂xu + e u∂xū = 0, x, t ∈ R, (1) fails to be uniformly continuous. We also approximate the nonlinear Schrodinger-Airy equation by the cubic nonlinear Schrodinger equation and prove ill-posedness in the more general case. This method was originally introduced by Christ, Colliander and Tao for the modified Korteweg-de Vries equation.