Marcia Scialom

Adiabatic switching for time‐dependent electric fields

In this work the scattering theory associated with the differential equation i(∂ψ/∂t)=(−Δ+e−E‖t‖g(t)x1 +q(x))ψ is considered, where x=(x1,x⊥)∈R×R2, E≥0, ω>0, α∈R, g(t), t∈R is continuous, periodic with mean value zero over a period, and q(x) approaches to zero sufficiently fast as ‖x‖→∞. In the case E>0, it is shown that the usual theory is adequate; however, a limit does not exist when E↓0. A modified theory is developed where the limit does exist as E↓0. Furthermore, the concepts of bound states and scattering states for E≥0 are discussed.

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

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.