Preprint A203/2003
A higher order nonlinear SchrÃ¶dinger equation with variable coefficients

Felipe Linares | Carvajal, Xavier

**Keywords: **
Schrodinger equation | mKdV equation

We study the initial value problem (IVP) associated to a
higher order nonlinear Schrodinger equation with variable coefficients. Under some regularity on its coefficients we establish local well-posedness for the IVP for data in
$H^s(\R)$, $s\ge 1/4$, improving previous results \cite{[C1]}.
The main ingredient in our proof is an estimate of the maximal function associated to the linear solution similar to the sharp one obtained for linear solutions of the Schrodinger and Korteweg-de Vries equations.