On the Cauchy- and periodic boundary value problem for a certain class of derivative nonlinear Schroedinger equations

Abstract

The Cauchy- and periodic boundary value problem for the nonlinear Schroedinger equations in $n$ space dimensions [u_t - iΔu = (\nabla \bar{u})^β, |β|=m \ge 2, u(0)=u_0 \in H^{s+1}_x] is shown to be locally well posed for $s>s_c:=\frac{n}{2}-\frac{1}{m-1}$, $s\ge 0$. In the special case of space dimension $n=1$ a global $L^2$-result is obtained for NLS with the nonlinearity $N(u)={\partial }_x({\bar{u}}^2)$. The proof uses the Fourier restriction norm method.