Local well-posedness for the biharmonic nonlinear Schrödinger equation in low dimensions

In this paper we consider the 4th order (or biharmonic) nonlinear Schrödinger (NLS) equation in dimensions 1, 2, and 3, where the potential term is expressed as a power non-linearity (for any positive power) and the dispersion operator has the {\it fourth} order combined with the lower {\it second} order. The fourth order NLS equation has recently attracted attention of researchers since quartic solitons have been experimentally obtained in optics, and thus, mathematical theory of solutions to the 4th order NLS equation in physical dimensions $N=1,2,3$ is timely to develop. In this work we show the local well-posedness of the 4th order or bi-harmonic NLS equation on a weighted subset of a Sobolev space as well as in $H^s$ spaces.