Local Well‐Posedness for the Biharmonic Nonlinear Schrödinger Equation in Low Dimensions

ABSTRACT

In this paper we consider the fourth‐order (or biharmonic) nonlinear Schrödinger (NLS) equation in dimensions 1, 2, and 3, where the potential term is expressed as a power nonlinearity (for any positive power) and the dispersion operator has the fourth ‐order combined with the lower second ‐order. The fourth order NLS equation has recently attracted the attention of researchers since quartic solitons have been experimentally obtained in optics, and thus, the mathematical theory of solutions to the fourth‐order NLS equation in physical dimensions is timely to develop. In this work we show the local well‐possesses of the fourth‐order or biharmonic NLS equation on a weighted subset of a Sobolev space as well as in  spaces.