Pointwise Decay for Cubic Nonlinear Schrödinger Systems in Four Dimensions

Abstract

This thesis analyzes pointwise decay for the solution of systems of nonlinear Schrödinger equations. In particular, this work focuses on two and three-component systems in four dimensions with cubic nonlinearity. This research tries to understand and show how a solution of the cubic NLS system will decay. Learning the decay rate gives an understanding of how solutions behave as time goes to infinity. In the previous work of other researchers, the dispersive estimate of a system in dimensions $ n = 1,2 $ was established. This work was motivated by researching energy-critical cubic NLS systems in $n=4$ to fill a gap in previous research. In this thesis, the main method will be the bootstrap argument, inspired by \cite{fan2024decaying}.