In this talk, I will discuss a new physical-space approach to establishing the time decay and global asymptotics of solutions to variable-coefficient Schrödinger equation in (3+1)-dimensions. The result is applicable to possibly large, time-dependent, complex-valued coefficients under a general set of hypotheses. As an application, we are able to handle certain quasilinear cubic and Hartree-type nonlinearities, proving global existence together with global asymptotics. I will begin with a model problem and describe the construction of a good commutator. Time permitting, I will explain how to incorporate the good commutator with Ifrim--Tataru the method of testing by wave packets to obtain global asymptotics. This talk is based on upcoming work with Sung-Jin Oh and Federico Pasqualotto.