Dispersive Decay for the Free Schrodinger Equation and a Higher-Order Model

Abstract

This undergraduate capstone project studies the pointwise decay of solutions to two dispersive equations: the free Schrodinger equation iu_{t} = u_{xx} and the third-order model iu_{t} = u_{xx} − iu_{xxx}. Under suitable regularity assumptions on the initial data, we prove the estimates ∥u(·,t)∥L_{∞} ≲ t^{−1/2} and ∥u(·, t)∥L_{∞} ≲ t^{−1/3} respectively, via explicit Fourier analysis and a cutoff argument in frequency space. The numerical simulations confirm these rates and compare the theoretical bounds to the observed decay.