Global well-posedness of the Cauchy problem for the modified Whitham equations

This paper aims to show global existence and modified scattering for the solutions of the Cauchy problem to the modified Whitham equations for small, smooth and localized initial data. The main difficulties come from slow decay and non-homogeneity of the Fourier multiplier $(\sqrt{\tanh \xi/\xi})\xi$, which will be overcome by introducing an interaction multiplier theorem and estimating the weighted norms in the frequency space. When estimating the weighted norms, due to loss of derivatives, the energy estimate will be performed in the frequency space, and the absence of time resonance will be effectively utilized by extracting some good terms arising from integration by parts in time before the energy estimate.