Inverse scattering of Alfvén waves in three dimensional ideal magnetohydrodynamics

Abstract

The purpose of this paper is to solve the inverse scattering problem of nonlinear Alfvén waves governed by the three dimensional ideal incompressible magnetohydrodynamics (MHD) system. By bridging together geometric methods and weighted energy estimates, we establish a set of scattering isomorphisms to substantially strengthen our previous rigidity results. This answer is consistent with the physical intuition that Alfvén waves behave exactly in the same manner as their scattering fields that are detected by faraway observers. The novelty of the present work is twofold: for one thing, the relationship between Alfvén waves emanating from the plasma and their scattering fields at infinities is explored to the best; for another thing, the null structure inherent in MHD equations is thoroughly examined, especially when we estimate the pressure term.