Ideal MHD. Part III: Inverse scattering of Alfv\'en waves in three dimensional ideal magnetohydrodynamics

TL;DR

This paper addresses the inverse scattering problem for nonlinear Alfvén waves in three-dimensional ideal MHD, establishing new mathematical isomorphisms that connect plasma waves with their far-field scattering data.

Contribution

It introduces novel geometric and energy estimate methods to relate Alfvén waves to their scattering fields, advancing the understanding of wave behavior in ideal MHD.

Findings

Established scattering isomorphisms for Alfvén waves

Enhanced rigidity results connecting plasma waves and scattering fields

Analyzed null structures and pressure estimates in MHD equations

Abstract

The purpose of this paper is to solve the inverse scattering problem of nonlinear Alfv\'en waves governed by the three dimensional ideal incompressible MHD system. Bridging together geometric methods and weighted energy estimates, we establish a couple of scattering isomorphisms to substantially strengthen our previous rigidity results. This answer is consistent with the physical intuition that Alfv\'en waves behave exactly in the same manner as their scattering fields detected by the faraway observers. The novelty of the present work is twofold: for one thing, the relationship between Alfv\'en 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.