Quantitative finite-time approximation of scattering fields for Alfvén waves in ideal MHD

In this paper, we establish a quantitative finite-time approximation of scattering fields for nonlinear Alfvén waves in three-dimensional ideal incompressible magnetohydrodynamics near a strong constant magnetic background. Using the global weighted estimates, we prove that truncating the scattering profiles at time $T$ introduces an error in both the $L^{\mathrm{\infty }}$ norm and the $H^{N_{\ast }+1}$ norm of order $$O({\epsilon }^2(R+T{)}^{-\delta }),$$ where $\epsilon \in (0,1)$ measures the size of the initial perturbation, $R\ge 100$ is the scale parameter in the weighted energy, $\delta \in (0,2\mathrm{/}3)$ is the exponent related to weights, and $N_{\ast }+1$ is the Sobolev regularity order assumed for the initial data. The pointwise estimate follows from the decay of the nonlocal pressure along nonlinear characteristics, while the Sobolev estimate combines higher-order weighted pressure bounds with estimates for derivatives of the characteristic flows. Numerical experiments measure pointwise and Sobolev pressure-integral tails to verify the quadratic dependence of the initial perturbation amplitude.