Scattering for the Klein-Gordon-Zakharov system in two dimensions

TL;DR

This paper proves global solutions and detailed long-time behavior, including decay and scattering, for the Klein-Gordon-Zakharov system in two dimensions, revealing a dichotomy in scattering phenomena due to long-range effects.

Contribution

It introduces a novel nonlinear transformation and combines physical and frequency space methods to analyze the system's long-term dynamics in two dimensions.

Findings

Established global existence for small initial data.

Characterized sharp decay and scattering behavior.

Discovered a dichotomy between modified and linear scattering.

Abstract

We study the Klein-Gordon-Zakharov system in two spatial dimensions, an important model in plasma physics. For small, smooth, and spatially localized initial data, we establish the global existence of solutions and characterize their sharp long-time behavior, including sharp time decay and scattering properties. A particularly interesting phenomenon is that the Klein-Gordon component exhibits modified scattering for certain initial data, while for others it undergoes linear scattering-a dichotomy highlighting delicate long-range interaction effects. The major obstacles are lack of symmetry and weak decay of the solution in two dimensions. To overcome these, we introduce a novel nonlinear transformation of the wave component and reinterpret the nonlinear coupling as a perturbation of the mass term in the Klein-Gordon equation. The proof employs a combination of physical space and frequency space methods.