A survey on long range scattering for Schrödinger equation and Klein-Gordon equation with critical nonlinearity of non-polynomial type (Regularity and Asymptotic Analysis for Critical Cases of Partial Differential Equations)

Abstract

We summarize recent progress on long range scattering for nonlinear Schrödinger equation and nonlinear Klein-Gordon equation. We introduce a technique of extracting a resonant part, which has the same oscillation speed as its argument, from non-polynomial nonlinearities, and exhibit its two applications. Firstly, we consider nonlinear Schrödinger equation with a general nonlinearity of the critical order, and investigate the relation between the shape of the nonlinearity and a typical asymptotic behavior of small solutions. Secondly, we consider nonlinear Klein-Gordon equation with a gauge-invariant nonlinearity, and find an asymptotic behavior for both real-valued case and complex-valued case. A slight improvement is seen in the second application.