Analyticity of the scattering operator for semilinear dispersive equations

TL;DR

This paper introduces a general algorithm to prove the real analyticity of scattering operators for semilinear dispersive equations and demonstrates its application to Schrödinger, wave, and Klein-Gordon equations, linking it to inverse scattering and integrability.

Contribution

A novel algorithmic approach to establish the analyticity of scattering operators and compute their Taylor series coefficients for various semilinear dispersive equations.

Findings

Scattering operators are shown to be real analytic in specific cases.

Explicit computation of Taylor series coefficients is possible with the algorithm.

The approach connects scattering theory with inverse problems and integrability.

Abstract

We present a general algorithm to show that a scattering operator associated to a semilinear dispersive equation is real analytic, and to compute the coefficients of its Taylor series at any point. We illustrate this method in the case of the Schrodinger equation with power-like nonlinearity or with Hartree type nonlinearity, and in the case of the wave and Klein-Gordon equations with power nonlinearity. Finally, we discuss the link of this approach with inverse scattering, and with complete integrability.