Well-posedness of scattering data for the derivative nonlinear Schr\"odinger equation in $H^s(\mathbb{R})$

TL;DR

This paper establishes the well-posedness of scattering data for the derivative nonlinear Schrödinger equation in Sobolev spaces, detailing the structure and asymptotics of the transmission coefficient and its reciprocal.

Contribution

It introduces a novel analysis of the transmission coefficient's reciprocal as a sum of iterative integrals, advancing understanding of scattering data regularity and asymptotics in $H^s( )$.

Findings

Reciprocal of transmission coefficient expressed as iterative integrals

Logarithm of reciprocal written as connected iterative integrals

Asymptotic properties of initial iterative integrals analyzed

Abstract

We prove the well-posedness results of scattering data for the derivative nonlinear Schr\"odinger equation in $H^{s}(\mathbb{R})(s\geq\frac12)$. We show that the reciprocal of the transmission coefficient can be written as the sum of some iterative integrals, and its logarithm can be written as the sum of some connected iterative integrals. And we provide the asymptotic properties of the first few iterative integrals of the reciprocal of the transmission coefficient. Moreover, we provide some regularity properties of the reciprocal of the transmission coefficient related to scattering data in $H^{s}(\mathbb{R})$.