Semi-linearity of the non-linear Fourier transform of the defocusing NLS equation

TL;DR

This paper proves that the non-linear Fourier transform of the defocusing NLS equation on the circle is almost linear, differing only by smoother terms, which simplifies understanding its structure.

Contribution

It establishes the semi-linearity of the non-linear Fourier transform for the defocusing NLS, revealing a near-linear behavior with smoother correction terms.

Findings

Non-linear Fourier transform is linear up to smoother terms.

Provides a mathematical proof of semi-linearity.

Enhances understanding of NLS spectral analysis.

Abstract

In this paper we prove that the non-linear Fourier transform of the defocusing NLS equation on the circle is linear up to terms which are one order smoother.