Well-posedness for the $\bar\partial$-problem relevant to the AKNS spectral problem

TL;DR

This paper establishes the well-posedness of the Dbar problem related to the AKNS spectral problem, introducing a new integral operator and extending the Dbar dressing method for potential construction.

Contribution

It develops a decomposition technique and a new integral operator to prove existence and uniqueness of solutions, extending the Dbar dressing method for AKNS spectral problems.

Findings

Existence and uniqueness of solutions for the Dbar problem are proven.

A new integral operator controls convergence and ensures small norm conditions.

The Dbar dressing method is extended to construct AKNS potentials with Lipschitz continuity.

Abstract

The well-posedness for the Dbar problem associated with the AKNS spectral problem is considered. In general, the relevant Dbar equation with normalization condition is quivalent to an integral equation, where the kernel involves exponents $\mathrm{e}^{\pm2ikx}$ with physical variable $x$ as a parameter. We develop a decomposition technique to control the convergence of the integral by defining a new integral operator $RT_{\mathbb{C}}(k;x)$. The small norm condition of the operator is obtained to show that there exists a unique solution for the Dbar problem. Moreover, the Dbar dressing method is extended to construct the AKNS spectral problem and the potential construction is presented via the Dbar data. Prior estimates are given to show that the map from the Dbar data to the AKNS potential is Lipschitz continuous.