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

Junyi Zhu, Huan Liu

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.

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

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 with physical variable as a parameter. We develop a decomposition technique to control the convergence of the integral by defining a new integral operator . 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.
Paper Structure (5 sections, 12 theorems, 123 equations, 1 figure)

This paper contains 5 sections, 12 theorems, 123 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminary for the integral operator to the Dbar problem on bounded domain
  3. Dbar problem relevant to the AKNS spectral problem
  4. Potential reconstruction of AKNS spectral problem
  5. Conclusions and Discussions

Key Result

Theorem 1

INV1962 Let $G$ be a bounded domain, and $f(k)\in L^1(\bar{G})$, then the integral exist for all points k outside $\bar{G}$, $f\hat{T}_G(k)$ is holomorphic outside $\bar{G}$, and vanish at $k=\infty$.

Figures (1)

  • Figure 1: Case i. (left); case ii.(meddle);case iii. (right).

Theorems & Definitions (12)

  • Theorem 1
  • Theorem 2
  • Lemma 1
  • Theorem 3
  • Lemma 2
  • Lemma 3
  • Theorem 4
  • Corollary 1
  • Theorem 5
  • Lemma 4
  • ...and 2 more