Well-posedness to nonlinear Schr\"odinger-Gerdjikov-Ivanon equation

TL;DR

This paper extends the Riemann-Hilbert approach to establish well-posedness of the nonlinear Schrödinger-Gerdjikov-Ivanon equation, demonstrating Lipschitz continuity and existence of global solutions under certain conditions.

Contribution

It introduces a novel Riemann-Hilbert framework for analyzing the well-posedness of the NLS-GI equation, including potential reconstruction and global solution existence.

Findings

Lipschitz continuity of potential in specific Sobolev spaces

Construction of two Riemann-Hilbert problems with reflection coefficients

Existence of global solutions without eigenvalues or resonances

Abstract

The Riemann-Hilbert approach is extended to discuss the well-posedness of the nonlinear Schr\"odinger-Gerdjikov-Ivanon equation. The Lipschitz continuity of potential in $H^{2}(\mathbb{R})\cap H^{1,1}(\mathbb{R})$ to scattering data is obtained through direct scattering transform. Two Riemann-Hilbert problems are constructed, and two sets of the reflection coefficients, that is $r(k)$ and $r_\pm(z)$, are introduced. The Lipschitz continuity from the reflection coefficients $r_\pm(z)$ in $H^{1}(\mathbb{R})\cap L^{2,1}(\mathbb{R})$ to the potential is estimated via the potential reconstruction. Existence of global solutions of NLS-GI equation is considered by the Riemann-Hilbert problem without eigenvalues or resonances.