Well-posedness to nonlinear Schrödinger-Gerdjikov-Ivanon equation

TL;DR

This work extends the inverse scattering transform to the nonlinear Schrödinger-Gerdjikov-Ivanov equation by formulating a pair of Riemann–Hilbert problems connected to spectral data and establishing Lipschitz continuity of the direct and inverse scattering maps in Sobolev-weighted spaces. It derives explicit potential reconstruction formulas from RH data and proves local well-posedness for initial data in $H^{2}(\mathbb{R})\cap H^{1,1}(\mathbb{R})$, followed by global existence for data with no eigenvalues or resonances, with the solution map remaining Lipschitz in time. The approach leverages a gauge-adjusted RH framework to handle the derivative nonlinearity and provides quantitative bounds linking scattering data to the reconstructed potential. These results enhance the IST toolkit for higher-order derivative NLS-type equations and contribute to rigorous understanding of their long-time behavior.

Abstract

The Riemann-Hilbert approach is extended to discuss the well-posedness of the nonlinear Schrödinger-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.

Theorems & Definitions (15)

• Lemma 1
• Lemma 2
• Lemma 3
• Lemma 4
• Lemma 5
• Proposition 1
• Lemma 6
• Corollary 1
• Lemma 7
• Lemma 8