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

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.