Maps from the Dbar-data to the derivative NLS potentials

Abstract

We develop a decomposition technique to investigate maps from the Dbar data to the derivative NLS potentials. We define a new integral operator which is equivalent to a modified Dbar problem relevant to the derivative NLS spectral problem, including the Kaup-Newell spectral problem, the Chen-Lee-Liu spectral problem and the Gerdjikov-Ivanov spectral problem. The well-posedness for the integral operator with the kernel $\mathrm{e}^{ik^2xσ_3}$ is discussed via the decomposition technique. Under different constraint conditions, the derivative NLS potentials are reconstructed via the Dbar data. We discuss the symmetry condition of the Dbar data as well as the compatibility of the derivative NLS spectral problem with the modified Dbar problem. We show that the maps from the Dbar data to the derivative NLS potentials are Lipschitz continuous.