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

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.