Existence of global solutions for the nonlocal derivation nonlinear Schrödinger equation by the inverse scattering transform method

We address the existence of global solutions to the initial value problem for the integrable nonlocal derivative nonlinear Schrödinger equation in weighted Sobolev space $H^2(\mathbb{R})\cap H^{1,1}(\mathbb{R})$. The key to prove this result is to establish a bijectivity between potential and reflection coefficient by using the inverse scattering transform method in the form of the Riemann-Hilbert problem.