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

Abstract: We address the existence of global solutions to the initial value problem for the integrable nonlocal derivative nonlinear Schr\"{o}dinger 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.