Global solution and asymptotic behavior for the kinetic derivative NLS on $\mathbb{R}$

In this paper we investigate the global well-posedness and long-term behavior of solutions to the kinetic derivative nonlinear Schrödinger equation (KDNLS) on the real line. The equation incorporates both local cubic nonlinearities with derivative terms and a non-local term arising from the Hilbert transform, modeling interactions in plasma physics. We establish global existence for small initial data in the weighted Sobolev space $H^2\cap H^{1,1}$ and optimal time decay effect. Using energy methods and a frequency-localized gauge transformation, we overcome the difficulties posed by the non-local nonlinearities and provide a rigorous analysis of the asymptotic behavior. Our results also describe modified scattering phenomena with a suitable phase modification, showing that the solutions exhibit a precise asymptotic profile as $t\to \mathrm{\infty }$.