Local and global well-posedness for the kinetic derivative NLS on $\mathbb{R}$

We investigate the local and global well-posedness of the kinetic derivative nonlinear Schrödinger equation (KDNLS) on $\mathbb{R}$, described by $$i{\mathrm{\partial }}_{t}u+{\mathrm{\partial }}_x^{2}u=i\alpha {\mathrm{\partial }}_x(\mathrm{\mid }u{\mathrm{\mid }}^{2}u)+i\beta {\mathrm{\partial }}_x(H(\mathrm{\mid }u{\mathrm{\mid }}^2)u),$$ where $\alpha ,\beta \in \mathbb{R}$, and $H$ represents the Hilbert transformation. For KDNLS, the $L^2$ norm of a solution is decreasing (resp. increasing, conserved) when $\beta$ is negative (resp. positive, zero). Focusing on the Sobolev spaces $H^2$ and $H^2\cap H^{1,1}$, we establish local well-posedness via the energy method combined with gauge transformations to address resonant interactions in both cases of negative and positive $\beta$. For the dissipative case $\beta <0$, we further demonstrate global well-posedness by deriving an a priori bound in $H^2$.