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

We investigate the local and global well-posedness of the kinetic derivative nonlinear Schrödinger equation(KDNLS) on $${{\mathbb {R}}}$$ R , described by $$\begin{aligned} i\partial _t u + \partial _x^2 u = i\alpha \partial _x (|u|^2 u) + i\beta \partial _x ({\mathcal {H}}(|u|^2) u), \end{aligned}$$ i ∂ t u + ∂ x 2 u = i α ∂ x ( | u | 2 u ) + i β ∂ x ( H ( | u | 2 ) u ) , where $$\alpha , \beta \in \mathbb {R}$$ α , β ∈ R , and $${\mathcal {H}}$$ H represents the Hilbert transformation. For KDNLS, the $$L^2$$ L 2 norm of a solution is decreasing (resp. increasing, conserved) when $$\beta $$ β is negative (resp. positive, zero). Focusing on the Sobolev spaces $$H^2$$ H 2 and $$H^2 \cap H^{1,1}$$ H 2 ∩ 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 $$\beta $$ β . For the dissipative case $$\beta