Ill-posedness in $H^s$ for a defocusing power-type derivative Schrödinger equation with lower-order linear perturbations

This work studied the one-dimensional defocusing power-type derivative Schrödinger equation with lower-order linear perturbations

$$\mathrm{i}{u}_t+u_{xx}-\mathrm{i}|u{|}^k{u}_x+\alpha u_x+\beta u=0,(t,x)\in \mathbb{R}\times \mathbb{R},$$

where $k\ge 2$ and $\alpha ,\beta \in \mathbb{R}$ are constants. An explicit family of solitary traveling-wave solutions is first constructed within an exactly integrable traveling-wave reduction, and their $H^s$ regularity and parameter dependence are characterized. A traveling-wave-based ill-posedness mechanism is then implemented: two solutions associated with nearby parameter sets are produced so that their initial data are arbitrarily close in $H^s$, while their profiles remain separated by a uniform positive lower bound in $H^s$ at some positive time. As a result, the solution flow map fails to be uniformly continuous below a certain regularity threshold. These results indicate that the presence of lower-order linear perturbations does not improve the low-regularity stability threshold for this DNLS-type equation.