Derivative NLS around L∞ backgrounds, such as dark solitons, is unconditionally locally well-posed in H^s for s>3/4 under suitable background conditions.
Abdallah, On the well-posedness of dispersive–dissipative one dimensional equations with non decaying initial data , Monatsh
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Local well-posedness for the derivative nonlinear Schr\"odinger equation with nonvanishing boundary conditions
Derivative NLS around L∞ backgrounds, such as dark solitons, is unconditionally locally well-posed in H^s for s>3/4 under suitable background conditions.