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.
Cazenave, Semilinear Schr¨ odinger Equations, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, 2003
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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.