Local well-posedness for the derivative nonlinear Schrödinger equation with nonvanishing boundary conditions

We consider the derivative nonlinear Schrödinger equation on the real line, with a background function $\psi (t,x)\in L^{\mathrm{\infty }}({\mathbb{R}}^2)$ that satisfies suitable conditions. Such a function may, for example, be a non-decaying solution of the equation, such as a dark soliton. By developing the energy method with correction terms, we prove that the Cauchy problem for perturbations around such an $L^{\mathrm{\infty }}$ function is unconditionally locally well-posed in $H^s(\mathbb{R})$ for $s>3\mathrm{/}4$. As a byproduct, we also establish local well-posedness in the Zhidkov space.