laws associated with it have been obtained explicitly using the bi-Hamiltonian structure and the multisoliton solutions have been derived by applying the dressing method \[4\]. Another remarkable feature of the FL equation is that it is the first negative flow of the integrable hierarchy of the derivative NLS equation \[2, 5\].

In a previous study \[6\] which is referred to as I hereafter, the two different expressions of the bright $N$-soliton solution of the FL equation have been obtained by a direct method which does not recourse to the IST and their properties have been explored in detail. Here, we construct the dark $N$-soliton solution of the FL equation on the background of a plane wave. Explicitly, we consider the boundary condition $$u \rightarrow \rho\,{\rm exp}\left\{{\rm i}\left(\kappa x-\omega t+\phi^{(\pm)}\right)\right\}, \quad x \rightarrow \pm\infty, \eqno(1.2)$$ where $\rho(>0)$ and $\kappa$ are real constants representing the amplitude and wavenumber, respectively, $\phi^{(\pm)}$ are real phase constants and the angular frequency $\omega=\omega(\kappa)$ obeys the dispersion relation $\omega=1/\kappa+2\rho^2.$ Note that the plane wave given in (1.2) is an exact solution of the FL equation. As will be discussed later, the possible values of $\kappa$ must be restricted to assure the existence of the soliton solutions. A 