detail, showing that both the dark and bright solitons appear on the nonzero background which reduce to algebraic solitons in specific limits. Last, we perform the asymptotic analysis of the two- and $N$-soliton solutions for large time and clarify their structure and dynamics.

[*PACS:*]{} 05.45.Yv; 42.81.Dp; 02.30.Jr

[*Keywords:*]{} derivative nonlinear Schrödinger equation; dark soliton; direct method of solution

The Fokas-Lenells derivative nonlinear Schrödinger (NLS) equation (FL equation shortly) is a completely integrable nonlinear partial differential equation (PDE) which has been derived as an integrable generalization of the NLS equation using bi-Hamiltonian methods \[1\]. In the context of nonlinear optics, the FL equation models the propagation of nonlinear light pulses in monomode optical fibers when certain higher-order nonlinear effects are taken into account \[2\]. We employ the following equation which can be derived from its original version by a simple change of variables combined with a gauge transformation \[2\]: $$u_{xt}=u-2{\rm i}|u|^2u_x. \eqno(1.1)$$ Here, $u=u(x,t)$ is a complex-valued function of $x$ and $t$, and subscripts $x$ and $t$ appended to $u$ denote partial differentiations. The complete integrability of the FL equation has been demonstrated by means of the inverse scattering transform (IST) method \[3\]. Especially, a Lax pair and a few conservation 