  We present a simple and constructive method to find $N$-soliton solutions of
the equation suggested by Davydova and Lashkin to describe the dynamics of
nonlinear ion-cyclotron waves in a plasma and subsequently known (in a more
general form and as applied to nonlinear optics) as the Fokas-Lenells equation.
Using the classical inverse scattering transform approach, we find bright
$N$-soliton solutions, rational $N$-soliton solutions, and $N$-soliton
solutions in the form of a mixture of exponential and rational functions.
Explicit breather solutions are presented as examples. Unlike purely algebraic
constructions of the Hirota or Darboux type, we also give a general expression
for arbitrary initial data decaying at infinity, which contains the
contribution of the continuous spectrum (radiation).
