We consider the one-dimensional cubic fractional nonlinear Schr\"odinger
equation i∂tu−(−Δ)σu+∣u∣2u=0, where σ∈(21,1) and the operator (−Δ)σ is the fractional Laplacian of
symbol ∣ξ∣2σ. Despite of lack of any Galilean-type invariance, we
construct a new class of traveling soliton solutions of the form
u(t,x)=e−it(∣k∣2σ−ω2σ)Qω,k(x−2tσ∣k∣2σ−2k),k∈R,ω>0 by a rather involved variational argument