  In this paper, we consider the initial value problem for both of the
defocusing and focusing Kundu-Eckhaus (KE) equation with non-zero boundary
conditions (NZBCs) at infinity by inverse scattering transform method. The
solutions of the KE equation with NZBCs can be reconstructed in terms of the
solution of an associated $2 \times 2$ matrix Riemann-Hilbert problem (RHP). In
our formulation, both the direct and the inverse problems are posed in terms of
a suitable uniformization variable which allows us to develop the IST on the
standard complex plane instead of a two-sheeted Riemann surface or the cut
plane with discontinuities along the cuts. Furthermore, on the one hand, we
obtain the N-soliton solutions with simple pole of the defocusing and focusing
KE equation with the NZBCs, especially, the explicit one-soliton solutions are
given in details. And we prove that the scattering data $a(\zeta)$ of the
defocusing KE equation can only have simple zeros. On the other hand, we also
obtain the soliton solutions with double pole of the focusing KE equation with
NZBCs. And we show that the double pole solutions can be viewed as some proper
limit of the two simple pole soliton solutions. Some dynamical behaviors and
typical collisions of the soliton solutions of both of the defocusing and
focusing KE equation are shown graphically.
