  In this article, we focus on investigating the focusing Kundu-Eckhaus
equation with nonzero boundary condition. A appropriate two-sheeted Riemann
surface is introduced to map the spectral parameter $k$ into a single-valued
parameter $z$. Starting from the Lax pair of Kundu-Eckhaus equation,two kind of
Jost solutions are construed. Further their asymptotic, analyticity, symmetries
as well as spectral matrix are detailed analyzed. It is shown that the solution
of Kundu-Eckhaus equation with nonzero boundary condition can characterized
with a matrix Riemann-Hilbert problem. Then a formula of $N$-soliton solutions
is derived by solving Riemann-Hilbert problem. As applications, the first-order
explicit soliton solution is obtained.

Exactly Solvable and Integrable Systems