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The Sasa-Satsuma equation with non-vanishing boundary conditions

We concentrate on inverse scattering transformation for the Sasa-Satsuma equation with $3\times 3$ matrix spectral and nonzero boundary condition in this article. To circumvent multi valuedness of eigenvalues, we introduce a suitable two-sheet Riemann surface to map the original spectral parameter $k$ into a single-valued parameter $z$. The analyticity of the Jost eigenfunctions and scattering coefficients of Lax pair for the SS equation are analyzed in details. According to the analyticity of eigenfunctions and scattering coefficients, the $z$-complex plane is divided into four analytic regions $D_j, \ j=1, 2, 3, 4$. Since the second column of Jost eigenfunctions is analytic in $D_{j}, \ j=1, 2, 3, 4$, but in upper-half or lower-half plane, we introduce certain auxiliary eigenfunctions which are necessary for deriving the analytic eigenfunctions in $D_{j}$. We find that for the eigenfunctions, scattering coefficients and the auxiliary eigenfunctions all possess three kinds of symmetries, which characterize the distribution of discrete spectrum. The asymptotic behaviors of eigenfunctions, auxiliary eigenfunctions and scattering coefficients are also systematically derived. Then a matrix Riemann-Hilbert problem with four kind jump conditions associated with the problem of nonzero asymptotic boundary conditions is established, from which $N$-soliton solutions is obtained via the corresponding reconstruction formulae. The reflectionless soliton solutions are explicitly given. As application of the $N$-soliton formula, we present three kinds of single-soliton solutions according to the distribution of discrete spectrum.