We concentrate on the inverse scattering transformation for the Sasa-Satsuma equation with $3\times 3$ matrix spectrum problem and a nonzero boundary condition. To circumvent the multi-value 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 the Lax pair for the Sasa-Satsuma equation are analyzed in detail. According to the analyticity of the eigenfunctions and the scattering coefficients, the $z$-complex plane is divided into four analytic regions of $D_j: \ j=1, 2, 3, 4$. Since the second column of Jost eigenfunctions is analytic in $D_{j}$, but in the 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 the eigenfunctions, the scattering coefficients and the auxiliary eigenfunctions all possess three kinds of symmetries; these characterize the distribution of the discrete spectrum. The asymptotic behaviors of eigenfunctions, auxiliary eigenfunctions and scattering coefficients are also systematically derived. Then a matrix Riemann-Hilbert problem with four kinds of jump conditions associated with the problem of nonzero asymptotic boundary conditions is established, from this $N$-soliton solutions are obtained via the corresponding reconstruction formulae. The reflectionless soliton solutions are explicitly given. As an application of the $N$-soliton formula, we present three kinds of single-soliton solutions according to the distribution of discrete spectrum.
[1] Zakharov V E, Shabat A B.Exact theory of two-dimensional self-focusing and one-dimensional selfmodulation of waves in nonlinear media. Sov Phys JETP, 1972, 34: 62-69
[2] Hasegawa A, Tappert F D.Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion. Appl Phys Lett, 1973, 23: 142-144
[3] Mollenauer L F, Stolen R H, Gordon J P.Experimental observation of picosecond pulse narrowing and solitons in optical fibers. Phys Rev Lett, 1980, 45(13): 1095-1098
[4] Kodama Y.Optical solitons in a monomode fiber. J Stat Phys, 1985, 39(5/6): 597-614
[5] Kodama Y, Hasegawa A.Nonlinear pulse propagation in a monomode dielectric guide. IEEE J Quantum Elect, 1987, 23(5): 510-524
[6] Anderson D, Lisak M.Nonlinear asymmetric self-phase modulation and self-steepening of pulses in long optical waveguides. Phys Rev A, 1983, 27(3): 1393-1398
[7] Chen H H, Lee Y C, Liu C S.Integrability of nonlinear Hamiltonian systems by inverse scattering method. Phys Scr, 1979, 20(3/4): 490-492
[8] Hirota R.Exact envelope soliton solutions of a nonlinear wave equation. J Math Phys, 1973, 14(7): 805-809
[9] Wadati M, Sogo K.Gauge transformations in soliton theory. J Phys Soc Jpn, 1983, 52(2): 394-398
[10] Sasa N, Satsuma J.New-type of soliton solutions for a higher-order nonlinear Schrüodinger equation. J Phys Soc Jpn, 1991, 60(2): 409-417
[11] Chen S H.Twisted rogue-wave pairs in the Sasa-Satsuma equation. Phys Rev E, 2013, 88(2): 023202
[12] Akhmediev N, Soto-Crespo J M, Devine N, et al. Rogue wave spectra of the Sasa-Satsuma equation. Phys D, 2015, 294: 37-42
[13] Soto-Crespo J M, Devine N, Hoffmann N P, et al. Rogue waves of the Sasa-Satsuma equation in a chaotic wave field. Phys Rev E, 2014, 90(3): 032902
[14] Ghosh S, Kundu A, Nandy S.Soliton solutions, Liouville integrability and gauge equivalence of Sasa- Satsuma equation. J Math Phys, 19999, 40(4): 1993-2000
[15] Gilson C, Hietarinta J, Nimmo J, et al.Sasa-Satsuma higher-order nonlinear Schrüodinger equation and its bilinearization and multisoliton solutions. Phys Rev E, 2003, 68(1): 016614
[16] Yang J, Kaup D J.Squared eigenfunctions for the Sasa-Satsuma equation. J Math Phys, 2009, 50(2): 023504
[17] Xu J, Fan E G.The unified transform method for the Sasa-Satsuma equation on the half-line. Proc Roy Soc A-Math Phy, 2013, 469(2159): 20130068
[18] Wu J P, Geng X G.Inverse scattering transform of the coupled Sasa-Satsuma equation by Riemann-Hilbert approach. Commun Theor Phys, 2017, 67(5): 527-534
[19] Liu H, Geng X G, Xue B.The Deift-Zhou steepest descent method to long-time asymptotics for the Sasa- Satsuma equation. J Differ Equ, 2018, 265(11): 5984-6008
[20] Yang B, Chen Y.High-order soliton matrices for Sasa-Satsuma equation via local Riemann-Hilbert problem. Nonlinear Anal: Real World Appl, 2019, 45: 918-941
[21] Zhang H Q, Wang Y, Ma W X.Binary Darboux transformation for the coupled Sasa-Satsuma equations. Chaos, 2017, 27(7): 073102
[22] Lüu X.Bright-soliton collisions with shape change by intensity redistribution for the coupled Sasa-Satsuma system in the optical fiber communications. Commun Nonlinear Sci Numer Simul, 2014, 19(11): 3969-3987
[23] Nakkeeran K, Porsezian K, Sundaram P S, et al.Optical solitons in N-coupled higher order nonlinear Schrüodinger equations. Phys Rev Lett, 1998, 80(7): 1425-1428
[24] Gardner C S, Greene J M, Kruskal M D, et al.Method for solving the Korteweg-de Vries equation. Phys Rev Lett, 1967, 19: 1095-1097
[25] Gardner C S, Greene J M, Kruskal M D, et al.Korteweg-de Vries equation and generalizations VI methods for exact solution. Comm Pure Appl Math, 1974, 27: 97-133
[26] Zakharov V E, Shabat A B.Interaction between solitons in a stable medium. Sov Phys-JETP, 1973, 37(5): 1627-1639
[27] Zakharov V E, Shabat A B.A scheme for integrating the nonlinear equations of mathematical physics by the method of the inverse scattering problem II. Funk Anal Pril, 1979, 13: 13-22
[28] Tovbis A, Venakides S, Zhou X.On semiclassical (zero dispersion limit) solutions of the focusing nonlinear Schrüodinger equation. Commun Pure Appl Math, 2004, 57(7): 877-985
[29] Xu J, Fan E G, Chen Y.Long-time asymptotic for the derivative nonlinear Schrüodinger equation with step-like initial value. Math Phys Anal Geom, 2013, 16(3): 253-288
[30] Jenkins R, McLaughlin K D T R. The semiclassical limit of focusing NLS for a family of square barrier initial data. Commun Pure Appl Math, 2014, 67(2): 246-320
[31] Yamane H.Long-time asymptotics for the defocusing integrable discrete nonlinear Schrüodinger equation. J Math Soc Jpn, 2014, 66(3): 765-803
[32] Buckingham R, Venakides S.Long-time asymptotics of the nonlinear Schrüodinger equation shock problem. Commun Pure Appl Math, 2007, 60(9): 1349-1414
[33] Tovbis A, El G A.Semiclassical limit of the focusing NLS: Whitham equations and the Riemann-Hilbert problem approach. Physica D, 2016, 333: 171-184
[34] Yang J K.Nonlinear Waves in Intergrable and Nonintergrable Systems. Philadelphia: SIAM Soc, 2010
[35] Biondini G, Kovač7;cič7;c G.Inverse scattering transform for the focusing nonlinear Schrüodinger equation with nonzero boundary conditions. J Math Phys, 2014, 55(3): 031506
[36] Kraus D, Biondini G, Kovač7;cič7;c G.The focusing Manakov system with nonzero boudary conditions. Nonlinearity, 2015, 28(9): 3101-3151
[37] Ablowitz M J, Biondini G, Prinari B.Inverse scattering transform for the integrable discrete nonlinear Schrüodinger equation with nonvanishing boundary conditions. Inverse Problems, 2007, 23(4): 1711-1758
[38] Prinari B, Biondini G, Trubatch A D.Inverse scattering transform for the multi-component nonlinear Schrüodinger equation with nonzero boundary conditions. Stud Appl Math, 2011, 126(3): 245-302
[39] Prinari B, Ablowitz M J, Biondini G.Inverse scattering transform for the vector nonlinear Schrüodinger equation with non-vanishing boundary conditions. J Math Phys, 2006, 47(6): 063508
[40] Biondini G, Mantzavinos D.Long-time asymptotics for the focusing nonlinear Schrüodinger equation with nonzero boundary conditions at infinity and asymptotic stage of modulational instability. Commun Pure Appl Math, 2017, 70(12): 2300-2365
[41] Zhang Z C, Fan E G. Inverse scattering transform for the Gerdjikov-Ivanov equation with nonzero boundary conditions. Z Angew Math Phys, 2020, 71(5): Art 149
[42] Wen L L, Fan E G.The Riemann-Hilbert approach to focusing Kundu-Eckhaus equation with non-zero boundary conditions. Mod Phys Lett B, 2020, 34(30): 2050332
