THE SASA-SATSUMA EQUATION ON A NON-ZERO BACKGROUND: THE INVERSE SCATTERING TRANSFORM AND MULTI-SOLITON SOLUTIONS

摘要

We concentrate on the inverse scattering transformation for the Sasa-Satsuma equation with 3×3 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 lowerhalf plane, we introduce certain auxiliary eigenfunctions which are necessary for deriving the analytic eigenfunctions in Dj. 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.