The unified transform method for the Sasa-Satsuma equation on the interval

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We present a Riemann-Hilbert problem formalism for the initial-boundary value problem for the Sasa-Satsuma(SS) equation on the finite interval. Assume that the solution existes, we show that this solution can be expressed in terms of the solution of a 3× 3 Riemann-Hilbert problem. The relevant jump matrices are explicitly given in terms of the three matrix-value spectral functions s(k), S(k) and Sₗ(k), which in turn are defined in terms of the initial values, boundary values at x=0 and boundary values at x=L, respectively. However, for a well-posed problem, only part of the boundary values can be prescribed, the remaining boundary data cannot be independently specified, but are determined by the so-called global relation. Here, we analyze the global relation to characterize the unknown boundary values in terms of the given initial and boundary data.