The initial-boundary value problem for the Sasa-Satsuma equation on a finite interval via the Fokas method

Abstract

In this paper, the initial-boundary value problem for the Sasa-Satsuma equation on the finite interval is studied by using the Fokas method (or called unified transform method). We show that the solution of this problem 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 u0(x), boundary values at x = 0, i.e, {g0(t), g1(t), g2(t)}, and boundary values at x = L, i.e, {f0(t), f1(t), f2(t)}, respectively. However, for a well-posed problem, only part of the boundary values can be prescribed, and the remaining boundary data cannot be independently specified but are determined by the so-called global relation. As the Lax pair is 3 × 3 and the derivative order of the Sasa-Satsuma equation is three, the analysis of the global relation is more complicated. We show that if we know the initial and partial boundary values {g0(t), f0(t), f1(t)}, then we can characterize the unknown boundary values {g1(t), g2(t), f2(t)} via a Gelfand-Levitan-Marchenko representation of the eigenfunctions {Φij(t, k), ϕij(t, k)} though analyzing the global relation.