Initial-boundary value problems of the coupled Sasa-Satsuma equation on the half-line via the Fokas method

Abstract

In this paper, we apply the Fokas unified transform method to study the initial-boundary value problems for the coupled Sasa-Satsuma equation with a \(5\times 5\) Lax pair on the half-line. The solution of the coupled Sasa-Satsuma equation is proved to be expressible in terms of the unique solution of a \(5\times 5\) matrix Riemann-Hilbert problem in the complex k-plane. The relevant jump matrix is formulated using the matrix spectral functions S(k) and s(k), which are determined by the initial values and all boundary values at \(x=0\), respectively. While introducing the foundational Riemann-Hilbert formalism, we further investigate the corresponding generalized Dirichlet-Neumann mapping through the lens of the global relation. Moreover, by utilizing the perturbation expansion, we obtain an effective characterization of the unknown boundary values.