Inverse scattering transform for the Sasa–Satsuma equation: multiple-pole case of N pairs

The inverse scattering transform is used to study the Sasa–Satsuma equation with the associated transmission coefficient consisting of N (an arbitrary finite number) pairs of higher-order poles. The direct problem is introduced by analyzing the discrete spectrum associated with N pairs of multiple zeros, and the inverse problem is characterized in terms of a \(3\times 3\) $3\times 3$ matrix Riemann–Hilbert problem equipped with several residue conditions at N pairs of multiple poles. In the reflectionless case, it is shown that N-multipole solutions of the Sasa–Satsuma equation can be constructed by a linear algebraic system, whose solution is unique. As an application, several explicit solutions of the Sasa-Satsuma equation are obtained by solving the linear algebraic system, including higher-order single-humped and double-humped solitons, higher-order breathers, and others.