Solitons with varying amplitudes and/or varying velocities in a variable-coefficient mixed spectral generalized Sasa–Satsuma equation revealed through the Riemann–Hilbert method

The integrable Sasa–Satsuma (SS) equation, a higher-order Nonlinear Schrödinger (NLS)-type model, has important applications in the field of optics. In this study, we present a generalized SS equation with variable coefficients derived from a mixed spectral problem, abbreviated as vcmsgSSE, and construct its N-soliton solution using the Riemann–Hilbert (RH) method. First, we provide the Lax pair associated with the vcmsgSSE and perform a spectral analysis on it, which allows us to derive a solvable RH problem. Then, by solving this RH problem we construct an explicit expression for N-soliton solution of the vcmsgSSE. Finally, under a special reduction, we obtain the specific one-soliton solution and two-soliton solution of the vcmsgSSE, and use them to illustrate graphically the N-soliton solution with varying amplitude and/or varying velocity corresponds to the physical background of non-uniform medium assumed by the vcmsgSSE.