Constructions of higher-order N -soliton solutions for massive Thirring model

We investigate the inverse scattering problem for the massive Thirring model, focusing particularly on cases where the transmission coefficient exhibits N pairs of higher-order poles. Our methodology involves transforming initial data into scattering data via the direct scattering problem. Utilizing two parameter transformations, we examine the asymptotic properties of the Jost functions at both vanishing and infinite parameters, yielding two equivalent spectral problems. We subsequently devise a mapping that translates the obtained scattering data into a 2 × 2 matrix Riemann–Hilbert (RH) problem, incorporating several residue conditions at N pairs of multiple poles. Additionally, we construct an equivalent pole-free RH problem and demonstrate the existence and uniqueness of its solution. In the reflectionless case, the higher-order N-soliton solutions can be reconstructed by resolving two linear algebraic systems. Furthermore, we present a visual demonstration of the interaction dynamics among higher-order multi-soliton solutions. This contributes to a deeper understanding of the structure of higher-order multi-solitons and offers valuable insights for the design of related physical experiments.