The massive Thirring model in Bragg grating: Soliton molecules, breather-positon and semirational solutions

The massive Thirring model, which describes pulse propagation in Bragg grating, is systematically investigated through Darboux transformation. Based on the Darboux transformation, we study soliton molecules, breather-positons, and semirational solutions for the massive Thirring model in this paper. What we present includes the following main results: (1) the general multiple soliton molecules (SMs) interaction, namely, $M$ SMs interact with $(N-M)$ solitons ($0\le M\le N$); (2) higher-order breather-positon solutions whose center region exhibit rogue waves’ patterns; and (3) semirational solutions with arbitrary $M$th-order rogue waves and $(N-M)$-breathers. Finally, the generating mechanisms and related dynamics of those obtained nonlinear localized waves are discussed in details.