N-fold generalized Darboux transformation and asymptotic analysis of the degenerate solitons for the Sasa-Satsuma equation in fluid dynamics and nonlinear optics

In this paper, the Sasa-Satsuma equation in fluid dynamics and nonlinear optics is investigated. Starting from the first-order Darboux transformation, we construct an N-fold generalized Darboux transformation (GDT), where N is a positive integer. Through the obtained N-fold GDT, we derive three kinds of the semirational solutions, which describe the second-order degenerate solitons, third-order degenerate solitons and interaction between the second-order degenerate solitons and one soliton, respectively. We graphically illustrate the above three kinds of semirational solutions and investigate them through the asymptotic analysis, from which we find that the characteristic lines of the semirational solutions are composed of the straight lines and curves. Expressions of the characteristic lines, positions, amplitudes, slopes, positions and phase shifts of the asymptotic solitons are presented through the asymptotic analysis. The above discussions might be extended to the higher-order solitons, and to the relevant analysis on the degenerate breathers.