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

Abstract

In this paper, the Sasa-Satsuma equation, which is applied to the dynamics of the deep water waves, and the pulse propagation in the optical fibers and generally in the dispersive nonlinear media, is investigated. Starting from the first-order Darboux transformation, we construct an N -fold generalized Darboux transformation (GDT) for the Sasa-Satsuma equation, 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, the third-order degenerate solitons, and the 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.

Multi-pole solitons and breathers for a nonlocal Lakshmanan-Porsezian-Daniel equation with non-zero boundary conditions

“…The degenerate soliton solutions which are so called the multi-pole soliton solution in the terminology of inverse scattering transform is the remarkable property for the NLS equation [25,26]. The muilti-pole solution for the Hirota equation [27][28][29], the Sasa-Satsuma equation [30], and the LPD equation [31][32][33] have been investigated.…”

“…Eliminating the contributions of the simple poles equation (30) and the asymptotic behaviors of M 1± , one has…”

Letter to the Editor on HFF 32, 138 (2022); 32, 2282 (2022); 33, 965 (2023) and 34, 1189 (2024) for the recent shallow-water studies

“…Shallow-water special cases of system (1) have been seen in Ying and Lou (2000), Li and Zhang (2004), Ma et al (2015), Zhao and Han (2015), Kassem and Rashed (2019), (2022), Gao et al (2023a) as well as Liu et al (2023). Additionally, fluids from all over the Solar System have been observed and discussed (Lainey et al, 2024;Neish et al, 2024;Cheng et al, 2022Cheng et al, , 2023aCheng et al, , 2023bCheng et al, , 2024Feng et al, 2023;Gao, 2024aGao, , 2024dGao et al, 2023c;Shen et al, 2023cShen et al, , 2023dWu et al, 2023b;Zhou et al, 2023aZhou et al, , 2023bZhou et al, , 2024.…”

“…(2022), Gao et al (2023a) as well as Liu et al (2023). Additionally, fluids from all over the Solar System have been observed and discussed (Lainey et al , 2024; Neish et al , 2024; Cheng et al , 2022, 2023a, 2023b, 2024; Feng et al , 2023; Gao, 2024a, 2024d; Gao et al , 2023c; Shen et al , 2023c, 2023d; Wu et al , 2023b; Zhou et al , 2023a, 2023b, 2024).…”

Letter to the Editor: Thinking of the oceanic shallow water in the light of a (2+1)-dimensional generalized dispersive long-wave system related to HFF 33, 3272; 33, 965 and 32, 2282

“…By the way, with the view of investigating the oceanic shallow water (Cheng et al , 2023; Shen et al , 2023a, 2023b, 2023c; NOAA, 2023; Gao, 2023a, 2023b, 2023c; Wu et al ., 2023c; Zhou et al ., 2023a, 2023b, 2023c; Feng et al ., 2023; Liu et al ., 2019, 2021; Zayed, 2014; Liu et al , 2018; Gao et al , 2023b, 2023c), people have presented some other dispersive-type systems, e.g., a (2+1)-dimensional generalized modified dispersive water-wave system modeling some nonlinear and dispersive long gravity waves traveling along two horizontal directions in the shallow water of uniform depth (Gao et al , 2023b), a variable-coefficient generalized dispersive water-wave system modeling certain long-weakly nonlinear and weakly dispersive surface waves of variable depth in the shallow water (Zayed, 2014; Liu et al , 2018) and a variable-coefficient dispersive-wave system modeling certain long gravity waves in a shallow ocean (Gao et al , 2023c).…”