N-Fold generalized Darboux transformation and semirational solutions for the Gerdjikov-Ivanov equation for the Alfvén waves in a plasma

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

“…Next, motivated by Refs. [24][25][26], n spectral parameters are used to construct an N -fold GDT for Eq. ( 1), where n is a positive integer and n ≤ N .…”

“…Without loss of generality, we assume that 0 < |ω| < ρ. For Solutions (24), if η is fixed, α can be expressed as…”

Modulation instability, localized wave solutions of the modified Gerdjikov–Ivanov equation with anomalous dispersion

“…Rogue waves (RWs) with a variety of properties have been uncovered as a hot research topic in recent years. It first appeared in the deep ocean in the form of an extremely destructive wave, surprisingly, RWs have also been found in various physical systems, such as optical fiber systems, water fluids, plasma, Bose-Einstein condensates and even financial systems [11][12][13][14][15][16][17][18][19], etc., which are currently more popular. There is a consensus in the current study of RWs that they are produced by tuning to modulation instability (MI) waves, and the breather solution involved in them usually come from instabilities with small amplitude perturbations, the magnitude of which has the potential to affect the degree of catastrophe suffered.…”

“…where m i , (i = 0, 1, 2, ...) represent the highest-order derivative term of the Taylor expansion of T and Φ in the above system for n different spectral parameters µ i , in actual calculation, the relationship between m i and N is N = n + n i=1 m i . Subsequently, with the help of Cramer's rule, A (2j) ,B (2j+1) , (j = 0, 1, ..., N − 1) can be determined by solving the system of algebraic equations (13). If the appropriate spectral parameters µ i can be The collision of localized waves to the Gerdjikov-Ivanov equation selected so that the determinant of the coefficient matrix in the system ( 13) is nonzero, then T can be uniquely determined by system (13).…”

“…Subsequently, with the help of Cramer's rule, A (2j) ,B (2j+1) , (j = 0, 1, ..., N − 1) can be determined by solving the system of algebraic equations (13). If the appropriate spectral parameters µ i can be The collision of localized waves to the Gerdjikov-Ivanov equation selected so that the determinant of the coefficient matrix in the system ( 13) is nonzero, then T can be uniquely determined by system (13). It also proves that the new Darboux matrix T , which consists of the coefficients of Taylor's expansion and the corresponding derivative terms, still satisfies Theorem.…”