A binary Darboux transformation for multicomponent NLS equations and their reductions

N-Fold Darboux Transformation for the Classical Three-Component Nonlinear Schrödinger Equations and Its Exact Solutions

“…Let n ∈ N, I n denote the identity matrix of size; n, α 1 and α 2 are different constants. The matrix spectral problems of the NLS equations read [30]…”

“…when p j = q j = 0, 2 ≤ j ≤ n, the spectral problem in (1) becomes the standard AKNS spectral problem [31]. Now, we consider the system as follows [30]:…”

“…In [30], the classical three-component NLS equations belong to the category of evolutionary equations, for the solution of which the method of the inverse scattering problem is successfully applied.…”

A second-order three-wave interaction system and its rogue wave solutions

“…The three potentials u, v, w in (1.1) are complex functions of two real independent variables x and t, the constants c j and ĉ j satisfy a certain relationship between them. System (1.1) is completely integrable in the sense that it admits a Lax pair representation, which was discovered by Zakharov, Manakov [4] and Kaup [2], from which some explicit solutions of the three-wave interaction system were obtained by using the inverse scattering transformation, Darboux transformation and other approaches [5][6][7][8][9][10][11][12][13][14][15][16][17][18]. It was proved that the finitedimensional Hamiltonian system [20] associated with the three-wave interaction system is completely integrable in the Liouville sense and the involutive solutions of the three-wave interaction system were given with the aid of the nonlinearization of Lax pair [19].…”

X-type Soliton, Resonance Y-type Soliton and Hybrid Solutions of a (2+1)-dimensional Hirota-Satsuma-Ito System for the Shallow Water Waves

“…As some NLEEs, shallow-water equations have been used to depict the horizontal structure of an atmosphere and the evolution of an incompressible fluid [23]. There have been some methods to solve the NLEEs, including the Lie symmetry approach, Bäcklund transformation, Darboux transformation, Hirota method, and so on [24][25][26][27][28][29][30].…”