Lax pair, binary Darboux transformation and dark solitons for the three-component Gross–Pitaevskii system in the spinor Bose–Einstein condensate

Darboux transformation, bright and dark-bright solitons of an N-coupled high-order nonlinear Schrödinger system in an optical fiber

“…Combining Eqs. (4), ( 5) and ( 6), we get U [1] T [1] = T [1] x + T [1] U, V [1] T [1] = T [1] t + T [1] V.…”

“…All the dark solitons u [1] f and bright solitons u [1] e move at the same velocity. We find that |b j u j / b 2 j + µ 2 1 | 2 < |u j | 2 and √ ξµ 1 > 0, which imply that the amplitudes of u [1] f is lower than the background, but the amplitude of u [1] e is higher than the background. This also verified the previous N 1 components u [1] f are dark solitons and the rest N − N 1 components u [1] e are bright solitons.…”

“…where I is the (2N +1)th-order identity matrix, W [1] is a (2N +1)th-order square matrix, Φ satisfies Lax Pair (4) for the initial potential (q 1 , q 2 , ..., q N ) T (the superscript T denotes the conjugate transpose), Φ [1] satisfies Lax pair (4) for the new potential (q…”

“…where U [1] , V [1] have the same forms with U, V , respectively, except that the r j , s j in the matrices U, V are replaced with r [1] j , s [1] j in the matrices U [1] , V [1] . Combining Eqs.…”

“…, where φ h (h = 1, 2, ..., N + 1) is the function of x and t. If Φ is a eigenfunction of Lax Pair (4) with λ = λ 1 , then (φ * 2 , − φ * 1 , ..., 0) T , (φ 2 , 0, − φ * 1 , ..., 0) T , ..., (φ * N +1 , 0, ..., − φ * 1 , 0) T and (φ N +1 , 0, ..., − φ * 1 ) T are other eigenfunctions of Lax Pair (4) with λ = λ * 1 . We choose W [1] = H [1] J [1] (H [1] ) −1 ,…”