Casorati Determinant Form of Dark Soliton Solutions of the Discrete Nonlinear Schrödinger Equation

Abstract

It is shown that the N-dark soliton solutions of the integrable discrete nonlinear Schrödinger (IDNLS) equation are given in terms of the Casorati determinant. The conditions for reduction, complex conjugacy and regularity for the Casorati determinant solution are also given explicitly. The relationship between the IDNLS and the relativistic Toda lattice is discussed.

Bilinearisation-reduction approach to the nonlocal discrete nonlinear Schrödinger equations

“…Here we remark that for the continuous focusing NLS equation iq t = q xx + |q| 2 q where |q| 2 = qq * , its double Wronskian solution was given in [11] in 1983, while for the discrete NLS equation (3), although there were many discussions on its solutions [12][13][14], surprisingly, it seems there is no explicit double Casoratian form that was presented as its solutions. Besides, (18) does not provide a solution with nonzero background for equation (3) with δ = 1, i.e.…”

Isoperimetric deformations of curves on the Minkowski plane

“…The system of bilinear equations (4.9)-(4.11) are included in the discrete two-dimensional Toda lattice hierarchy [12,17,18,19,20,21,22,23,24]. A typical example of f .…”

Explicit solutions to the semi-discrete modified KdV equation and motion of discrete plane curves

“…To this end, we construct such formulas that express the determinants in the Plücker relations in terms of derivative or shift of discrete variable of ρ k l,m (u, v; y) by using the linear relations of the entries. For details of the technique, we refer to [9,[18][19][20][21].…”