Integrable semi-discretization of the coupled nonlinear Schrödinger equations

Abstract

Abstract. A system of semi-discrete coupled nonlinear Schrödinger equations is studied. To show the complete integrability of the model with multiple components, we extend the discrete version of the inverse scattering method for the single-component discrete nonlinear Schrödinger equation proposed by Ablowitz and Ladik. By means of the extension, the initial-value problem of the model is solved. Further, the integrals of motion and the soliton solutions are constructed within the framework of the extension of the inverse scattering method. †

An Alternative Approach to Integrable Discrete Nonlinear Schrödinger Equations

“…Comprehensive accounts of its scattering theory were given in [6] and [24]. In [6,24] the matrix equations (1.1a), (1.1b) were studied in detail using the IST.…”

“…Comprehensive accounts of its scattering theory were given in [6] and [24]. In [6,24] the matrix equations (1.1a), (1.1b) were studied in detail using the IST. In [1,25] the defocussing N = M = 1 problem was studied for potentials not vanishing as n → ±∞.…”

“…This forward difference scheme leads to the spectral problem (1.2) which lacks forward-backward symmetry and requires the assumption that the square matrix of order N + M describing the transition v n → v n+1 is invertible for each n ∈ Z. In fact, the authors of [6,24] found it useful to essentially "scalarize" the nonlinear evolution system (1.1a), (1.1b) by requiring the potentials {U n } ∞ n=−∞ and {W n } ∞ n=−∞ to also satisfy the condition U n W n = W n U n = c n I N , n∈ Z, (1.4) where N = M and, for each n ∈ Z, c n ( = 1) is an unknown complex number. Assuming (1.4), in the focusing case the scattering matrix, when multiplied by a weight matrix, appears to be symplectically unitary [10].…”

“…Without assuming condition (1.4), a priori information on the IDNLS solution is required. After the N -soliton and breather solutions to (1.1a), (1.1b) under condition (1.4) were derived before in terms of solutions to N × N linear systems [24,Eq. (3.43)], breather solutions were constructed before by using the Hirota method [7], and one and two soliton solutions were derived by various methods [5], an extensive family of IDNLS solutions in terms of triplets of matrices parametrizing the Marchenko kernel was derived in [10,11].…”

“…The time evolution of the reflection coefficients, norming constants, and Marchenko kernels associated with the discrete Zakharov-Shabat system (1.2) leading to IDNLS solutions has been explained in detail in [6], also without assuming condition (1.4). The renormalized Marchenko equations given in [6,10,24] have the form…”