Marchenko Equations and Norming Constants of the Matrix Zakharov-Shabat System

Abstract

A method is given to construct globally analytic (in space and time) exact solutions to the focusing cubic nonlinear Schrödinger equation on the line. An explicit formula and its equivalents are presented to express such exact solutions in a compact form in terms of matrix exponentials. Such exact solutions can alternatively be written explicitly as algebraic combinations of exponential, trigonometric and polynomial functions of the spatial and temporal coordinates.

Symmetries for exact solutions to the nonlinear Schrödinger equation

“…In previous papers [3,[6][7][8] we presented a method to construct exact solutions to (1.1) that are globally analytic on the entire xt-plane and decay exponentially as x → ±∞ at each fixed t ∈ R. This has been achieved by using a matrix triplet (A, B, C), where all eigenvalues of A have positive real parts. A similar method was applied to the Korteweg-de Vries equation on the half-line [4].…”

Inverse Scattering on the Half-Line for ZS-AKNS Systems with Integrable Potentials

“…In particular, Dirac systems with discontinuous coefficients and spectral parameter in the boundary conditions were studied in [14,32,62] and systems for nonself-adjoint Dirac operators were discussed in [5,60]. Recently, Demontis and van der Mee in [17][18][19] studied the problem of reconstructing non-self-adjoint 326 R. O. Hryniv and S. S. Manko IEOT matrix Zakharov-Shabat and AKNS systems. An interesting approach to inverse scattering analysis of more general systems was suggested by Beals and Coifman [8][9][10] and Beals, Deift, and Tomei [11].…”

“…part (1) of the above theorem) was incomplete. One of the motivation for this work was to suggest the approach that would (after suitable adaptation to the matrix case) fill in the gap in the proof of the paper [17]. Also, we give a rigorous proof of continuity of the inverse scattering transform.…”

“…The most essential component of the inverse scattering map requires solving the corresponding Marchenko equation; this is done by a convergent successive approximation method and thus provides the basis for a reconstruction algorithm. We also note that the case p = 1 for matrix potentials of arbitrary size was discussed in [17]; however, the analysis of the inverse scattering transform in that paper contains some inaccuracies which we had to fix.…”

“…In the paper [17] the authors considered an inverse scattering problem for a general matrix Dirac-type system with integrable potentials. However, the proof of the fact that the reconstructed potential is integrable (cf.…”