Novel Formulation of Inverse Scattering and Characterization of Scattering Data

In this article we formulate the direct and inverse scattering theory for the focusing matrix Zakharov-Shabat system as the construction of a 1, 1-correspondence between focusing potentials with entries in L1(R) and Marchenko integral kernels, given the fact that these kernels encode the usual scattering data (one reflection coefficient, the discrete eigenvalues with positive imaginary part, and the corresponding norming constants) faithfully. In the reflectionless case, we solve the Marchenko equations explicitly using matrix triplets and obtain focusing matrix NLS solutions in closed form. 1. Direct and inverse scattering theory. Consider the focusing matrix nonlinear Schrödinger (NLS) equation iut + uxx + 2uu †u = 0, (1) where u = u(x, t) is an m × n matrix function depending on position x ∈ R and time t ∈ R and the dagger indicates the matrix conjugate transpose. By means of the inverse scattering transform (IST), (1) is associated with the focusing matrix Zakharov-Shabat problem iJ ∂X ∂x − V (x, t)X(λ, x; t) = λX(λ, x; t), (2) where J = ( Im 0m×n 0n×m −In ) , V (x, t) = ( 0m×m iu(x, t) iu(x, t)† 0n×n ) , the potential u(x, t) has its entries in L(R; dx) for each t ∈ R, and λ is a spectral parameter. For background material we refer to the standard sources (e.g., [2, 1, 11, 13]). Let us introduce the (m+ n)×m and (m+ n)× n Jost functions from the right ψ(λ, x) and ψ(λ, x), the (m+ n)×m and (m+ n)× n Jost solutions from the left φ(λ, x) and φ(λ, x), and the (m+n)×(m+n) Jost matrices Ψ(λ, x) and Φ(λ, x) from 2000 Mathematics Subject Classification. Primary: 35Q55, 37K15.