Naruyoshi Asano

Non‐self‐adjoint Zakharov–Shabat operator with a potential of the finite asymptotic values. I. Direct spectral and scattering problems

The Zakharov and Shabat equation for the scattering problem is studied: The estimates, analytical properties, and asymptotic expansions of the Jost solution are presented for a general class of the potentials Q(x) not vanishing at infinity. The existence of the similarity transformation is also shown. For Q(x) vanishing at infinity, the continuous part of the spectrum doubly degenerates. However, nonvanishing (finite) asymptotic values of Q(x) dissolve the degeneracy completely. The expansion theorem is given in C02(R) and for a class of Q(x) we prove that the Zakharov and Shabat equation yields a non‐self‐adjoint spectral operator in the Hilbert space in the sense of Dunford and Schwartz.

Non‐self‐adjoint Zakharov–Shabat operator with a potential of the finite asymptotic values. II. Inverse problem

The inverse spectral and scattering problem of the Zakharov–Shabat (ZS) operator is studied. The similarity transformation between ZS operators is examined when their potentials Q(x) have the common nonvanishing asymptotic values Q± at the infinity. The Marchenko equation is derived from the Parseval equation. We give the necessary as well as the sufficient condition of the scattering data for the potential of the specified class.

Fredholm Determinant Solution for the Inverse Scattering Transform of the N × N Zakharov-Shabat Equation

We give an explicit solution of the Riemann-Hilbert problem with pole singularity for the NxN Zakharov-Shabat equation under the assumption of the existence and the uniqueness of the solution. For a general class of the scattering data the solution of the Marchenko integral equation which was derived in our previous article is exhibited in a quite simple form in terms of the Fredholm deter. minant and its first minors of the integral kernel. It is shown that the potential and the Jost solution derived in this way surely compose the Zakharov-Shabat equation.

Coordinate transformations for singular perturbation problem

Singular perturbation methods are refomulated with the aid of Lies invariant transformation group. Singularities are analized through the expansion in small parameter and in order to modefy the singularity new independent variables are introduced as the canonical coordinates for the projectable group . Several examples of the algebraic and the differential equations are shown.