Darboux System: Liouville Reduction and an Explicit Solution

For a Darboux system in ?³, we introduce a class of solutions for which an auxiliary second-order linear problem satisfies the factorization condition. We show that this reduction provides the (local) solvability of the Darboux system, and present an explicit solution to this problem for two types of dependent variables. We also construct explicit formulas for the Lamé coefficients and solutions to the associated linear problem. The previously known reduction to a weakly nonlinear system is shown to be a particular case of the approach proposed.     

Scattering problem for the differential operator ∂ x ∂ y +1+a(x,y)∂ y +a(x,y)

The scattering problem for the two-dimensional Klein — Gordon differential operator with variable coefficients is studied in the framework of the resolvent approach. Jost solutions, retarded and advanced solutions, and spectral data are introduced, and their properties are described. The inverse scattering problem is formulated.V. A. Steklov Mathematics Institute, Russian Academy of Sciences (e-mail:POGREB@QFT.MIAN.SU). Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 102, No. 2, pp. 163–182, February, 1995.     

An Ablowitz-Ladik system with a discrete potential: I. Extended resolvent

An Ablowitz-Ladik linear system with the potential taking the values 0 or 1 is considered. The extended resolvent of this system is constructed, and the singularities of this operator are analyzed in detail. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 119, No. 1, pp. 20–33, April, 1999.     

Towards an inverse scattering theory for two-dimensional nondecaying potentials

The inverse scattering method is considered for the nonstationary Schrödinger equation with the potentialu (x ₁,x ₂) nondecaying in a finite number of directions in thex plane. The general resolvent approach, which is particularly convenient for this problem, is tested using a potential that is the Bäcklund transformation of an arbitrary decaying potential and that describes a soliton superimposed on an arbitrary background. In this example, the resolvent, Jost solutions, and spectral data are explicitly constructed, and their properties are analyzed. The characterization equations satisfied by the spectral data are derived, and the unique solution of the inverse problem is obtained. The asymptotic potential behavior at large distances is also studied in detail. The obtained resolvent is used in a dressing procedure to show that with more general nondecaying potentials, the Jost solutions may have an additional cut in the spectral-parameter complex domain. The necessary and sufficient condition for the absence of this additional cut is formulated.