Search

In this paper we study the semiclassical behavior of the scattering data of a non-self-adjoint Dirac operator with a real, positive, multi-humped, fairly smooth but not necessarily analytic potential decaying at infinity. We provide the rigorous semiclassical analysis of the Bohr-Sommerfeld condition for the location of the eigenvalues, the norming constants, and the reflection coefficient. [ABSTRACT FROM AUTHOR]

Inverse scattering problem for the operator representing sum of the operator of the third derivative on semi-axis and of the operator of multiplication by a real function is studied in this paper. Properties of Jost solutions of such an operator are studied and it is shown that these Jost solutions are solutions of the Riemann boundary value problem on a system of rays. The main system of linear singular integral equations is derived. This system is equivalent to the solution of inverse scattering problem. [ABSTRACT FROM AUTHOR]

Direct and inverse scattering problem for an operator with non-local potential is solved in the paper. The method is based on the Riemann boundary value problem on a bundle of three straight lines. Description of scattering problem data is given. [ABSTRACT FROM AUTHOR]

The work is devoted to solving the Cauchy problem for the modified Korteweg-de Vries–sine-Gordon (mKdV–sG) equation with an additional term in the class of rapidly decreasing functions. The problem is solved by the inverse scattering method. Several examples are given illustrating the application of the obtained results. [ABSTRACT FROM AUTHOR]

It is well-known that spectral singularities play a certain critical role in the spectral theory of nonselfadjoint differential operators. Their existence is accompanied by specific phenomenon which are new in the sense that they do not occur either in the spectral theory of self-adjoint or normal operators. So, in this paper, we examine spectrum and spectral singularities of the impulsive quadratic pencil of Schrödinger operators. In particular, we study the dependence of the structure of discrete spectrum and spectral singularities of these operators of the behaviour of the potential functions at infinity. We present some conditions on potential functions which guarantee that the impulsive quadratic pencil of Schrödinger operators have a finite number of eigenvalues and spectral singularities with finite multiplicities. Later, we discuss the properties of principal functions corresponding to the spectral singularities. [ABSTRACT FROM AUTHOR]

The work is devoted to solving the Cauchy problem for the sine-Gordon equation with an additional term and a self-consistent source in the class of rapidly decreasing functions. The problem is solved by the inverse scattering method. Several special cases of the sine-Gordon equation with an additional term are given, which can be integrated using the inverse scattering method, for example, the loaded sine-Gordon equation. Several examples are given to illustrate the application of the obtained results. [ABSTRACT FROM AUTHOR]

We derive dispersion estimates for solutions of a one‐dimensional discrete Dirac equations with a potential. In particular, we improve our previous result, weakening the conditions on the potential. To this end we also provide new results concerning scattering for the corresponding perturbed Dirac operators which are of independent interest. Most notably, we show that the reflection and transmission coefficients belong to the Wiener algebra. [ABSTRACT FROM AUTHOR]

In this paper, we explore the Jost solutions and the scattering matrix of the impulsive difference Dirac systems (IDDS) on the whole axis and study their analytic and asymptotic properties. Furthermore, characteristic properties of the scattering matrix of the IDDS have been examined. [ABSTRACT FROM AUTHOR]

In this paper, we study the Jost solutions of the impulsive Dirac systems (IDS) on entire axis and examine analytic and asymptotic properties of these solutions. Furthermore, we obtain a general form of the scattering matrix of the IDS and its characteristic properties. Finally, we also compare the similar properties for the IDS with the mass on entire axis with an example. [ABSTRACT FROM AUTHOR]

In this paper, we present a rigorous study of the direct scattering problem that arises from the complete integrability of the Benjamin-Ono (BO) equation. In particular, we establish existence, uniqueness, and asymptotic properties of the Jost solutions to the scattering operator in the Fokas{Ablowitz inverse scattering transform (IST). Formulas relating different scattering coefficients are proven, together with their asymptotic behavior with respect to the spectral parameter. This work is an initial step toward the construction of general solutions to the BO equation by IST. [ABSTRACT FROM AUTHOR]

Our goal is to find an asymptotic behavior as n → ∞ of the orthogonal polynomials P n (z) defined by Jacobi recurrence coefficients a n (off-diagonal terms) and b n (diagonal terms). We consider the case a n → ∞ , b n → ∞ in such a way that ∑ a n − 1 < ∞ (that is, the Carleman condition is violated) and γ n : = 2 − 1 b n (a n a n − 1) − 1 ∕ 2 → γ as n → ∞. In the case | γ | ≠ 1 asymptotic formulas for P n (z) are known; they depend crucially on the sign of | γ | − 1. We study the critical case | γ | = 1. The formulas obtained are qualitatively different in the cases | γ n | → 1 − 0 and | γ n | → 1 + 0. Another goal of the paper is to advocate an approach to a study of asymptotic behavior of P n (z) based on a close analogy of the Jacobi difference equations and differential equations of Schrödinger type. [ABSTRACT FROM AUTHOR]