Alexei Rybkin

SOME NEW AND OLD ASYMPTOTIC REPRESENTATIONS OF THE JOST SOLUTION AND THE WEYL m -FUNCTION FOR SCHRÖDINGER OPERATORS ON THE LINE

For the general one-dimensional Schrodinger operator − d 2 / dx 2 + q ( x ) with real q ∈ L 1 (ℝ), this paper presents a new series representation of the Jost solution which, in turn, implies a new asymptotic representation of the Weyl m -function for locally summable q . This representation is then applied to smooth potentials q to obtain Weyl m -function power asymptotics. The condition q ( N ) ∈ L 1 ( x 0 , x 0 +δ), for N ∈ ℕ 0 , allows one to derive the ( N +1) term for almost all x ∈ [ x 0 , x 0 +δ), thereby refining a relevant result by Danielyan, Levitan and Simon.

Meromorphic solutions to the KdV equation with non-decaying initial data supported on a left half line

We show that under the Korteweg–de Vries flow an initial profile supported on (−∞, 0) from a very broad function class (without any decay assumption) instantaneously evolves into a meromorphic function with no poles on the real line. Our treatment is based on a suitable modification of the inverse scattering transform and a detailed investigation of the Titchmarsh–Weyl m-function. As a by-product, we improve some related results of others.

ON THE TRACE APPROACH TO THE INVERSE SCATTERING PROBLEM IN DIMENSION ONE

We present an elementary procedure to derive certain trace relations for Schrodinger operators in dimension one relating potentials with some scattering data. In this way, we obtain some new trace type formulas as well as known ones previously studied by Gesztesy, Holden, Simon, and others, our a priori hypothesis on potentials being minimal. We also establish sharp conditions on absolute summability of the trace formulas.

On a trace formula of the Buslaev–Faddeev type for a long-range potential

We propose an approach to obtaining new trace formulas of the Gel’fand–Levitan–Buslaev–Faddeev type, valid for Hilbert–Schmidt perturbations. In this way we obtain a new trace formula for Schrodinger operators on the half-line with long-range potentials.

Higher order trace formulas of the Buslaev–Faddeev-type for the half-line Schrödinger operator with long-range potentials

We deal with trace formulas for half-line Schrodinger operators with long-range potentials. We generalize the Buslaev–Faddeev trace formulas to the case of square integrable potentials. The exact relation between the number of the trace formulas and the number of integrable derivatives of the potential is also given. The main results are optimal.

Spatial Analyticity of Solutions to Integrable Systems. I. The KdV Case

We are concerned with the Cauchy problem for the KdV equation for nonsmooth locally integrable initial profiles qs which are, in a certain sense, essentially bounded from below and q(x) = O(e−cxϵ ), x → + ∞, with some positive c and ϵ. Using the inverse scattering transform, we show that the KdV flow turns such initial data into a function which is (1) meromorphic (in the space variable) on the whole complex plane if ϵ > 1/2, (2) meromorphic on a strip around the real line if ϵ = 1/2, and (3) Gevrey regular if ϵ <1/2. Note that qs need not have any decay or pattern of behavior at − ∞.

A NEW INTERPOLATION FORMULA FOR THE TITCHMARSH-WEYL m-FUNCTION

For the Titchmarsh-Weyl m-function of the half-line Schrodinger operator with Dirichlet boundary conditions we put forward a new interpolation formula which allows one to reconstruct the m-function from its values on a certain infinite set of points for a broad class of potentials.

On the spectral L2 conjecture, 3/2-Lieb-Thirring inequality and distributional potentials

Let H=−∂x2+V(x) be a properly defined Schrodinger operator on L2(R) with real potentials of the form V(x)=q(x)+p′(x) (the derivative is understood in the distributional sense) with some p,q∊L2(R). We prove that the absolutely continuous spectrum of H fills (0,∞) which was previously proven by Deift-Killip for V∊L2(R). We also refine the 3/2-Lieb-Thirring inequality.

Runup of Nonlinear Long Waves in Trapezoidal Bays: 1-D Analytical Theory and 2-D Numerical Computations

Long nonlinear wave runup on the coasts of trapezoidal bays is studied analytically in the framework of one-dimensional (1-D) nonlinear shallow-water theory with cross-section averaging, and is also studied numerically within a two-dimensional (2-D) nonlinear shallow water theory. In the 1-D theory, it is assumed that the trapezoidal cross-section channel is inclined linearly to the horizon, and that the wave flow is uniform in the cross-section. As a result, 1-D nonlinear shallow-water equations are reduced to a linear, semi-axis variable-coefficient 1-D wave equation by using the generalized Carrier–Greenspan transformation [Carrier and Greenspan (J Fluid Mech 1:97–109, 1958)] recently developed for arbitrary cross-section channels [Rybkinet al. (Ocean Model 43–44:36–51, 2014)], and all characteristics of the wave field can be expressed by implicit formulas. For detailed computations of the long wave runup process, a robust and effective finite difference scheme is applied. The numerical method is verified on a known analytical solution for wave runup on the coasts of an inclined parabolic bay. The predictions of the 1-D model are compared with results of direct numerical simulations of inundations caused by tsunamis in narrow bays with real bathymetries.

On positive type initial profiles for the KdV equation

We show that the KdV flow evolves any real locally integrable initial profile q of the form q = r′ + r2, where r ∈ Lloc, r|R+ = 0 into a meromorphic function with no real poles.