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Alexei Rybkin

Publications and source records attributed to Alexei Rybkin.

23 records · Page 2Linked to original sources
arXiv2011

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 q's which are, in a certain sense, essentially bounded from below and q(x)=O(e^{-cx^{ε}}),x\rightarrow+\infty, 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 q's need not have any decay or pattern of behavior at -\infty.

nlin.SI↗
arXiv2011

A Haar-type Approximation and a New Numerical Schema for the Korteweg-de Vries Equation

We discuss a new numerical schema for solving the initial value problem for the Korteweg-de Vries equation for large times. Our approach is based upon the Inverse Scattering Transform that reduces the problem to calculating the reflection coefficient of the corresponding Schrödinger equation. Using a step-like approximation of the initial profile and a fragmentation principle for the scattering data, we obtain an explicit recursion formula for computing the reflection coefficient, yielding a high resolution KdV solver. We also discuss some generalizations of this algorithm and how it might be improved by using Haar and other wavelets.

math.SP↗
arXiv2011

On the determinant formula in the inverse scattering procedure with a partially known steplike potential

We are concerned with the inverse scattering problem for the full line Schrödinger operator $-\partial_x^2+q(x)$ with a steplike potential $q$ a priori known on $\Reals_+=(0,\infty)$. Assuming $q|_{\Reals_+}$ is known and short range, we show that the unknown part $q|_{\Reals_-}$ of $q$ can be recovered by {equation*} q|_{\Reals_-}(x)=-2\partial_x^2\log\det(1+(1+\mathbb{M}_x^+)^{-1}\mathbb{G}_x), {equation*} where $\mathbb{M}_x^+$ is the classical Marchenko operator associated to $q|_{\Reals_+}$ and $\mathbb{G}_x$ is a trace class integral Hankel operator. The kernel of $\mathbb{G}_x$ is explicitly constructed in term of the difference of two suitably defined reflection coefficients. Since $q|_{\Reals_-}$ is not assumed to have any pattern of behavior at $-\infty$, defining and analyzing scattering quantities becomes a serious issue. Our analysis is based upon some subtle properties of the Titchmarsh-Weyl $m$-function associated with $\Reals_-$.

math-ph↗
arXiv2010

The Hirota τ-function and well-posedness of the KdV equation with an arbitrary step like initial profile decaying on the right half line

We are concerned with the Cauchy problem for the KdV equation on the whole line with an initial profile V_0 which is decaying sufficiently fast at +\infty and arbitrarily enough (i.e., no decay or pattern of behavior) at -\infty. We show that this system is completely integrable in a very strong sense. Namely, the solution V(x,t) admits the Hirota τ-function representation V(x,t)=-2\partial_{x}^2 logdet(I+M_{x,t}) where M_{x,t} is a Hankel integral operator constucted from certain scattering and spectral data suitably defined in terms of the Titchmarsh-Weyl m-functions associated with the two half-line Schrödinger operators corresponding to V_0. We show that V(x,t) is real meromorphic with respect to x for any t>0. We also show that under a very mild additional condition on V_0 representation implies a strong well-posedness of the KdV equation with such V_0's. Among others, our approach yields some relevant results due to Cohen, Kappeler, Khruslov, Kotlyarov, Venakides, Zhang and others.

nlin.SI↗