# KdV hierarchy via Abelian coverings and operator identities

@article{Eichinger2019KdVHV, title={KdV hierarchy via Abelian coverings and operator identities}, author={Benjamin Eichinger and Tom Vandenboom and Peter Yuditskii}, journal={Transactions of the American Mathematical Society, Series B}, year={2019} }

We establish precise spectral criteria for potential functions
V
V
of reflectionless Schrödinger operators
L
V
=
−
∂
x
2
+
V
L_V = -\partial _x^2 + V
to admit solutions to the Korteweg–de Vries (KdV) hierarchy with
V
V
as an initial value. More generally, our methods extend the classical study of algebro-geometric solutions for the KdV hierarchy to noncompact Riemann surfaces by defining generalized Abelian integrals and analogues of the Baker… Expand

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#### References

SHOWING 1-10 OF 63 REFERENCES

Construction of KdV flow I. Tau-Function via Weyl Function

- Mathematics, Physics
- Zurnal matematiceskoj fiziki, analiza, geometrii
- 2018

Sato introduced the tau-function to describe solutions to a wide class of completely integrable differential equations. Later Segal-Wilson represented it in terms of the relevant integral operators… Expand

The absolutely continuous spectrum of Jacobi matrices

- Mathematics, Physics
- 2007

I explore some consequences of a groundbreaking result of Breimesser and Pearson on the absolutely continuous spectrum of one-dimensional Schr"odinger operators. These include an Oracle Theorem that… Expand

Almost periodicity in time of solutions of the KdV equation

- Mathematics, Physics
- Duke Mathematical Journal
- 2018

We study the Cauchy problem for the KdV equation $\partial_t u - 6 u \partial_x u + \partial_x^3 u = 0$ with almost periodic initial data $u(x,0)=V(x)$. We consider initial data $V$, for which the… Expand

Soliton Equations and their Algebro-Geometric Solutions

- Mathematics, Physics
- 2003

As a partner to Volume 1: Dimensional Continuous Models, this monograph provides a self-contained introduction to algebro-geometric solutions of completely integrable, nonlinear, partial… Expand

On the Direct Cauchy Theorem in Widom Domains: Positive and Negative Examples Peter Yuditskii

- Mathematics
- 2010

We discuss several questions which remained open in our joint work with M. Sodin “Almost periodic Jacobi matrices with homogeneous spectrum, infinite-dimensional Jacobi inversion, and Hardy spaces of… Expand

Counterexamples to the Kotani-Last Conjecture for Continuum Schr\"odinger Operators via Character-Automorphic Hardy Spaces

- Mathematics, Physics
- 2014

The Kotani-Last conjecture states that every ergodic operator in one space dimension with non-empty absolutely continuous spectrum must have almost periodic coefficients. This statement makes sense… Expand

Almost periodic Jacobi matrices with homogeneous spectrum, infinite dimensional Jacobi inversion, and hardy spaces of character-automorphic functions

- Mathematics
- 1997

All three subjects reflected in the title are closely intertwined in the paper.LetJE be a class of Jacobi matrices acting inl2(ℤ) with a homogeneous spectrumE (see Definition 3.2) and with diagonal… Expand

Almost periodic Sturm-Liouville operators with Cantor homogeneous spectrum

- Mathematics
- 1995

Being based on an infinite-dimensional real version of the Jacobi inversion problem [24], we establish the direct generalization of the well-known properties of finite-band Sturm-Liouville operators… Expand

Integrals of Nonlinear Equations of Evolution and Solitary Waves

- Mathematics
- 1968

In Section 1 we present a general principle for associating nonlinear equations evolutions with linear operators so that the eigenvalues of the linear operator integrals of the nonlinear equation. A… Expand

Kotani–Last problem and Hardy spaces on surfaces of Widom type

- Mathematics, Physics
- 2012

This is a small theory of non almost periodic ergodic families of Jacobi matrices with purely (however) absolutely continuous spectrum. The reason why this effect may happen is that under our… Expand