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P. G. Grinevich

Publications and source records attributed to P. G. Grinevich.

At least 37 records · Page 2Linked to original sources
arXiv2015

On the s-meromorphic OD operators

We consider linear spectral-meromorphic (s-meromorphic) OD operators at the real axis such that all local solutions to the eigenvalue problems are meromorphic for all $λ$. By definition, rank one algebro-geometrical operator $L$ admit an OD operator $A$ such that $[L,A]=0$ and rank of this commuting pair is equal to one. All of them are s-meromorphic. In particular, second order ``singular soliton'' operators satisfy to this condition. Operator $L^+$ formally adjoint to s-meromorphic operator $L$ is also s-meromorphic. For singular eigenfunctions of operators $L,L^+$ following scalar product $ =\int_R f\bar{g}dx$ is well-defined such that $ = $ avoiding isolated singular points. For the case $L=L^+$ this formula defines indefinite inner product on the spaces of singular functions $f,g\in F_L$ associated with operator $L$. They are $C^{\infty}$ outside of singularities and have isolated singularities of the same type as eigenfunctions $Lf=λf$. Every s-meromorphic operator can be approximated by algebro-geometric rank one operators in any finite interval

math-ph↗
arXiv2015

Nonlocality and the inverse scattering transform for the Pavlov equation

As in the case of soliton PDEs in 2+1 dimensions, the evolutionary form of integrable dispersionless multidimensional PDEs is non-local, and the proper choice of integration constants should be the one dictated by the associated Inverse Scattering Transform (IST). Using the recently made rigorous IST for vector fields associated with the so-called Pavlov equation $v_{xt}+v_{yy}+v_xv_{xy}-v_yv_{xx}=0$, in this paper we establish the following. 1. The non-local term $\partial_x^{-1}$ arising from its evolutionary form $v_{t}= v_{x}v_{y}-\partial^{-1}_{x}\,\partial_{y}\,[v_{y}+v^2_{x}]$ corresponds to the asymmetric integral $-\int_x^{\infty}dx'$. 2. Smooth and well-localized initial data $v(x,y,0)$ evolve in time developing, for $t>0$, the constraint $\partial_y {\cal M}(y,t)\equiv 0$, where ${\cal M}(y,t)=\int_{-\infty}^{+\infty} \left[v_{y}(x,y,t) +(v_{x}(x,y,t))^2\right]\,dx$. 3. Since no smooth and well-localized initial data can satisfy such constraint at $t=0$, the initial ($t=0+$) dynamics of the Pavlov equation can not be smooth, although, as it was already established, small norm solutions remain regular for all positive times. We expect that the techniques developed in this paper to prove the above results, should be successfully used in the study of the non-locality of other basic examples of integrable dispersionless PDEs in multidimensions.

nlin.SI↗
arXiv2015

Singular Soliton Operators and Indefinite Metrics

The singular real second order 1D Schrodinger operators are considered here with such potentials that all local solutions near singularities to the eigenvalue problem are meromorphic for all values of the spectral parameter. All algebro-geometrical or "singular finite-gap" potentials satisfy to this condition. A Spectral Theory is constructed here for the periodic and rapidly decreasing cases in the special classes of functions with singularities and indefinite inner product. It has a finite number of negative squares if the unimodular Bloch multipliers are fixed in the periodic case, and in the rapidly decreasing case. The time dynamics provided by the KdV hierarchy preserves this number. The right analog of Fourier Transform for the Riemann Surfaces preserving remarkable multiplicative properties of the ordinary (i.e. genus zero) Fourier Transform based on the standard exponential basis, leads to such operators as it was shown in our previous works.

math-ph↗
arXiv2014

Spectral Meromorphic Operators and Nonlinear Systems

We study here class of 1D spectral-meromorphic (s-meromorphic) OD operators $L=\partial_x^n+\sum_{n-2\geq i\geq 0}a_{n-2-i}\partial_x^i$ with meromorphic coefficients $a_j$ near $x\in R$ such that all eigenfunctions $Lψ=αψ$ are $x$--meromorphic near $x\in R$ for all $α$. Symmetric $s$-meromorphic operators are self-adjoint with respect to indefinite inner product well-defined for some special spaces of singular functions. In particular, all algebraic operators $L$--i.e. operators entering Burchnall-Chaundy-Krichever (BChK) rank one commutative rings -- are s-meromorphic. For KdV system corresponding algebraic operator $L=-\partial_x^2+u(x,t)$ is called singular finite gap, singular soliton or algebrogeometric Schrodinger operator. This special case was already studied by the present authors in the recent works.

math.FA↗
arXiv2014

The Cauchy problem for the Pavlov equation

Commutation of multidimensional vector fields leads to integrable nonlinear dispersionless PDEs arising in various problems of mathematical physics and intensively studied in the recent literature. This report is aiming to solve the scattering and inverse scattering problem for integrable dispersionless PDEs, recently introduced just at a formal level, concentrating on the prototypical example of the Pavlov equation, and to justify an existence theorem for global bounded solutions of the associated Cauchy problem with small data.

nlin.SI↗
arXiv2013

Discrete SL2 Connections and Self-Adjoint Difference Operators on the Triangulated 2-manifold

Discretization Program of the famous Completely Integrable Systems and associated Linear Operators was developed in 1990s. In particular, specific properties of the second order difference operators on the triangulated manifolds and equilateral triangle lattices were studied in the works of S.Novikov and I.Dynnikov since 1996. They involve factorization of operators, the so-called Laplace Transformations, new discretization of Complex Analysis and new discretization of $GL_n$ connections on the triangulated $n$-manifolds. The general theory of the new type discrete $GL_n$ connections was developed. However, the special case of $SL_n$-connections (and unimodular $SL_n^{\pm}$ connections such that $\det A=\pm 1$) was not selected properly. As we prove in this work, it plays fundamental role (similar to magnetic field in the continuous case) in the theory of self-adjoint discrete Schrodinger operators for the equilateral triangle lattice in $\RR^2$. In Appendix~1 we present a complete characterization of rank 1 unimodular $SL_n^{\pm}$ connections. Therefore we correct a mistake made in the previous versions of our paper (we wrongly claimed that for $n>2$ every unimodular $SL_n^{\pm}$ Connection is equivalent to the standard Canonical Connection). Using communications of Korepanov we completely clarify connection of classical theory of electric chains and star-triangle with discrete Laplace transformation on the triangle lattices

math-ph↗
arXiv2012

Faddeev eigenfunctions for multipoint potentials

We present explicit formulas for the Faddeev eigenfunctions and related generalized scattering data for multipoint potentials in two and three dimensions. For single point potentials in 3D such formulas were obtained in an old unpublished work of L.D. Faddeev. For single point potentials in 2D such formulas were given recently by the authors in arXiv:1110.3157 .

math-ph↗
arXiv2012

Holomorphic eigenfunctions of the vector field associated with the dispersionless Kadomtsev-Petviashvili equation

Vector fields naturally arise in many branches of mathematics and physics. Recently it was discovered that Lax pairs for many important multidimensional integrable partial differential equations (PDEs) of hydrodynamic type (also known as dispersionless PDEs) consist of vector field equations. These vector fields have complex coefficients and their analytic, in the spectral parameter, eigenfunctions play an important role in the formulations of the direct and inverse spectral transforms. In this paper we prove existence of eigenfunctions of the basic vector field associated with the celebrated dispersionless Kadomtsev-Petviashvili equation, which are holomorphic in the spectral parameter $λ$ in the strips $|\Imλ|> C_0$.

math-ph↗
arXiv2009

Calculation of Topological Charge of Real Finite-Gap sine-Gordon solutions using Theta-functional formulae

The most basic characteristic of x-quasiperiodic solutions u(x,t) of the sine-Gordon equation u_{tt}-u_{xx}+\sin u=0 is the topological charge density denoted $\bar n$. The real finite-gap solutions u(x,t) are expressed in terms of the Riemann theta-functions of a non-singular hyperelliptic curve $Γ$ and a positive generic divisor D of degree g on $Γ$, where the spectral data $(Γ, D)$ must satisfy some reality conditions. The problem addressed in note is: to calculate $\bar n$ directly from the theta-functional expressions for the solution u(x,t). The problem is solved here by introducing what we call the multiscale or elliptic limit of real finite-gap sine-Gordon solutions. We deform the spectral curve to a singular curve, for which the calculation of topological charges reduces to two special easier cases.

math-ph↗
arXiv2008

Spectral conservation laws for periodic nonlinear equations of the Melnikov type

We consider the nonlinear equations obtained from soliton equations by adding self-consistent sources. We demonstrate by using as an example the Kadomtsev-Petviashvili equation that such equations on periodic functions are not isospectral. They deform the spectral curve but preserve the multipliers of the Floquet functions. The latter property implies that the conservation laws, for soliton equations, which may be described in terms of the Floquet multipliers give rise to conservation laws for the corresponding equations with self-consistent sources. Such a property was first observed by us for some geometrical flow which appears in the conformal geometry of tori in three- and four-dimensional Euclidean spaces (math/0611215).

math-ph↗
arXiv2006

Infinitesimal Darboux transformations of the spectral curves of tori in the four-space

We study the action of conformal transformations of the ambient space on the Dirac operator coming into the Weierstrass (or spinor) representation of a torus in the Euclidean four-space. It is showed that such an action generates a flow acting on the potential of the operator, that this flow is described by a nonlinear system of the Melnikov type and that it preserves the Floquet multipliers of the Dirac operator with double-periodic potential. However this flow is only almost isospectral since it does not preserve the spectral curve in general and its action may result in adding or removing multiple points corresponding to the same multipliers. We demonstrate that in some important geometrical examples after reparameterization of the temporary variable such flows are governed by integrable systems on whiskered tori.

math.DG↗
arXiv2006

Newtonian dynamics in the plane corresponding to straight and cyclic motions on the hyperelliptic curve $μ^2=ν^n-1, n\in{\Bbb Z}$: ergodicity, isochrony, periodicity and fractals

We study the complexification of the one-dimensional Newtonian particle in a monomial potential. We discuss two classes of motions on the associated Riemann surface: the rectilinear and the cyclic motions, corresponding to two different classes of real and autonomous Newtonian dynamics in the plane. The rectilinear motion has been studied in a number of papers, while the cyclic motion is much less understood. For small data, the cyclic time trajectories lead to isochronous dynamics. For bigger data the situation is quite complicated; computer experiments show that, for sufficiently small degree of the monomial, the motion is generically periodic with integer period, which depends in a quite sensitive way on the initial data. If the degree of the monomial is sufficiently high, computer experiments show essentially chaotic behaviour. We suggest a possible theoretical explanation of these different behaviours. We also introduce a one-parameter family of 2-dimensional mappings, describing the motion of the center of the circle, as a convenient representation of the cyclic dynamics; we call such mapping the center map. Computer experiments for the center map show a typical multi-fractal behaviour with periodicity islands. Therefore the above complexification procedure generates dynamics amenable to analytic treatment and possessing a high degree of complexity.

nlin.CD↗
arXiv2003

The initial boundary value problem on the segment for the Nonlinear Schrödinger equation; the algebro-geometric approach. I

This is the first of a series of papers devoted to the study of classical initial-boundary value problems of Dirichlet, Neumann and mixed type for the Nonlinear Schrödinger equation on the segment. Considering proper periodic discontinuous extensions of the profile, generated by suitable point-like sources, we show that the above boundary value problems can be rewritten as nonlinear dynamical systems for suitable sets of algebro-geometric spectral data, generalizing the classical Dubrovin equations. In this paper we consider, as a first illustration of the above method, the case of the Dirichlet problem on the segment with zero-boundary value at one end, and we show that the corresponding dynamical system for the spectral data can be written as a system of ODEs with algebraic right-hand side.

nlin.SI↗
arXiv2003

Topological Phenomena in the Real Periodic Sine-Gordon Theory

The set of real finite-gap Sine-Gordon solutions corresponding to a fixed spectral curve consists of several connected components. A simple explicit description of these components obtained by the authors recently is used to study the consequences of this property. In particular this description allows to calculate the topological charge of solutions (the averaging of the $x$-derivative of the potential) and to show that the averaging of other standard conservation laws is the same for all components.

math-ph↗
arXiv2001

Topological Charge of the real periodic finite-gap Sine-Gordon solutions

An effective description of the inverse spectral data corresponding to the real periodic and quasiperiodic solutions for the sine-gordon equation is obtained. In particular, the explicit formula for the so-called topological charge of the solutions is found and proved. As it was understood already 20 years ago, it is very hard to extract any formula for this quantity from the Theta-functional expressions. A new method was developed by the authors for this goal. In the appendix 3 an analog of the Fourier-Laurent integral transform on the Riemann surfaces is defined, based on the constructions of the Soliton Theory. It is a natural continuous analog of the discrete Krichever-Novikov bases developed in the late 80s for the needs of the Operator Quantization of String Theory.

math-ph↗
arXiv2000

String equation--2. Physical solution

This paper is a continuation of the paper by S.P.Novikov in Funct.Anal.Appl., v.24(1990), No 4, pp 196-206. String equation is by definition the equation $[L,A]=1$ for the coefficients of two linear ordinary differential operators $L$ and $A$. For the ``double scaling limit'' of the matrix model we always have $L=-\partial_x^2+u(x)$, $A$ is some differential operator of the odd order $2k+1$. In the first nontrivial case $k=1$ we have the Painelevé-1 (P-1) equation. Only special real ``separatrix'' solutions of P-1 are important in the quantum field theory. By the conjecture of Novikov these ``physical'' solutions, which are analytically exceptional probably have much stronger symmetry then the other solutions but it is not proved until now. Two asymptotic methods were developed in the previous paper -- nonlinear semiclassics (or the Bogolubov-Whitham averaging method) and the linear semiclassics for the ``Isomonodromic'' method. The nonlinear semiclassics gives a good approximation for the general (``non-physical'') solutions of P-1 but fails in the ``physical'' case. In our paper the linear semiclasics for the ``physical'' solutions of the P-1 equations is studied. In particular connection between the semiclassics on Riemann surfaces and Hamiltonian foliations on these surfaces is established.

solv-int↗
arXiv1997

Flag Spaces in KP Theory and Virasoro Action on \det D_j and Segal-Wilson τ-Function

It is well-known that the algebra of vector fields on the circle acts on the space of Riemann surfaces with a marked point and a local parameter at this point. We show that this action has a natural realization in the soliton theory, indeed it coincides with the action of some non-isospectral Kadomtsev-Petviashvili symmetries on the finite-gap solutions. A technique based on the so-called Cauchy-Baker-Akhiezer kernel is developed. The deformations of the τ-function corresponding to the Baker-Akhiezer forms of tensor weight j generate representations of the Virasoro algebra with a central charge 6j^2-6j+1. A system including the Kadomtsev-Petviashvili hierarchy and the Toda lattice simultaneously is considered. The Virasoro representations corresponding to such a system explicitly depend on an extra discrete time t_0. The tau-function for this system is defined in terms of infinite dimensional flag spaces, generalizing the grassmanians.

math-ph↗