Sergey Tsarev

Siberian Federal University · Institute of Space and Information Technology

In this paper, we show that the classical discrete orthogonal univariate polynomials (namely, Hahn polynomials on an equidistant lattice with unit weights) of sufficiently high degrees have extremely small values near the endpoints (we call this property as "rapid decay near the endpoints of the discrete lattice". We demonstrate the importance of t...

Standard models of ionospheric delays have errors of order 1--8 TECU (standard total electron content units). On the basis of the free interpolation framework we propose a new simple model of the slant TEC distributions approximating slant TEC distributions obtained from the three-dimensional ionospheric models NeQuick2 and IRI-2016 with RMS error...

We study local and global approximations of smooth nets of curvature lines and smooth conjugate nets by discrete nets (circular nets and planar quadrilateral nets, respectively) with edges of order ϵ. Both smooth and discrete geometries are described by integrable systems. It is shown that one can obtain an order ϵ² approximation globally with poin...

We investigate algebraic properties of weakly commutative triples, appearing in the theory of integrable nonlinear partial differential equations. Algebraic technique of skew fields of formal pseudodifferential operators as well as skew Ore fields of fractions are applied to this problem, relating weakly commutative triples to commuting elements of...

A new method to identify all sufficiently long repeating nucleotide substrings in one or several DNA sequences is proposed. The method based on a specific gauge applied to DNA sequences that guarantees identification of the repeating substrings. The method allows the matching substrings to contain a given level of errors. The gauge is based on the...

A new method to identify all sufficiently long repeating substrings in one or several symbol sequences is proposed. The method is based on a specific gauge applied to symbol sequences that guarantees identification of the repeating substrings. It allows the matching of substrings to contain a given level of errors. The gauge is based on the develop...

In this paper we present a construction of multiparametric families of two-dimensional metrics with a polynomial first integral of arbitrary degree in momenta. Such integrable geodesic flows are described by solutions of some semi-Hamiltonian hydrodynamic-type system. We give a constructive algorithm for the solution of the derived hydrodynamic-typ...

The linear stability of mechanical equilibrium in a two-layer system formed by different phases of the same binary mixture is investigated. The temperature difference is applied to the layers by heating and cooling the opposite rigid boundaries. In the state of mechanical equilibrium, the applied temperature gradient induces concentration gradients...

By the Moutard transformation method we construct two-dimensional Schrodinger operators with real smooth potential decaying at infinity and with a multiple positive eigenvalue. These potentials are rational functions of spatial variables and their sines and cosines.

We consider non-stationary dynamical systems with one-and-a-half degrees of freedom. We are interested in algorithmic construction of rich classes of Hamilton's equations with the Hamiltonian H=p^2/2+V(x,t) which are Liouville integrable. For this purpose we use the method of hydrodynamic reductions of the corresponding one-dimensional Vlasov kinet...

We demonstrate how the Moutard transformation of two-dimensional Schrodinger operators acts on the Faddeev eigenfunctions on the zero energy level and present some explicitly computed examples of such eigenfunctions for smooth fast decaying potentials of operators with non-trivial kernel and for deformed potentials which correspond to blowing up so...

We investigate second-order quasilinear equations of the form fijuxixj = 0, where u is a function of n independent variables x1, …, xn, and the coefficients fij depend on the first-order derivatives p1 = ux1, …, pn = uxn only. We demonstrate that the natural equivalence group of the problem is isomorphic to SL(n + 1, R), which acts by projective tr...

We consider the Moutard transformation which is a two-dimensional version of the well-known Darboux transformation. We give an algebraic interpretation of the Moutard transformation as a conjugation in an appropriate ring and the corresponding version of the algebro-geometric formalism for two-dimensional Schroedinger operators. An application to s...

We give the basic definitions and some theoretical results about hyperdeterminants, introduced by A. Cayley in 1845. We prove integrability (understood as 4d-consistency) of a nonlinear difference equation defined by the 2x2x2-hyperdeterminant. This result gives rise to the following hypothesis: the difference equations defined by hyperdeterminants...

Darboux’s classical results about transformations of second-order hyperbolic equations by means of differential substitutions are extended to the case of parabolic equations of the form Lu = (D x 2 + a(x, y)D x + b(x, y)D y + c(x, y))u = 0. We prove a general theorem that provides a way to determine admissible differential substitutions for such pa...

We construct a family of two-dimensional stationary Schrödinger operators with rapidly decaying smooth rational potentials and nontrivial L2 kernels. We show that some of the constructed potentials generate solutions of the Veselov-Novikov equation that decay rapidly at infinity, are nonsingular at t = 0, and have singularities at finite times t ≥...

The Novikov-Veselov equation, which is a two dimensional gernalization of the Korteweg-de Varies (KdV) equation that determines deformation of the two-dimensional Schrödinger operator, is presented. The equation with nontrivial dynamics can be obtained for stationary solutions and systems. All rational solitons of the KdV equation are obtained from...

We classify all integrable three-dimensional scalar discrete affine linear equations Q 3=0 on an elementary cubic cell of the lattice \mathbb Z3{\mathbb Z}^3 . An equation Q 3=0 is called integrable if it may be consistently imposed on all three-dimensional elementary faces of the lattice \mathbb Z4{\mathbb Z}^4 . Under the natural requirement...

For hyperbolic first-order systems of linear partial differential equations (master equations), appearing in description of kinetic processes in physics, biology and chemistry we propose a new procedure to obtain their complete closed-form non-stationary solutions. The methods used include the classical Laplace cascade method as well as its recent...

We start with elementary algebraic theory of factorization of linear ordinary differential equations developed in the period 1880-1930. After exposing these classical results we sketch more sophisticated algorithmic approaches developed in the last 20 years. The main part of this paper is devoted to modern generalizations of the notion of factoriza...

It is proved that there exists an infinite involutive family of integrals of hydrodynamic type for diagonal Hamiltonian systems of quasilinear equations; the completeness of the family is also proved, and a basis for it is constructed for Whitham's equation. Higher integrals and symmetries of these systems are found.

We study local and global approximations of smooth nets of curvature lines and smooth conjugate nets by respective discrete nets (circular nets and planar quadrilateral nets) with infinitesimal quads. It is shown that choosing the points of discrete nets on the smooth surface one can obtain second-order approximation globally. Also a simple geometr...

In this note using Moutard transformations we show how explicit examples of two-dimensional Schrodinger operators L = � +u(x,y) with fast decaying potential and multidimensional L2-kernel may be constructed. In the explicit examples below the potential u(x,y) and square-summable solutions (x,y) of the equation L = 0 are smooth rational functions of...

This booklet (in Russian) exposes some classical and modern methods of integrability of linear and nonlinear partial dufferential equations.

Using Moutard transformations we show how explicit examples of two-dimensional Schroedinger operators with fast decaying potential and multidimensional $L_2$-kernel may be constructed

We describe a method of obtaining closed-form complete solutions of certain second-order linear partial differential equations with more than two independent variables. This method generalizes the classical method of Laplace transformations of second-order hyperbolic equations in the plane and is based on an idea given by Ulisse Dini in 1902.

We characterize non-degenerate Lagrangians of the form such that the corresponding Euler-Lagrange equations are integrable by the method of hydrodynamic reductions. The integrability conditions constitute an over-determined system of fourth order PDEs for the Lagrangian density f, which is in involution and possesses interesting differential-geome...

A partial proof of the van Hoeij-Abramov conjecture on the algorithm of definite rational summation is given. The results obtained underlie an algorithm for finding a wide class of sums of the form åk = 0n - 1 R( k,n )\sum\limits_{k = 0}^{n - 1} {R\left( {k,n} \right)} .

We give a new procedure for generalized factorization and construction of the complete solution of strictly hyperbolic linear partial differential equations or strictly hyperbolic systems of such equations in the plane. This procedure generalizes the classical theory of Laplace transformations of second-order equations in the plane.

We present a partial proof of van Hoeij-Abramov conjecture about the algorithmic possibility of computation of finite sums of rational functions. The theoretical results proved in this paper provide an algorithm for computation of a large class of sums $ S(n) = \sum_{k=0}^{n-1}R(k,n)$.

A D-finite system is a finite set of linear homogeneous partial differential equations in several independent and dependent variables, whose solution space is of finite dimension. Let L be a D-finite system with rational function coefficients. We present an algorithm for computing all hyperexponential solutions of L, and an algorithm for computing...

The description of the Egorov hydrodynamic type systems is presented in language of conservation laws. Under extra conditions of semisimplicity and homogeneity tri-Hamiltonian structures of such systems are described. Even and odd cases are related with flat and metric of constant curvature, respectively. Some important and well-known examples like...

We prove a simple condition under which the metric corresponding to a diagonalizable semi-Hamiltonian hydrodynamic type system belongs to the class of Egorov (potential) metrics. For Egorov diagonal hydrodynamic type systems satisfying natural semisimplicity and homogeneity conditions, we prove necessary and sufficient conditions under which the th...

We present an algorithm for factoring a zero-dimensional left ideal in the ring Q(x, y) [∂x, ∂y], i.e. factoring a linear homogeneous partial differential system whose coefficients belong to Q(x, y), and whose solution space is finite-dimensional over Q. The algorithm computes all the zero-dimensional left ideals containing the given ideal. It gene...

We give an improved algorithm for factorization, i.e. decomposition of such systems with rational function coefficients into lowerorder systems. This algorithm is based on the recent algorithm of Z. Li and F. Schwarz for construction of hyperexponential solutions. An analogue of the Loewy-Ore theory of factorization is exposed. We also prove an ana...

Using a new definition of the generalized factorization of linear partial differential operators, we discuss possible generalizations of the Darboux integrability of nonlinear partial differential equations.

A remarkable parallelism between the theory of integrable systems of first-order quasilinear PDEs and some old results in the differential geometry of orthogonal curvilinear coordinates, conjugate nets, Laplace equations and their Bianchi-Bäcklund transformations is exposed. The applications of these results range from models in topological field t...

We consider Benney's equations, and their reductions to systems with finitely many Riemann invariants. The equations describing these reductions were given in [5] and a construction of a class of their solutions was briefly described there. Here we discuss the properties of these equations in more detail, and investigate the relationship between th...

In this paper we solve the following problem: Suppose we have a sphere without any marked point and a compass. Is it possible to build two points on the sphere which are at the distance from a "pole" to the "equator" of the sphere using ONLY compass? No additional construction on any other geometric object (e.g. a plane) are allowed (otherwise the...

We give a decision procedure for existence of factorixations (tlec:ompositions int.0 lower order ODEs) of nonlinear orcli-nary differential cquat.ions y(") = F(z: y(x), y', . : y'"-") for the general cast (,equations with arbitrary locally niero-morphic F). The prol)lcm of factorization for the CRSC of rational F is discussed.

Using a new definition of generalized divisors we prove that the lattice of such divisors for a given linear partial differential operator is modular and obtain analogues of the well-known theorems of the Loewy-Ore theory of factorization of linear ordinary differential operators. Possible applications to factorized Gröbner bases computations in th...

Using a new definition of generalized divisors we prove that the lattice of such divisors for a given linear partial dif- ferential operator is modular and obtain analogues of the well-known theorems of the Loewy-Ore theory of factoriza- tion of linear ordinary differential operators. Possible ap- plications to factorized Grobner bases computations...

We discuss the problem of manipulation of expressions in-volving indefinite functions, integration operators and in-verses of linear ordinary differential operators (LODO). Us-ing Loewy-Ore formal theory we obtain some subnormal forms for such expressions and algorithms for verification of identities involving such expressions.

Factorization of a linear ordinary differential operator is an important tool for analyzing and solving differential equations. The paper discusses classical and some recent results on nonuniqueness of the complete factorization (i.e., factorization into irreducible components) of linear ordinary differential operators, as well as the character of...

The usual superposition formulas for Baecklund transformations of (2+1)-dimensional integrable systems include quadratures unlike the well known case of (1+1)-dimensional inegrable systems where the fourth solution is found with algebraic operations. In the present paper we show how in the case of (2+1)-dimensional integrable systems one can find a...

We discuss the problem of exhaustive e n umeration of all possible factorizations for a given linear ordinary diierential operator. A theoretical investigation of topological and com-binatorial obstacles to uniform description of factors which include arbitrary parameters and a complete algorithm for enumeration of all (discrete and parameterized)...

The reductions of the Benney moment equations to systems of finitely many partial dif-ferential equations are discussed. Several families of these are constructed explicitly. These must all satisfy a compatibility condition, and the reduced equations are diagonalisable and semi-Hamiltonian. By imposing a further constraint, of scaling and Galilean...

It is shown that a novel 2 + 1-dimensional system recently introduced by Konopelchenko and Rogers contains as a specialization the Zakharov-Manakov matrix triad system. The latter, in turn, in its scalar version yields a classical system investigated by Darboux in connection with conjugate coordinate systems. This Darboux system, in a 1 + 1-dimensi...

This article discusses problems that appear during factorization of linear ordinary differential operators with the Beke-Schwarz method.

The general theory of hamiltonian systems of hydrodynamic type was developed by B.A.Dubrovin and S.P.Novikov [7],[8]. Here we study only one-dimensional hamiltonian systems possessing a complete set of Riemann invariants arising in the theory of multiphase averaging of completely integrable equations (see [2], [4], [13], [20], [32]), for example th...

We discuss the problems arising in the Beke-Schwarz approach to the factorization of linear ordinary differential operators.

Remarkable parallelism between the theory of integrable systems of first-order quasilinear PDE and some old results in projective and affine differential geometry of conjugate nets, Laplace equations, their Bianchi-Baecklund transformations is exposed. These results were recently applied by I.M.Krichever and B.A.Dubrovin to prove integrability of s...

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This thesis was published (in abridged form, but with essential additions) in my paper "The geometry of harniltonian systems of hydrodynamic type. The generalized hodograph method", Mathematics of the USSR-Izvestiya, 1991, v. 37, no. 2, p. 397-419 (available as a supplementary file here)

We classify all integrable 3-dimensional scalar discrete affine linear equations Q3 = 0 on an elementary cubic cell of the lattice Z3. An equation Q3 = 0 is called integrable if it may be consistently imposed on all 3-dimensional elementary faces of the lattice Z4. Under the natural requirement of invariance of the equation under the action of the...