Integrable elliptic pseudopotentials

We construct integrable pseudopotentials with an arbitrary number of fields in terms of an elliptic generalization of hypergeometric functions in several variables. These pseudopotentials are multiparameter deformations of ones constructed by Krichever in studying the Whitham-averaged solutions of the KP equation and yield new integrable (2+1)-dimensional systems of hydrodynamic type. Moreover, an interesting class of integrable (1+1)-dimensional systems described in terms of solutions of an elliptic generalization of the Gibbons-Tsarev system is related to these pseudopotentials.     

Birkhoff strata of the Grassmannian Gr<Superscript>(2)</Superscript>: Algebraic curves

We study algebraic varieties and curves arising in the Birkhoff strata of the Sato Grassmannian Gr ⁽²⁾ . We show that the big cell Σ ₀ contains the tower of families of the normal rational curves of all odd orders. The strata Σ _2n, n = 1, 2, 3,..., contain hyperelliptic curves of genus n and their coordinate rings. The strata Σ _2n+1 , n = 0, 1, 2, 3,..., contain (2m + 1, 2m + 3) plane curves for n = 2m, 2m − 1 (m ≥ 2) and also (3, 4) and (3, 5) curves in Σ ₃ and Σ ₅ . Curves in the strata Σ _2n+1 have zero genus.     

Coexistence of small and large amplitude limit cycles of polynomial differential systems of degree four

A class of degree four differential systems that have an invariant conic x ² + Cy ² = 1, C ∈ ℝ, is examined. We show the coexistence of small amplitude limit cycles, large amplitude limit cycles, and invariant algebraic curves under perturbations of the coefficients of the systems.     

Uniformization of jacobi varieties of trigonal curves and nonlinear differential equations

We obtain an explicit realization of the Jacobi and Kummer varieties for trigonal curves of genusg (gcd(g,3)=1) of the form

On the generalized Chaplygin system

We consider two polynomial bi—Harnilt0nian structures for the generalized integrable Chaplygin system on the sphere S ² with an additional integral of fourth order in momenta. An explicit procedure for finding variables of separation, separation relations, and transformation of the corresponding algebraic curves of genus two is considered in detail. Bibliography: 21 titles.     

Embeddings of the groupL(2,13) in groups of lie typeE 6

In this paper we show there is exactly one conjugacy class of subgroups ofE ₆(ℂ) isomorphic toL(2, 13) with each of the characters 13+14 and 1+12+14 on a 27-dimensional module forE ₆. The one with character 13+14 is a subgroup of the irreducible closed subgroup of typeG ₂. There is a unique conjugacy class for each of the three algebraic conjugate characters 1+12+14. Our arguments have applications to fields of characteristic prime to |L(2, 13)|. Dedicated to John Thompson for his keen interest in broad areas of mathematics and in mathematicians     

Integrable Models of (1+1)-Dimensional Dilaton Gravity Coupled to Scalar Matter

We describe a class of explicitly integrable models of (1+1)-dimensional dilaton gravity coupled to scalar fields in sufficient detail. The equations of motion of these models reduce to systems of Liouville equations with energy and momentum constraints. We construct the general solution of the equations and constraints in terms of chiral moduli fields explicitly and briefly discuss some extensions of the basic integrable model. These models can be related to higher-dimensional supergravity theories, but we mostly consider them independently of such interpretations. We also briefly review other integrable models of two-dimensional dilaton gravity. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 146, No. 1, pp. 115–131, January, 2006.     

Rational torsion of J0(N) for hyperelliptic modular curves and families of Jacobians of genus 2 and genus 3 curves with a rational point of order 5,7 or 10

We describe a way of constructing Jacobians of hyperelliptic curves of genus g ≥ 2, defined over a number field, whose Jacobians have a rational point of order of some (well chosen) integer l ≥ g + 1; the method is based on a polynomial identity. Using this approach we construct new families of genus 2 curves defined over — which contain the modular curves X₀(31) (and X₀(22) as a by-product) and X₀(29), the Jacobians of which have a rational point of order 5 and 7 respectively. We also construct a new family of hyperelliptic genus 3 curves defined over —, which contains the modular curve X₀(41), the Jacobians of which have a rational point of order 10. Finally we show that all hyperelliptic modular curves X₀(N) with N a prime number fit into the described strategy, except for N = 37 in which case we give another explanation. The authors thank the FNR (project FNR/04/MA6/11) for their support.     

A Decomposition Theorem for Compact-Valued Henstock Integral

We prove that if X is a separable Banach space, then a measurable multifunction Γ : [0, 1] → ck(X) is Henstock integrable if and only if Γ can be represented as Γ = G + f, where G : [0, 1] → ck(X) is McShane integrable and f is a Henstock integrable selection of Γ.     

Drinfeld Modules of Rank 1 and Algebraic Curves with Many Rational Points

 We construct algebraic curves C defined over a finite prime field such that the number of -rational points of C is large relative to the genus of C. The methods of construction are based on the relationship between algebraic curves and their function fields, as well as on narrow ray class extensions obtained from Drinfeld modules of rank 1. Received 21 July 1997; in revised form 5 February 1998     

Drinfeld Modules of Rank 1 and Algebraic Curves with Many Rational Points

 We construct algebraic curves C defined over a finite prime field such that the number of -rational points of C is large relative to the genus of C. The methods of construction are based on the relationship between algebraic curves and their function fields, as well as on narrow ray class extensions obtained from Drinfeld modules of rank 1.     

Spin chain related to the quantum superalgebra slq(1¦1)

We consider an integrable system with R-matrix related to the algebra sl _q(1 | 1). The Hamiltonian of the system is constructed, and its spectrum is found by means of the algebraic Bethe ansatz. The symmetry algebra of the chain is written out. The partition function of the model on the lattice with domain wall boundary conditions is calculated. Bibliography: 12 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 325, 2005, pp. 146–162.     

Completeness of orthogonal systems in asymmetric spaces with sign-sensitive weight

We study the problem on the completeness of orthogonal systems in asymmetric spaces with sign-sensitive weight. Theorems of general form are obtained. In particular, the necessary and sufficient conditions on α, β, q ₁, and q ₂ for which the known orthogonal systems are everywhere dense in asymmetric spaces L _(α,β);q ([0, 1]) are found. Theorem. Let α, β, q ₁, q ₂ ∈ [1,+∞]. The following orthogonal systems are dense in asymmetric spaces L _(α,β);q ([0, 1]) if and only if either max{α, β, q ₁, q ₂} < + ∞ or max {α, β} < +∞, q ₁ = q ₂ = +∞: trigonometric, algebraic, Haar’s system, Meyer’s system of wavelets, Shannon-Kotel’nikov wavelets, Stromberg and Lemarie-Battle wavelets, the Walsh system, and the Franklin system. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 24, Dynamical Systems and Optimization, 2005.     

A dynamical system connected with an inhomogeneous 6-vertex model

A completely integrable dynamical system in discrete time is studied by methods of algebraic geometry. The system is associated with factorization of a linear operator acting in the direct sum of three linear spaces into a product of three operators, each acting nontrivially only in the direct sum of two spaces, and subsequently reversing the order of the factors. There exists a reduction of the system, which can be interpreted as a classical field theory in the 2+1-dimensional space-time, whose integrals of motion coincide, in essence, with the statistical sum of an inhomogeneous 6-vertex free-fermion model on the 2-dimensional kagome lattice (here the statistical sum is a function of two parameters). This establishes a connection with the “local,” or “generalized,” quantum Yang-Baxter equation. Bibliography:10 titles. Dedicated to L. D. Faddeev on the occasion of his 60th birthday Published inZapiski Nauchnykh Seminarov POMI, Vol. 215, 1994, pp. 178–196. Translated by I. G. Korepanov.     

The region of values of the system {ƒ(z1), …, ƒ(zn)} in the class of typically real functions. II

The paper studies the region of values D_m,1(T) of the system {ƒ(z₁), ƒ(z₂), …, ƒ(zm), ƒ(r)}, m e 1, where z_j (j = 1, 2, …,m) are arbitrary fixed points of the disk U = {z: |z| < 1} with Im zj ≠ 0 (j = 1, 2, …,m), and r, 0 < r < 1, is fixed, in the class T of functions ƒ(z) = z+a₂z²+ ⋯ regular in the disk U and satisfying in the latter the condition Im ƒ(z) Imz > 0 for Im z ≠ 0. An algebraic characterization of the set D_m,1(T) in terms of nonnegative-definite Hermitian forms is given, and all the boundary functions are described. As an implication, the region of values of ƒ(z_m) in the subclass of functions from the class T with prescribed values ƒ(z_k) (k = 1, 2, …,m − 1) and ƒ(r) is determined. Bibliography: 5 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 323, 2005, pp. 24–33. Original article submitted June 13, 2005.     

Prime Fano threefolds and integrable systems

For a general K3 surface S of genus g, with 2 ≤ g ≤ 10, we prove that the intermediate Jacobians of the family of prime Fano threefolds of genus g containing S as a hyperplane section, form generically an algebraic completely integrable Hamiltonian system. The first author is partially supported by grant MI1503/2005 of the Bulgarian Foundation for Scientific Research.     

Symplectic Lefschetz fibrations of curves as gonal coverings

We show that after a finite base change every symplectic Lefschetz fibration ${f \colon X \rightarrow B}$ of genus g >  3 curves over a closed oriented surface becomes a finite covering of degree ${\frac{g}{2} + 1}$ or ${\frac{g}{2} + \frac{3}{2}}$ of a family of spheres over a Riemann surface, with a branch locus admitting complex algebraic curves as local models. In the case of fibers of genus 4, it is shown that after a 2:1 base change the family admits a trigonal covering to a symplectic ruled surface, with symplectic branch locus.     

IntegrableN-dimensional systems on the Hopf algebra andq-deformations

We construct the class of integrable classical and quantum systems on the Hopf algebras describing n interacting particles. We obtain the general structure of an integrable Hamiltonian system for the Hopf algebra A(g) of a simple Lie algebra g and prove that the integrals of motion depend only on linear combinations of k coordinates of the phase space, 2·ind g≤k≤g·ind g, whereind g andg are the respective index and Coxeter number of the Lie algebra g. The standard procedure of q-deformation results in the quantum integrable system. We apply this general scheme to the algebras sl(2), sl(3), and o(3, 1). An exact solution for the quantum analogue of the N-dimensional Hamiltonian system on the Hopf algebra A(sl(2)) is constructed using the method of noncommutative integration of linear differential equations. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 124, No. 3, pp. 373–390, September, 2000     

Geometry and multidimensional soliton equations

The connection between the differential geometry of curves and (2+1)-dimensional integrable systems is established. The Zakharov equation, the modified Veselov-Novikov equation, the modified Kortewegde Vries equation, etc., are equivalent in the Lakshmanan sense to (2+1)-dimensional spin systems. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 118, No. 3, pp. 441–451, March, 1999.