相关论文: Double Elliptic Dynamical Systems From Generalized…
We obtain B\"acklund transformations and integrable time discretization of the recently introduced deformed Ruijsenaars-Schneider many-body system which is the dynamical system for poles of elliptic solutions to the Toda lattice with…
This thesis presents our results on Liouville integrable systems of Calogero-Ruijsenaars type: 1. We prove an explicit formula providing canonical spectral coordinates for the rational Calogero-Moser system. 2. We explore action-angle…
In this note we further develop the duality between supersymmetric gauge theories in various dimensions and elliptic integrable systems such as Ruijsenaars-Schneider model and periodic intermediate long wave hydrodynamics. These models…
We study algebro-geometric (finite-gap) and elliptic solutions of fully discretized KP or 2D Toda equations. In bilinear form they are Hirota's difference equation for $\tau$-functions. Starting from a given algebraic curve, we express the…
The approach of multi-dimensional SUSY Quantum Mechanics is used in an explicit construction of exactly solvable 3-body (and quasi-exactly-solvable $N$-body) matrix problems on a line. From intertwining relations with time-dependent…
The two full body problem concerns the dynamics of two spatially extended rigid bodies (e.g. rocky asteroids) subject to mutual gravitational interaction. In this note we deduce the Euler-Poincare and Hamiltonian equations of motion using…
We construct a general class of Calabi--Yau threefolds from fiber products of rational elliptic surfaces with section, generalizing a construction of Schoen to include all Kodaira fiber types. The resulting threefolds each have two elliptic…
We study elliptic solutions of the recently introduced Toda lattice with the constraint of type B and derive equations of motion for their poles. The dynamics of poles is given by the deformed Ruijsenaars-Schneider system. We find its…
We consider a hierarchy of classical Liouville completely integrable models sharing the same (linear) $r$--matrix structure obtained through an $N$--th jet--extension of $\mathfrak{su}(2)$ rational Gaudin models. The main goal of the…
The classical (dynamical) $R$-matrices for the 2- and 3-body Calogero-Moser models with elliptic potentials are given. The 3-body case has an interesting nontrivial structure that goes beyond the known ansatz for momentum independent…
In this study, we introduce a new class of rational elliptic 3-folds, which we refer to as "1/2 Calabi-Yau 3-folds". We construct elliptically fibered Calabi-Yau 3-folds by utilizing these rational elliptic 3-folds. The construction yields…
The new integrable systems associated to the space of elliptic branched coverings are constructed. The relationship of these systems with elliptic Schlesinger's system is described. For the standard two-fold elliptic coverings the…
We consider K3 surfaces which are double cover of rational elliptic surfaces. The former are endowed with a natural elliptic fibration, which is induced by the latter. There are also other elliptic fibrations on such K3 surfaces, which are…
We prove doubling inequalities for solutions of elliptic systems with an iterated Laplacian as diagonal principal part and for solutions of the Lame' system of isotropic linearized elasticity. These inequalities depend on global properties…
A detailed review of the $p,q$-duality for Calogero system and its generalizations is given. For the first time, we present some of elliptic-trigonometric Hamiltonians dual to the elliptic Ruijsenaars Hamiltonians (i.e.…
We formulate the problem of finding self-dual Hamiltonians (associated with integrable systems) as deformations of free systems given on various symplectic manifolds and discuss several known explicit examples, including recently found…
The paper aims at constructing two different solutions to an elliptic system $$ u \cdot \nabla u + (-\Delta)^m u = \lambda F $$ defined on the two dimensional torus. It can be viewed as an elliptic regularization of the stationary Burgers…
We give an analytical description of the locus of the two-gap elliptic potentials associated with the corresponding flow of the Calogero--Moser system. We start with the description of Treibich--Verdier two--gap elliptic potentials. The…
We revisit the secular 3D planetary three-body problem aiming to provide a unified formalism for studying the structure of the phase space for progressively higher values of the mutual inclination $i_{mut}$ between the two planets' orbits.…
In this paper we discuss four methods of proving modularity of Calabi--Yau threefolds with $h^{12}=1$: existence of elliptic ruled surfaces inside (Hulek-Verrill), correspondence with a product of an elliptic curve and a K3 surface…