Non-integrability vs. integrability in pentagram maps

We revisit recent results on integrable cases for higher-dimensional generalizations of the 2D pentagram map: short-diagonal, dented, deep-dented, and corrugated versions, and define a universal class of pentagram maps, which are proved to possess projective duality. We show that in many cases the pentagram map cannot be included into integrable flows as a time-one map, and discuss how the corresponding notion of discrete integrability can be extended to include jumps between invariant tori. We also present a numerical evidence that certain generalizations of the integrable 2D pentagram map are non-integrable and present a conjecture for a necessary condition of their discrete integrability.     

Topological monodromy as an obstruction to Hamiltonization of nonholonomic systems: Pro or contra?

The phenomenon of a topological monodromy in integrable Hamiltonian and nonholonomic systems is discussed. An efficient method for computing and visualizing the monodromy is developed. The comparative analysis of the topological monodromy is given for the rolling ellipsoid of revolution problem in two cases, namely, on a smooth and on a rough plane. The first of these systems is Hamiltonian, the second is nonholonomic. We show that, from the viewpoint of monodromy, there is no difference between the two systems, and thus disprove the conjecture by Cushman and Duistermaat stating that the topological monodromy gives a topological obstruction for Hamiltonization of the rolling ellipsoid of revolution on a rough plane.     

Derivation of invariant varieties of periodic points from singularity confinement in the case of Toda map

We have shown in [1] that the invariant varieties of periodic points (IVPP) of all periods of some higher dimensional rational maps can be derived, iteratively, from the singularity confinement (SC). We generalize this algorithm, in this paper, to apply to any birational map, which has more invariants than the half of the dimension.     

On one-point functions for sinh-Gordon model at finite temperature

Using fermionic basis we conjecture the exact formulae for the expectation values of local fields in sinh-Gordon model. The conjecture is checked against previously known results.     

Darboux transformation and explicit solutions for the integrable sixth-order KdV equation for nonlinear waves

Korteweg-de Vries (KdV)-type equations can describe some physical phenomena in fluids, nonlinear optics, quantum mechanics, plasmas, etc. In this paper, with the aid of symbolic computation, the integrable sixth-order KdV equation is investigated. Darboux transformation (DT) with an arbitrary parameter is presented. Explicit solutions are derived with the DT. Relevant properties are graphically illustrated, which might be helpful to understand some physical processes in fluids, plasmas, optics and quantum mechanics.     

On the singular sector of the Hermitian random matrix model in the large N limit

The one-cut case of the Hermitian random matrix model in the large N limit is considered. Its singular sector in the space of coupling constants is analyzed from the point of view of the hodograph equations of the underlying dispersionless Toda hierarchy. A deep connection with the singular sector of the hodograph equations of the 1-layer Benney (classical long wave equation) hierarchy is stablished. This property is a consequence of the fact that the hodograph equations for both hierarchies describe the critical points of solutions of Euler-Poisson-Darboux equations.     

Beltrami方程组弱解分量函数的弱单调性

该文引入一类新的函数空间, 并借助于此空间, 研究了 A -调和方程很弱解的弱单调性, 并得到了空间Beltrami方程组弱解分量函数的弱单调性.     

A cohomological obstruction for global quasi-bi-Hamiltonian fields

We introduce the notion of integrating factor for a 1-form which is an inner product of a vector fields and a 2-form, and the notion of weakly bi-Hamiltonian field also, which is locally quasi-bi-Hamiltonian. A cohomological class in some first cohomology space is associated to such vector fields when this is weakly bi-Hamiltonian and defined relatively to the above 1-form. This class is a cohomological obstruction to the existence of a global integrating factor for the 1-form.     

Jacobi structures of evolutionary partial differential equations

In this paper we introduce the notion of infinite dimensional Jacobi structure to describe the geometrical structure of a class of nonlocal Hamiltonian systems which appear naturally when applying reciprocal transformations to Hamiltonian evolutionary PDEs. We prove that our class of infinite dimensional Jacobi structures is invariant under the action of reciprocal transformations that only change the spatial variable. The main technical tool is in a suitable generalization of the classical Schouten–Nijenhuis bracket to the space of the so called quasi-local multi-vectors, and a simple realization of this structure in the framework of supermanifolds. These constructions are used to compute the Lichnerowicz–Jacobi cohomologies and to prove a Darboux theorem for Jacobi structures with hydrodynamic leading terms. We also introduce the notion of bi-Jacobi structures, and consider the integrability of a system of evolutionary PDEs that possesses a bi-Jacobi structure.