Matteo Petrera

On Miura Transformations and Volterra-Type Equations Associated with the Adler-Bobenko-Suris Equations

We construct Miura transformations mapping the scalar spectral problems of the integrable lattice equations belonging to the Adler-Bobenko-Suris (ABS) list into the dis- crete Schrodinger spectral problem associated with Volterra-type equations. We show that the ABS equations correspond to Backlund transformations for some particular cases of the discrete Krichever-Novikov equation found by Yamilov (YdKN equation). This enables us to construct new generalized symmetries for the ABS equations. The same can be said about the generalizations of the ABS equations introduced by Tongas, Tsoubelis and Xenitidis. All of them generate Backlund transformations for the YdKN equation. The higher order generalized symmetries we construct in the present paper confirm their integrability.

Continuous symmetries of the lattice potential KdV equation

In this paper we present a set of results on the integration and on the symmetries of the lattice potential Korteweg–de Vries (lpKdV) equation. Using its associated spectral problem we construct the soliton solutions and the Lax technique enables us to provide infinite sequences of generalized symmetries. Finally, using a discrete symmetry of the lpKdV equation, we construct a large class of non-autonomous generalized symmetries.

ON INTEGRABILITY OF HIROTA-KIMURA TYPE DISCRETIZATIONS

We give an overview of the integrability of the Hirota-Kimura discretizationmethod applied to algebraically completely integrable (a.c.i.) systems with quadratic vector fields. Along with the description of the basic mechanism of integrability (Hirota-Kimura bases), we provide the reader with a fairly complete list of the currently available results for concrete a.c.i. systems.

The lattice Schwarzian KdV equation and its symmetries

In this paper, we present a set of results on the symmetries of the lattice Schwarzian Korteweg-de Vries (1SKdV) equation. We construct the Lie point symmetries and, using its associated spectral problem, an infinite sequence of generalized symmetries and master symmetries. We finally show that we can use master symmetries of the 1SKdV equation to construct non-autonomous non-integrable generalized symmetries.

Separation of variables and Backlund transformations for the symmetric Lagrange top

We construct the one- and two-point integrable maps (Backlund transformations) for the symmetric Lagrange top. We show that the Lagrange top has the same algebraic Poisson structure that belongs to the sl(2) Gaudin magnet. The two-point map leads to a real time discretization of the continuous flow. Therefore, it provides an integrable numerical scheme for integrating the physical flow. We illustrate the construction by a few pictures of the discrete flow calculated in MATLAB.

On integrability of Hirota-Kimura type discretizations. Experimental study of the discrete Clebsch system

R. Hirota and K. Kimura discovered integrable discretizations of the Euler and the Lagrange tops, given by birational maps. Their method is a specialization to the integrable context of a general discretization scheme introduced by W. Kahan and applicable to any vector field with a quadratic dependence on phase variables. According to a proposal by T. Ratiu, discretizations of Hirota–Kimura type can be considered for numerous integrable systems of classical mechanics. Due to a remarkable and not well understood mechanism, such discretizations seem to inherit the integrability for all algebraically completely integrable systems. We introduce an experimental method for a rigorous study of integrability of such discretizations. Application of this method to the Hirota–Kimura-type discretization of the Clebsch system leads to the discovery of four functionally independent integrals of motion of this discrete-time system, which turn out to be much more complicated than the integrals of the continuous-time system. Further, we prove that every orbit of the discrete-time Clebsch system lies in an intersection of four quadrics in the six-dimensional phase space. Analogous results hold for the Hirota–Kimura-type discretizations for all commuting flows of the Clebsch system, as well as for the so(4) Euler top.

Algebraic Extensions of Gaudin Models

Abstract We perform a Inönü–Wigner contraction on Gaudin models, showing how the integrability property is preserved by this algebraic procedure. Starting from Gaudin models we obtain new integrable chains, that we call Lagrange chains, associated to the same linear r-matrix structure. We give a general construction involving rational, trigonometric and elliptic solutions of the classical Yang-Baxter equation. Two particular examples are explicitly considered: the rational Lagrange chain and the trigonometric one. In both cases local variables of the models are the generators of the direct sum of N interacting tops.

What is integrability of discrete variational systems

We propose a notion of a pluri-Lagrangian problem, which should be understood as an analogue of multi-dimensional consistency for variational systems. This is a development along the line of research of discrete integrable Lagrangian systems initiated in 2009 by Lobb and Nijhoff, however, having its more remote roots in the theory of pluriharmonic functions, in the Z-invariant models of statistical mechanics and their quasiclassical limit, as well as in the theory of variational symmetries going back to Noether. A d-dimensional pluri-Lagrangian problem can be described as follows: given a d-form on an m-dimensional space (called multi-time, m>d), whose coefficients depend on a sought-after function x of m independent variables (called field), find those fields x which deliver critical points to the action functionals for any d-dimensional manifold Σ in the multi-time. We derive the main building blocks of the multi-time Euler–Lagrange equations for a discrete pluri-Lagrangian problem with d=2, the so-called corner equations, and discuss the notion of consistency of the system of corner equations. We analyse the system of corner equations for a special class of three-point two-forms, corresponding to integrable quad-equations of the ABS list. This allows us to close a conceptual gap of the work by Lobb and Nijhoff by showing that the corresponding two-forms are closed not only on solutions of (non-variational) quad-equations, but also on general solutions of the corresponding corner equations. We also find an example of a pluri-Lagrangian system not coming from a multi-dimensionally consistent system of quad-equations.

S. V. Kovalevskaya system, its generalization and discretization

We consider an integrable three-dimensional system of ordinary differential equations introduced by S. V. Kovalevskaya in a letter to G. Mittag-Leffler. We prove its isomorphism with the three-dimensional Euler top, and propose two integrable discretizations for it. Then we present an integrable generalization of the Kovalevskaya system, and study the problem of integrable discretization for this generalized system.

Multi-time Lagrangian 1-forms for families of Bäcklund transformations: Toda-type systems

General Lagrangian theory of discrete one-dimensional integrable systems is illustrated by a detailed study of Backlund transformations for Toda-type systems. Commutativity of Backlund transformations is shown to be equivalent to the consistency of the system of discrete multi-time Euler–Lagrange equations. The precise meaning of the commutativity in the periodic case, when all maps are double-valued, is established. It is shown that the gluing of different branches is governed by the so-called superposition formulas. The closure relation for the multi-time Lagrangian 1-form on solutions of the variational equations is proved for all Toda-type systems. Superposition formulas are instrumental for this proof. The closure relation was previously shown to be equivalent to the spectrality property of Backlund transformations, i.e., to the fact that the derivative of the Lagrangian with respect to the spectral parameter is a common integral of motion of the family of Backlund transformations. We relate this integral of motion to the monodromy matrix of the zero curvature representation which is derived directly from equations of motion in an algorithmic way. This serves as further evidence in favor of the idea that Backlund transformations serve as zero curvature representations for themselves.