The Pentagram Map: A Discrete Integrable System

The pentagram map is a projectively natural transformation defined on (twisted) polygons. A twisted polygon is a map from \mathbb Z{\mathbb Z} into \mathbbRP²{{\mathbb{RP}}^2} that is periodic modulo a projective transformation called the monodromy. We find a Poisson structure on the space of twisted polygons and show that the pentagram map relative to this Poisson structure is completely integrable. For certain families of twisted polygons, such as those we call universally convex, we translate the integrability into a statement about the quasi-periodic motion for the dynamics of the pentagram map. We also explain how the pentagram map, in the continuous limit, corresponds to the classical Boussinesq equation. The Poisson structure we attach to the pentagram map is a discrete version of the first Poisson structure associated with the Boussinesq equation. A research announcement of this work appeared in [16].     

On the Integrability of q-Oscillators Based on Invariants of Discrete Fourier Transforms

A discretization of the quantum mechanical phase space is presented in the context of q-noncommutative structures. We give two generalizations of the Heisenberg algebra in the arising lattice phase space. In contrast to ordinary quantum mechanics, there is, a priori, no systematic approach to an integrable oscillator Hamiltonian in lattice quantum mechanics. This is the central obstacle to deal with in this Letter. To do so, we show how in general the integrability of the harmonic oscillator is related to the Fourier transform between momentum and space variables. This will be done in both cases, the continuous and the discrete one. As an application, we finally obtain an integrable lattice Hamiltonian for the harmonic oscillator with generalized Hermite eigenfunctions.     

Integrable nonlinear evolution partial differential equations in 4 + 2 and 3 + 1 dimensions

The derivation and solution of integrable nonlinear evolution partial differential equations in three spatial dimensions has been the holy grail in the field of integrability since the late 1970s. The celebrated Korteweg-de Vries and nonlinear Schr?dinger equations, as well as the Kadomtsev-Petviashvili (KP) and Davey-Stewartson (DS) equations, are prototypical examples of integrable evolution equations in one and two spatial dimensions, respectively. Do there exist integrable analogs of these equations in three spatial dimensions? In what follows, I present a positive answer to this question. In particular, I first present integrable generalizations of the KP and DS equations, which are formulated in four spatial dimensions and which have the novelty that they involve complex time. I then impose the requirement of real time, which implies a reduction to three spatial dimensions. I also present a method of solution.     

Nonlinear nonautonomous discrete dynamical systems from a general discrete isomonodromy problem

Nonlinear nonautonomous discrete dynamical systems (DDS) whose continuum limits are the well-known Painlevé equations, have recently arisen in models of quantum gravity. The Painlevé equations are believed integrable because each is the isomonodromy condition for an associated linear differential equation. However, not every DDS with an integrable continuum limit is necessarily integrable. Which of the many discrete versions of the Painlevé equations inherit their integrability is not known. How to derive all their integrable discrete versions is also not known. We provide a systematic method of attacking these questions by giving a general discrete isomonodromy problem. Discrete versions of the first and second Painlevé equations are deduced from this general problem.     

Discretizing constant curvature surfaces via loop group factorizations: The discrete sine- and sinh-Gordon equations

The sine- and sinh-Gordon equations are the harmonic map equations for maps of the (Lorentz) plane into the 2-sphere. Geometrically they correspond to the integrability equations for surfaces of constant Gauss and constant mean curvature. There is a well-known dressing action of a loop group on the space of harmonic maps. By discretizing the vacuum solutions we obtain via the dressing action completely integrable discretizations (in both variables) of the sine- and sinh-Gordon equations. For the sine-Gordon equation we get Hirota's discretization. Since we work in a geometric context we also obtain discrete models for harmonic maps into the 2-sphere and discrete models of constant Gauss and mean curvature surfaces.     

Localized coherent structures of (2+1) dimensional generalizations of soliton systems

We briefly review the recent progress in obtaining (2+1) dimensional integrable generalizations of soliton equations in (1+1) dimensions. Then, we develop an algorithmic procedure to obtain interesting classes of solutions to these systems. In particular using a Painlevé singularity structure analysis approach, we investigate their integrability properties and obtain their appropriate Hirota bilinearized forms. We identify line solitons and from which we introduce the concept of ghost solitons, which are patently boundary effects characteristic of these (2+1) dimensional integrable systems. Generalizing these solutions, we obtain exponentially localized solutions, namely the dromions which are driven by the boundaries. We also point out the interesting possibility that while the physical field itself may not be localized, either the potential or composite fields may get localized. Finally, the possibility of generating an even wider class of localized solutions is hinted by using curved solitons.     

Hamiltonian theory of integrable generalizations of the nonlinear Schrödinger equation

A method for constructing integrable systems and their Bäcklund transformations is proposed. The case of integrable generalizations of the nonlinear Schrödinger equation in the one-dimensional case and the possibility of extending the method to higher dimensions are discussed in detail. The existence of Bäcklund transformations of a definite type in the systems considered is used as a criterion of integrability. This leads to “gauge fixing” — the number of physically different integrable systems is strongly diminished. The method can be useful in constructing the admissible nonlinear terms in some models of quantum field theory, e.g., in Ginzburg-Landau functionals.     

Bargmann Symmetry Constraint for a Family of Liouville Integrable Differential-Difference Equations

A family of integrable differential-difference equations is derived from a new matrix spectral problem. The Hamiltonian forms of obtained differential-difference equations are constructed. The Liouville integrability for the obtained integrable family is proved. Then, Bargmann symmetry constraint of the obtained integrable family is presented by binary nonliearization method of Lax pairs and adjoint Lax pairs. Under this Bargmann symmetry constraints, an integrable symplectic map and a sequences of completely integrable finite-dimensional Hamiltonian systems in Liouville sense are worked out, and every integrable differential-difference equations in the obtained family is factored by the integrable symplectic map and a completely integrable finite-dimensional Hamiltonian system.     

Integrable deformations of Lotka-Volterra systems

The Hamiltonian structure of a class of three-dimensional (3D) Lotka-Volterra (LV) equations is revisited from a novel point of view by showing that the quadratic Poisson structure underlying its integrability structure is just a real three-dimensional Poisson-Lie group. As a consequence, the Poisson coalgebra map Δ(2) that is given by the group multiplication provides the keystone for the explicit construction of a new family of 3N-dimensional integrable systems that, under certain constraints, contain N sets of deformed versions of the 3D LV equations. Moreover, by considering the most generic Poisson-Lie structure on this group, a new two-parametric integrable perturbation of the 3D LV system through polynomial and rational perturbation terms is explicitly found.     

Integrability of the Rabi model

The Rabi model is a paradigm for interacting quantum systems. It couples a bosonic mode to the smallest possible quantum model, a two-level system. I present the analytical solution which allows us to consider the question of integrability for quantum systems that do not possess a classical limit. A criterion for quantum integrability is proposed which shows that the Rabi model is integrable due to the presence of a discrete symmetry. Moreover, I introduce a generalization with no symmetries; the generalized Rabi model is the first example of a nonintegrable but exactly solvable system.     

The pentagram integrals for Poncelet families

The pentagram map is now known to be a discrete integrable system. We show that the integrals for the pentagram map are constant along Poncelet families. That is, if $P_1$ and $P_2$ are two polygons in the same Poncelet family, and $f$ is a monodromy invariant for the pentagram map, then $f\left(P_1\right)=f\left(P_2\right)$. Our proof combines complex analysis with an analysis of the geometry of a degenerating sequence of Poncelet polygons.     

Integrable dissipative nonlinear second order differential equations via factorizations and Abel equations

We emphasize two connections, one well known and another less known, between the dissipative nonlinear second order differential equations and the Abel equations which in their first-kind form have only cubic and quadratic terms. Then, employing an old integrability criterion due to Chiellini, we introduce the corresponding integrable dissipative equations. For illustration, we present the cases of some integrable dissipative Fisher, nonlinear pendulum, and Burgers–Huxley type equations which are obtained in this way and can be of interest in applications. We also show how to obtain Abel solutions directly from the factorization of second order nonlinear equations.     

Integrability of fermionic extensions of the Korteweg-de Vries equation

We investigate in detail the integrability of fermionic extensions of the Korteweg-de Vries equation by using the prolongation structure method. The integrable one- and two-parameter systems are obtained. The previously known integrable systems are their special cases.     

Hyperbolic reduction

The notions of weak Darboux integrability and hyperbolic reduction are introduced, and their potential is gauged as a means of extending the range of application of geometric methods for solving hyperbolic partial differential equations. For directness, our work is expressed in local coordinates and formulated for semilinear hyperbolic systems in two independent variables. The theory is applied to the study of 1+1-wave maps into surfaces of revolution. It is shown that the differential system for any such wave map may be viewed as an integrable extension of a certain scalar, semilinear, hyperbolic partial differential equation which is explicitly constructed. Using this we discover a new integrable wave map system for which hyperbolic reduction leads to a large family of explicit wave maps.     

On the quantization of the Kowalevskaya top

The equations of motion of a heavy top can be integrated for three different combinations of the parameters of the system. Historically, the discovery of these three integrable cases is attributed to Euler, Lagrange and Kowalevskaya, respectively. While the quantization of the first two cases can be performed in a straightforward way, the quantum integrability of the Kowalevskaya top is far from trivial. We show here how one can recover quantum integrability for this case as well.     

New integrable cases of a Cremona transformation: a finite-order orbits analysis

We analyse the properties of a particular birational mapping of two variables (Cremona transformation) depending on two free parameters ( and ), associated with the action of a discrete group of non-linear (birational) transformations on the entries of a q × q matrix. This mapping originates from the analysis of birational transformations obtained from very simple algebraic calculations, namely taking the inverse of q × q matrices and permuting some of the entries of these matrices. It has been seen to yield weak chaos and integrability. We have found new integrable cases of this Cremona transformation, corresponding to the values of = 0 when , besides the already known values = 0 and = −1, and also arbitrary when = 0. For these cases, one has a foliation of the parameter space in elliptic curves. We give the equations of these elliptic curves. Based on this very example we show how one can find these integrability cases of the Cremona transformation and actually integrate it using a method based on the systematic study of the finite-order conditions of the Cremona transformation. The method is shown to be efficient and straightforward. The various integrability cases are revisited using many different representations of this very mapping (birational transformations, recursion in one variable, …).     

Quantum and classical spin clusters: disappearance of quantum numbers and Hamiltonian chaos

We present a direct link between manifestations of classical Hamiltonian chaos and quantum nonintegrability effects as they occur in quantum invariants. In integrable classical Hamiltonian systems, analytic invariants (integrals of the motion) can be constructed numerically by means of time averages of dynamical variables over phase-space trajectories, whereas in near-integrable models such time averages yield nonanalytic invariants with qualitatively different properties. Translated into quantum mechanics, the invariants obtained from time averages of dynamical variables in energy eigenstates provide a topographical map of the plane of quantized actions (quantum numbers) with properties which again depend sensitively on whether or not the classical integrability condition is satisfied. The most conspicuous indicator of quantum chaos is the disappearance of quantum numbers, a phenomenon directly related to the breakdown of invariant tori in the classical phase flow. All results are for a system consisting of two exchange-coupled spins with biaxial exchange and single-site anisotropy, a system with a nontrivial integrability condition.     

Generalized Toda Mechanics Associated with Loop Algebras (L)(Cr) and (L)(Dr) and Their Reductions

We construct a class of integrable generalization of Toda mechanics with long-range interactions. These systems are associated with the loop algebras L(C_r) and L(D_r) in the sense that their Lax matrices can be realized in terms of the c=0 representations of the affine Lie algebras C⁽¹⁾_r and D⁽¹⁾_r and the interactions pattern involved bears the typical characters of the corresponding root systems. We present the equations of motion and the Hamiltonian structure. These generalized systems can be identified unambiguously by specifying the underlying loop algebra together with an ordered pair of integers (n,m). It turns out that different systems associated with the same underlying loop algebra but with different pairs of integers (n₁,m₁) and (n₂,m₂) with n₂<n₁ and m₂<m₁ can be related by a nested Hamiltonian reduction procedure. For all nontrivial generalizations, the extra coordinates besides the standard Toda variables are Poisson non-commute, and when either $n$ or m≥3, the Poisson structure for the extra coordinate variables becomes some Lie algebra (i.e. the extra variables appear linearly on the right-hand side of the Poisson brackets). In the quantum case, such generalizations will become systems with noncommutative variables without spoiling the integrability.     

Competing Species: Integrability and Stability

Abstract

We examine the classical model of two competing species for integrability in terms of analytic functions by means of the Painlevé analysis. We find that the governing equations are integrable for certain values of the essential parameters of the system. We find that, for all integrable cases with the nontrivial equilibrium point in the physically acceptable region, the nontrivial equilibrium point is stable.     

On the Structure of the B¨acklund Transformations for the Relativistic Lattices

Abstract

The B¨acklund transformations for the relativistic lattices of the Toda type and their discrete analogues can be obtained as the composition of two duality transformations. The condition of invariance under this composition allows to distinguish effectively the integrable cases. Iterations of the B¨acklund transformations can be described in the terms of nonrelativistic lattices of the Toda type. Several multifield generalizations are presented.