By studying various rational integrable maps on $\mathbf{\hat C}^d$ with $p$ invariants, we show that periodic points form an invariant variety of dimension $\ge p$ for each period, in contrast to the case of nonintegrable maps in which they are isolated. We prove the theorem: {\it `If there is an invariant variety of periodic points of some period, there is no set of isolated periodic points of other period in the map.'}
We propose a new approach to studying electrical networks interpreting the Ohm law as the operator which solves certain Local Yang-Baxter equation. Using this operator and the medial graph of the electrical network we define a vertex integrable statistical model and its boundary partition function. This gives an equivalent description of electrical networks We show that in the important case of an electrical network on the standard graph introduced by Curtis, Ingerman and Morrow, the response matrix of an electrical network, its most important feature, and the boundary partition function of ou…
The theory of representations of quivers and of their preprojective algebras are reviewed. In particular, moduli spaces of representations of these algebras, quiver varieties and reflection functor are described. The proof that the bijection between moduli spaces induced by the reflection functor is an isomorphism of symplectic affine varieties is presented. Hamiltonian systems with complete flows on the quiver varieties are introduced. An application of the reflection functor to these systems is described. The results on the case of the cyclic quiver are reviewed and a role of the reflection …
We study some classical integrable systems naturally associated with multiplicative quiver varieties for the (extended) cyclic quiver with $m$ vertices. The phase space of our integrable systems is obtained by quasi-Hamiltonian reduction from the space of representations of the quiver. Three families of Poisson-commuting functions are constructed and written explicitly in suitable Darboux coordinates. The case $m=1$ corresponds to the tadpole quiver and the Ruijsenaars-Schneider system and its variants, while for $m>1$ we obtain new integrable systems that generalise the Ruijsenaars-Schneider …
This is a survey of recent progress on the irreducibility of Fermi varieties, rigidity results and embedded eigenvalue problems of discrete periodic Schrödinger operators.
For periodic graph operators, we establish criteria to determine the overlaps of spectral band functions based on Bloch varieties. One criterion states that for a large family of periodic graph operators, the irreducibility of Bloch varieties implies no non-trivial periods for spectral band functions. This particularly shows that spectral band functions of discrete periodic Schrödinger operators on $\mathbb{Z}^d$ have no non-trivial periods, answering a question asked by Mckenzie and Sabri [Quantum ergodicity for periodic graphs. arXiv preprint arXiv:2208.12685 (2022)].
First, we establish the relation between the associated varieties of modules over Kac-Moody algebras \hat{g} and those over affine W-algebras. Second, we prove the Feigin-Frenkel conjecture on the singular supports of G-integrable admissible representations. In fact we show that the associated variates of G-integrable admissible representations are irreducible G-invariant subvarieties of the nullcone of g, by determining them explicitly. Third, we prove the C_2-cofiniteness of a large number of simple W-algebras, including all minimal series principal W-algebras and the exceptional W-algebras …
In a recent paper by the author (K. Yagasaki, Nonintegrability of the restricted three-body problem, submitted for publication), a technique was developed for determining whether nearly integrable systems are not meromorphically Bogoyavlenskij-integrable such that the first integrals and commutative vector fields also depend meromorphically on the small parameter. Here we continue to demonstrate the technique for some classes of dynamical systems. In particular, we consider time-periodic perturbations of single-degree-of-freedom Hamiltonian systems and discuss a relationship of the technique w…
In our previous work we have shown that the invariant varieties of periodic points (IVPP) of all periods of the 3 dimensional Lotka-Volterra map can be derived, iteratively, from the singularity confinement (SC). The method developed there can be applied to any integrable maps of dimension $d$ only when the number of the invariants $p$ equals to $d-1$. We propose, in this note, a new algorithm of the derivation which can be used in the cases ${d\over 2}\le p\le d-2$. Applying this algorithm to the 3 point Toda map, we derive a series of its IVPP's.
We provide a constructive method designed in order to control the stability of a given periodic orbit of a general completely integrable system. The method consists of a specific type of perturbation, such that the resulting perturbed system becomes a codimension-one dissipative dynamical system which also admits that orbit as a periodic orbit, but whose stability can be a-priori prescribed. The main results are illustrated in the case of a three dimensional dissipative perturbation of the harmonic oscillator, and respectively Euler's equations form the free rigid body dynamics.
In the present note, we uncover a remarkable connection between the length of periodic orbit of a classical particle enclosed in a class of 2-dimensional planar billiards and the energy of a quantum particle confined to move in an identical region with infinitely high potential wall on the boundary. We observe that the quantum energy spectrum of the particle is in exact one-to-one correspondence with the spectrum of the amplitude squares of the periodic orbits of a classical particle for the class of integrable billiards considered. We have established the results by geometric constructions an…
Integrable curl-force Hamiltonians: bi-Hamiltonian structure, separability, and periodic orbits
Advanced Institute for Materials Research, Tohoku University（东北大学物质材料研究高等研究院）；Department of Mathematics, City St George’s, University of London（伦敦大学城市圣乔治学院数学系）
In this paper we study the Klein-Gordon (KG) lattice with periodic boundary conditions. It is an $N$ degrees of freedom Hamiltonian system with linear inter-site forces and nonlinear on-site potential, which here is taken to be of the $ϕ^4$ form. First, we prove that the system in consideration is non-integrable in Liuville sense. The proof is based on the Morales-Ramis theory. Next, we deal with the resonant Birkhoff normal form of the KG Hamiltonian, truncated to order four. Due to the choice of potential, the periodic KG lattice shares the same set of discrete symmetries as the periodic Fer…
The main goal of the first part of the paper is to show that the Fermi curve of a two-dimensional periodic Schrödinger operator with nonnegative potential whose points parameterize the Bloch solutions of the Shrödinger equation at the zero energy level is a smooth $M$-curve. Moreover, it is shown that the poles of the Bloch solutions are located on the fixed ovals of an antiholomorphic involution so that each but one oval contains precisely one pole. The topological type is stable until, at some value of the deformation parameter, the zero level becomes an eigenlevel for the Schrödinger operat…
We demonstrate the existence of special phantom excitations for open and periodically closed integrable systems at the example of the $XXZ$ Heisenberg spin chain. The phantom excitations do not contribute to the energy of the Bethe state and correspond to special solutions to the Bethe Ansatz equations with infinite "phantom" Bethe roots. The phantom Bethe roots lead to degeneracies between different magnetization sectors in the periodic case and to the appearance of spin helix states (SHS), i.e. periodically modulated states of chiral nature in both open and closed systems. For the periodic c…
Abel's quadratures for integrable Hamiltonian systems are defined up to a group law of the corresponding Abelian variety $A$. If $A$ is isogenous to a direct product of Abelian varieties $A\cong A_1\times\cdots\times A_k$, the group law can be used to construct various Lax matrices on the factors $A_1,\ldots,A_k$. As an example, we discuss 2-dimensional reducible Abelian variety $A=E_+\times E_-$, which is a product of 1-dimensional varieties $E_\pm$ obtained by Euler in his study of the two fixed centres problem, and the Lax matrices on the factors $E_\pm$.
We study the existence of real-analytic first integrals and real-analytic integrability for perturbations of integrable systems in the sense of Bogoyavlenskij including non-Hamiltonian ones. We especially assume that there exists a family of periodic orbits on a regular level set of the first integrals having a connected and compact component and give sufficient conditions for nonexistence of the same number of real-analytic first integrals in the perturbed systems as the unperturbed ones and for their real-analytic nonintegrability near the level set such that the first integrals and commutat…
We study the relationship between singularities of finite-dimensional integrable systems and singularities of the corresponding spectral curves. For the large class of integrable systems on matrix polynomials, which is a general framework for various multidimensional spinning tops, as well as Beauville systems, we prove that if the spectral curve is nodal, then all singularities on the corresponding fiber of the system are non-degenerate. We also show that the type of a non-degenerate singularity can be read off from the behavior of double points on the spectral curve under an appropriately de…
We study transverse stability and instability of one-dimensional small-amplitude periodic traveling waves of a generalized Kadomtsev-Petviashvili equation with respect to two-dimensional perturbations, which are either periodic or square-integrable in the direction of the propagation of the underlying one-dimensional wave and periodic in the transverse direction. We obtain transverse instability results in KP-fKdV, KP-ILW, and KP-Whitham equations. Moreover, assuming the spectral stability of one-dimensional wave with respect to one-dimensional square-integrable periodic perturbations, we obta…
In (1+1)-dimensional quantum field theory, integrability is typically defined as the existence of an infinite number of local charges of different Lorentz spin, which commute with the Hamiltonian. A well known consequence of integrability is that scattering of particles is elastic and factorizable. These properties are the basis for the bootstrap program, which leads to the exact computation of S-matrices and form factors. We consider periodically-driven field theories, whose stroboscopic time-evolution is described by a Floquet Hamiltonian. It was recently proposed by Gritsev and Polkovnikov …
Recently, an integrable system of coupled (2+1)-dimensional nonlinear Schrodinger (NLS) equations was introduced by Fokas (eq. (18) in Nonlinearity 29}, 319324 (2016)). Following this pattern, two integrable equations [eqs.2 and 3] with specific parity-time symmetry are introduced here, under different reduction conditions. For eq. 2, two kinds of periodic solutions are obtained analytically by means of the Hirota's bilinear method. In the long-wave limit, the two periodic solutions go over into rogue waves (RWs) and semi-rational solutions, respectively. The RWs have a line shape, while the s…
We give a conjectural formula for sheaves supported on (irreducible) conormal varieties inside the cotangent bundle of the Grassmannian, such that their equivariant $K$-class is given by the partition function of an integrable loop model, and furthermore their $K$-theoretic pushforward to a point is a solution of the level $1$ quantum Knizhnik-Zamolodchikov equation. We prove these results in the case that the Lagrangian is smooth (hence is the conormal bundle to a subGrassmannian). To compute the pushforward to a point, or equivalently to the affinization, we simultaneously degenerate the Lag…
