Dimensional Integrable Systems Research Articles

Symplectic invariants for parabolic orbits and cusp singularities of integrable systems.

We discuss normal forms and symplectic invariants of parabolic orbits and cuspidal tori in integrable Hamiltonian systems with two degrees of freedom. Such singularities appear in many integrable systems in geometry and mathematical physics and can be considered as the simplest example of degenerate singularities. We also suggest some new techniques which apparently can be used for studying symplectic invariants of degenerate singularities of more general type.This article is part of the theme issue 'Finite dimensional integrable systems: new trends and methods'.

A survey on polynomial in momenta integrals for billiard problems.

In this paper, we give a short survey of recent results on the algebraic version of the Birkhoff conjecture for integrable billiards on surfaces of constant curvature. We also discuss integrable magnetic billiards. As a new application of the algebraic technique, we study the existence of polynomial integrals for the two-sided magnetic billiards introduced by Kozlov and Polikarpov.This article is part of the theme issue 'Finite dimensional integrable systems: new trends and methods'.

Asymptotic velocity for four celestial bodies.

Asymptotic velocity is defined as the Cesàro limit of velocity. As such, its existence has been proved for bounded interaction potentials. This is known to be wrong in celestial mechanics with four or more bodies. Here, we show for a class of pair potentials including the homogeneous ones of degree -α for α∈(0, 2), that asymptotic velocities exist for up to four bodies, dimension three or larger, for any energy and almost all initial conditions on the energy surface.This article is part of the theme issue 'Finite dimensional integrable systems: new trends and methods'.

Singular fibres of the Gelfand-Cetlin system on 𝔲(n).

In this paper, we show that every singular fibre of the Gelfand-Cetlin system on co-adjoint orbits of unitary groups is a smooth isotropic submanifold which is diffeomorphic to a two-stage quotient of a compact Lie group by free actions of two other compact Lie groups. In many cases, these singular fibres can be shown to be homogeneous spaces or even diffeomorphic to compact Lie groups. We also give a combinatorial formula for computing the dimensions of all singular fibres, and give a detailed description of these singular fibres in many cases, including the so-called (multi-)diamond singularities. These (multi-)diamond singular fibres are degenerate for the Gelfand-Cetlin system, but they are Lagrangian submanifolds diffeomorphic to direct products of special unitary groups and tori. Our methods of study are based on different ideas involving complex ellipsoids, Lie groupoids and also general ideas coming from the theory of singularities of integrable Hamiltonian systems.This article is part of the theme issue 'Finite dimensional integrable systems: new trends and methods'.

Convexity of the moment map image for torus actions on bm -symplectic manifolds.

We prove a convexity theorem for the image of the moment map of a Hamiltonian torus action on a bm -symplectic manifold.This article is part of the theme issue 'Finite dimensional integrable systems: new trends and methods'.

On the integrability of Birkhoff billiards.

In this survey, we provide a concise introduction to convex billiards and describe some recent results, obtained by the authors and collaborators, on the classification of integrable billiards, namely the so-called Birkhoff conjecture.This article is part of the theme issue 'Finite dimensional integrable systems: new trends and methods'.

Constructing (2+1)-dimensional supersymmetric integrable systems from the Hirota formalism in the superspace**Project supported by the National Natural Science Foundation of China (Grant Nos. 11605102, 11475052, 11675055, and 11626140).

The supersymmetric extensions of two integrable systems, a special negative Kadomtsev–Petviashvili (NKP) system and a (2+1)-dimensional modified Korteweg–de Vries (MKdV) system, are constructed from the Hirota formalism in the superspace. The integrability of both systems in the sense of possessing infinitely many generalized symmetries are confirmed by extending the formal series symmetry approach to the supersymmetric framework. It is found that both systems admit a generalization of type algebra and a Kac-Moody–Virasoro type subalgebra. Interestingly, the first one of the positive flow of the supersymmetric NKP system is another supersymmetric extension of the (2+1)-dimensional MKdV system. Based on our work, a hypothesis is put forward on a series of (2+1)-dimensional supersymmetric integrable systems. It is hoped that our work may develop a straightforward way to obtain supersymmetric integrable systems in high dimensions.

New integrable ( $$3+1$$ 3 + 1 )-dimensional systems and contact geometry

We introduce a novel systematic construction for integrable (3+1)-dimensional dispersionless systems using nonisospectral Lax pairs that involve contact vector fields. In particular, we present new large classes of (3+1)-dimensional integrable dispersionless systems associated to the Lax pairs which are polynomial and rational in the spectral parameter.

Erratum to: A Finite Dimensional Integrable System Arising in the Study of Shock Clustering

Erratum to: A Finite Dimensional Integrable System Arising in the Study of Shock Clustering

Hamiltonian and symplectic symmetries: An introduction

Classical mechanical systems are modeled by a symplectic manifold $(M,\omega)$, and their symmetries, encoded in the action of a Lie group $G$ on $M$ by diffeomorphisms that preserves $\omega$. These actions, which are called symplectic, have been studied in the past forty years, following the works of Atiyah, Delzant, Duistermaat, Guillemin, Heckman, Kostant, Souriau, and Sternberg in the 1970s and 1980s on symplectic actions of compact abelian Lie groups that are, in addition, of type, i.e. they also satisfy Hamilton's equations. Since then a number of connections with combinatorics, finite dimensional integrable Hamiltonian systems, more general symplectic actions, and topology, have flourished. In this paper we review classical and recent results on Hamiltonian and non Hamiltonian symplectic group actions roughly starting from the results of these authors. The paper also serves as a quick introduction to the basics of symplectic geometry.

Nonlocal symmetry and consistent Riccati expansion integrability of the (1+1)-dimensional integrable nonlinear dispersive-wave system

Under investigation in this paper is nonlocal symmetry, consistent Riccati expansion (CRE) integrability of the (1+1)-dimensional integrable nonlinear dispersive-wave system, which can be used to describes a bidirectional soliton for wave propagation. We construct the Bäcklund transformation and consider the truncated Painlevé expansion of the system. It’s Schwarzian form is derived, whose nonlocal symmetry is localized to provide the corresponding nonlocal group. Furthermore, we verify that the system is solvable via the CRE. Based on the CRE, we further present its soliton-cnoidal wave interaction solution in terms of Jacobi elliptic functions and the third type of incomplete elliptic integral.

Three New (2+1)-dimensional Integrable Systems and Some Related Darboux Transformations**Supported by the National Natural Science Foundation of China under Grant No. 11371361, the Shandong Provincial Natural Science Foundation of China under Grant Nos. ZR2012AQ011, ZR2013AL016, ZR2015EM042, National Social Science Foundation of China under Grant No. 13BJY026, the Development of Science and Technology Project under Grant No. 2015NS1048 and A Project

We introduce two operator commutators by using different-degree loop algebras of the Lie algebra A1, then under the framework of zero curvature equations we generate two (2+1)-dimensional integrable hierarchies, including the (2+1)-dimensional shallow water wave (SWW) hierarchy and the (2+1)-dimensional Kaup-Newell (KN) hierarchy. Through reduction of the (2+1)-dimensional hierarchies, we get a (2+1)-dimensional SWW equation and a (2+1)-dimensional KN equation. Furthermore, we obtain two Darboux transformations of the (2+1)-dimensional SWW equation. Similarly, the Darboux transformations of the (2+1)-dimensional KN equation could be deduced. Finally, with the help of the spatial spectral matrix of SWW hierarchy, we generate a (2+1) heat equation and a (2+1) nonlinear generalized SWW system containing inverse operators with respect to the variables x and y by using a reduction spectral problem from the self-dual Yang-Mills equations.

Lax operator algebras and integrable systems

A new class of infinite-dimensional Lie algebras, called Lax operator algebras, is presented, along with a related unifying approach to finite- dimensional integrable systems with a spectral parameter on a Riemann surface such as the Calogero–Moser and Hitchin systems. In particular, the approach includes (non-twisted) Kac–Moody algebras and integrable systems with a rational spectral parameter. The presentation is based on quite simple ideas about the use of gradings of semisimple Lie algebras and their interaction with the Riemann–Roch theorem. The basic properties of Lax operator algebras and the basic facts about the theory of the integrable systems in question are treated (and proved) from this general point of view. In particular, the existence of commutative hierarchies and their Hamiltonian properties are considered. The paper concludes with an application of Lax operator algebras to prequantization of finite-dimensional integrable systems. Bibliography: 51 titles.

A Finite Dimensional Integrable System Arising in the Study of Shock Clustering

In this work, we consider a finite dimensional Hamiltonian system that contains as a special case an exact discretization of the Lax equation for shock clustering. We characterize the generic coadjoint orbits of the underlying Lie group and establish the Liouville integrability of the system on such orbits. We also solve the Hamiltonian equation explicitly via Riemann–Hilbert factorization problems.

Some 2+1 Dimensional Super-Integrable Systems

Abstract In the article, we make use of the binormial-residue-representation (BRR) to generate super 2+1 dimensional integrable systems. Using these systems, we can deduce a super 2+1 dimensional AKNS hierarchy, which can be reduced to a super 2+1 dimensional nonlinear Schrödinger equation. In particular, two main results are obtained. One of them is a set of super 2+1 dimensional integrable couplings. The other one is a 2+1 dimensional diffusion equation. The Hamiltonian structure of the super 2+1 dimensional hierarchy is derived by using the super-trace identity.

Generalised (2+1)-dimensional Super MKdV Hierarchy for Integrable Systems in Soliton Theory

AbstractMuch attention has been given to constructing Lie and Lie superalgebra for integrable systems in soliton theory, which often have significant scientific applications. However, this has mostly been confined to (1+1)-dimensional integrable systems, and there has been very little work on (2+1)-dimensional integrable systems. In this article, we construct a class of generalised Lie superalgebra that differs from more common Lie superalgebra to generate a (2+1)-dimensional super modified Korteweg-de Vries (mKdV) hierarchy, via a generalised Tu scheme based on the Lax pair method where the Hamiltonian structure derives from a generalised supertrace identity. We also obtain some solutions of the (2+1)-dimensional mKdV equation using the G′/G2 method.

(2+1)-Dimensional mKdV Hierarchy and Chirp Effect of Rossby Solitary Waves

By constructing a kind of generalized Lie algebra, based on generalized Tu scheme, a new (2+1)-dimensional mKdV hierarchy is derived which popularizes the results of (1+1)-dimensional integrable system. Furthermore, the (2+1)-dimensional mKdV equation can be applied to describe the propagation of the Rossby solitary waves in the plane of ocean and atmosphere, which is different from the (1+1)-dimensional mKdV equation. By virtue of Riccati equation, some solutions of (2+1)-dimensional mKdV equation are obtained. With the help of solitary wave solutions, similar to the fiber soliton communication, the chirp effect of Rossby solitary waves is discussed and some conclusions are given.

Tetrahedron Equation and Quantum R Matrices for q-oscillator Representations of $${U_q(A^{(2)}_{2n})}$$ U q ( A 2 n ( 2 ) ) , $${U_q(C^{(1)}_{n})}$$ U q ( C n ( 1 ) ) and $${U_q(D^{(2)}_{n+1})}$$ U q ( D n + 1 ( 2 ) )

The intertwiner of the quantized coordinate ring $A_q(sl_3)$ is known to yield a solution to the tetrahedron equation. By evaluating their $n$-fold composition with special boundary vectors we generate series of solutions to the Yang-Baxter equation. Finding their origin in conventional quantum group theory is a clue to the link between two and three dimensional integrable systems. We identify them with the quantum $R$ matrices associated with the $q$-oscillator representations of $U_q(A^{(2)}_{2n})$, $U_q(C^{(1)}_n)$ and $U_q(D^{(2)}_{n+1})$.

Majorana edge modes in the Kitaev model

We study the Majorana modes, both equilibrium and Floquet, which can appear at the edges of the Kitaev model on the honeycomb lattice. We first present the analytical solutions known for the equilibrium Majorana edge modes for both zigzag and armchair edges of a semi-infinite Kitaev model and chart the parameter regimes of the model in which they appear. We then examine how edge modes can be generated if the Kitaev coupling on the bonds perpendicular to the edge is varied periodically in time as periodic $\delta$-function kicks. We derive a general condition for the appearance and disappearance of the Floquet edge modes as a function of the drive frequency for a generic $d$-dimensional integrable system. We confirm this general condition for the Kitaev model with a finite width by mapping it to a one-dimensional model. Our numerical and analytical study of this problem shows that Floquet Majorana modes can appear on some edges in the kicked system even when the corresponding equilibrium Hamiltonian has no Majorana mode solutions on those edges. We support our analytical studies by numerics for finite sized system which show that periodic kicks can generate modes at the edges and the corners of the lattice.

G/G gauged WZW-matter model, Bethe Ansatz for q-boson model and Commutative Frobenius algebra

We investigate the correspondence between two dimensional topological gauge theories and quantum integrable systems discovered by Moore, Nekrasov, Shatashvili. This correspondence means that the hidden quantum integrable structure exists in the topological gauge theories. We showed the correspondence between the G/G gauged WZW model and the phase model in JHEP 11 (2012) 146 (arXiv:1209.3800). In this paper, we study a one-parameter deformation for this correspondence and show that the G/G gauged WZW model coupled to additional matters corresponds to the q-boson model. Furthermore, we investigate this correspondence from a viewpoint of the commutative Frobenius algebra, the axiom of the two dimensional topological quantum field theory.