Two-dimensional Integrable Systems Research Articles

Higher gauge theory and integrability

In recent years, significant progress has been made in the study of integrable systems from a gauge theoretic perspective. This development originated with the introduction of four-dimensional Chern-Simons theory with defects, which provided a systematic framework for constructing two-dimensional integrable systems. In this article, we propose a novel approach to studying higher-dimensional integrable models employing techniques from higher category theory. Starting with higher Chern-Simons theory on the 4-manifold R×Y, we complexify and compactify the real line to CP1 and introduce the disorder defect ω=z−1dz. This procedure defines a holomorphic five-dimensional variant of higher Chern-Simons theory, which, when endowed with suitable boundary conditions, allows for the localization to a three-dimensional theory on Y. The equations of motion of the resulting model are equivalent to the flatness of a 2-connection (L,H), that we then use to construct the corresponding higher holonomies. We prove that these are invariants of homotopies relative boundary, which enables the construction of conserved quantities. The latter are labeled by both the categorical characters of a Lie crossed module and the infinite number of homotopy classes of surfaces relative boundary in Y. Moreover, we also demonstrate that the three-dimensional theory has left and right acting symmetries whose current algebra is given by an infinite-dimensional centrally extended affine Lie 2-algebra. Both of these conditions are direct higher homotopy analogs of the properties satisfied by the two-dimensional Wess-Zumino-Witten conformal field theory, which we therefore interpret as facets of integrable structures. Published by the American Physical Society 2024

Two-dimensional integrable systems with position-dependent mass via complex holomorphic functions

We study the relationship between integrable systems with a position-dependent mass (PDM) and complex holomorphic functions and the potential applications of the latter to deduce the former. For a prescribed mass term the associated complex function is derived. The complex function and related plane transformation are used to generate the PDM systems of three integrable Hénon–Heiles systems and a Holt system as well. We also figure out a holomorphic function, which ensures separability of the corresponding PDM systems in the polar-like coordinates. The holomorphic function together with Jacobi method have yielded a variety of generalized separable systems. At last we put forward an example of a family of separable systems to show that not all PDM systems can be deduced through some holomorphic function.

On soliton solutions and soliton interactions of Kulish–Sklyanin and Hirota–Ohta systems

In this paper we consider a simplest two-dimensional reduction of the remarkable three-dimensional Hirota-Ohta system. The Lax pair of the Hirota-Ohta system was extended to a Lax triad by adding extra third linear equation, whose compatibility conditions with the Lax pair of the Hirota-Ohta imply another remarkable systems: the Kulish-Sklyanin system (KSS) together with its first higher commuting flow, which we can call as vector complex MKdV. This means that any common particular solution of these both two-dimensional integrable systems yields a corresponding particular solution of the three-dimensional Hirota-Ohta system. Using the dressing Zakharov-Shabat method we derive the $N$-soliton solutions of these systems and analyze their interactions, i.e. derive explicitly the shifts of the relative center-of-mass coordinates and the phases as functions of the discrete eigenvalues of the Lax operator. Next we relate to these NLEE system of Hirota--Ohta type and obtain its $N$-soliton solutions

Graded discrete integrable systems and Darboux transformations

We present the Darboux transformations for a novel class of two-dimensional discrete integrable systems named as graded discrete integrable systems, which were firstly proposed by Fordy and Xenitidis within the framework of graded discrete Lax pairs very recently. In this paper, the graded discrete equations in the coprime case and their corresponding Lax pairs are derived from the discrete Gel’fand–Dikii hierarchy by applying a transformation of the independent variables. The construction of the Darboux tranformations is realised by considering the associated linear problems in the bilinear formalism for the graded lattice equations. We show that all these graded equations share a unified solution structure in our scheme.

Boundary conditions for General Relativity in three-dimensional spacetimes, integrable systems and the KdV/mKdV hierarchies

We present a new set of boundary conditions for General Relativity on AdS3, where the dynamics of the boundary degrees of freedom are described by two independent left and right members of the Gardner hierarchy of integrable equations, also known as the “mixed KdV-mKdV” hierarchy. This integrable system has the very special property that simultaneously combines both, the Korteweg-de Vries (KdV) and modified Korteweg-de Vries (mKdV) hierarchies in a single integrable structure. This relationship between gravitation in three-dimensional spacetimes and two-dimensional integrable systems is based on an extension of the recently introduced “soft hairy boundary conditions” on AdS3, where the chemical potentials are now allowed to depend locally on the dynamical fields and their spatial derivatives. The complete integrable structure of the Gardner system, i.e., the phase space, the Poisson brackets and the infinite number of commuting conserved charges, are directly obtained from the asymptotic analysis and the conserved surface integrals in the gravitational theory. These boundary conditions have the particular property that they can also be interpreted as being defined in the near horizon region of spacetimes with event horizons. Black hole solutions are then naturally accommodated within our boundary conditions, and are described by static configurations associated to the corresponding member of the Gardner hierarchy. The thermodynamic properties of the black holes in the ensembles defined by our boundary conditions are also discussed. Finally, we show that our results can be naturally extended to the case of a vanishing cosmological constant, and the integrable system turns out to be precisely the same as in the case of AdS3.

Integrable and Superintegrable Systems with Higher Order Integrals of Motion: Master Function Formalism

We construct two-dimensional integrable and superintegrable systems in terms of the master function formalism and relate them to Mielnik’s and Marquette’s construction in supersymmetric quantum mechanics. For two different cases of the master functions, we obtain two different two-dimensional superintegrable systems with higher order integrals of motion.

On reductions of the discrete Kadomtsev–Petviashvili-type equations

The reduction by restricting the spectral parameters k and on a generic algebraic curve of degree is performed for the discrete AKP, BKP and CKP equations, respectively. A variety of two-dimensional discrete integrable systems possessing a more general solution structure arise from the reduction, and in each case a unified formula for the generic positive integer is given to express the corresponding reduced integrable lattice equations. The obtained extended two-dimensional lattice models give rise to many important integrable partial difference equations as special degenerations. Some new integrable lattice models such as the discrete Sawada–Kotera, Kaup–Kupershmidt and Hirota–Satsuma equations in extended form are given as examples within the framework.

Characterizing classical periodic orbits from quantum Green's functions in two-dimensional integrable systems: Harmonic oscillators and quantum billiards.

A general method is developed to characterize the family of classical periodic orbits from the quantum Green's function for the two-dimensional (2D) integrable systems. A decomposing formula related to the beta function is derived to link the quantum Green's function with the individual classical periodic orbits. The practicality of the developed formula is demonstrated by numerically analyzing the 2D commensurate harmonic oscillators and integrable quantum billiards. Numerical analyses reveal that the emergence of the classical features in quantum Green's functions principally comes from the superposition of the degenerate states for 2D harmonic oscillators. On the other hand, the damping factor in quantum Green's functions plays a critical role to display the classical features in mesoscopic regime for integrable quantum billiards, where the physical function of the damping factor is to lead to the coherent superposition of the nearly degenerate eigenstates.

New integrable problems in the dynamics of particle and rigid body

In the present article, a new two-dimensional integrable system containing 17 free parameters is introduced. For giving certain values for these parameters, new integrable problems can be constructed, which generalize some known previous problems, and in some cases, we can restore some previous integrable problems. Two new integrable problems are announced, describing the motion in an Euclidean plane and on a pseudo-sphere. In the irreversible case, a new integrable problem in rigid body dynamics, which generalizes Goriachev–Chaplygin’s case (Varshav Univ Izvest 3:1–13, 1916), Yehia’s case (Mech Res Commun 23:423–427, 1996) and Elmandouh’s case (Acta Mech 226:2461–2472, 2015), is announced.

Ricci-flat spacetimes admitting higher rank Killing tensors

Ricci-flat spacetimes of signature (2,q) with q=2,3,4 are constructed which admit irreducible Killing tensors of rank-3 or rank-4. The construction relies upon the Eisenhart lift applied to Drach's two-dimensional integrable systems which is followed by the oxidation with respect to free parameters. In four dimensions, some of our solutions are anti-self-dual.

New approach to symmetries and singularities in integrable Hamiltonian systems

This article describes new results obtained in the theory of symmetries and singularities of integrable Hamiltonian systems, developed in recent years by the Fomenko school in Moscow State University. The Sadetovʼs proof of Mischenko–Fomenko conjecture, the correlation between the discriminant of the spectral curve and the bifurcation diagram and the theory of atoms for the two-dimensional integrable systems are discussed.

Gauge-invariant description of several (2+1)-dimensional integrable nonlinear evolution equations

We obtain new gauge-invariant forms of two-dimensional integrable systems of nonlinear equations: the Sawada-Kotera and Kaup-Kuperschmidt system, the generalized system of dispersive long waves, and the Nizhnik-Veselov-Novikov system. We show how these forms imply both new and well-known twodimensional integrable nonlinear equations: the Sawada-Kotera equation, Kaup-Kuperschmidt equation, dispersive long-wave system, Nizhnik-Veselov-Novikov equation, and modified Nizhnik-Veselov-Novikov equation. We consider Miura-type transformations between nonlinear equations in different gauges.

ASPECTS OF INTEGRABILITY IN $\mathcal{N} = 4$ SYM

Various recently developed connections between supersymmetric Yang–Mills theories in four dimensions and two-dimensional integrable systems serve as crucial ingredients in improving our understanding of the AdS/CFT correspondence. In this review, we highlight some connections between superconformal four-dimensional Yang–Mills theory and various integrable systems. In particular, we focus on the role of Yangian symmetries in studying the gauge theory dual of closed string excitations. We also briefly review how the gauge theory connects to Calogero models and open quantum spin chains through the study of the gauge theory duals of D3 branes and open strings ending on them. This invited review is based on a seminar given at the Institute of Advanced Study, Princeton.

Quasiseparation of variables in the Schrödinger equation with a magnetic field

We consider a two-dimensional integrable Hamiltonian system with a vector and scalar potential in quantum mechanics. Contrary to the case of a pure scalar potential, the existence of a second order integral of motion does not guarantee the separation of variables in the Schrödinger equation. We introduce the concept of “quasiseparation of variables” and show that in many cases it allows us to reduce the calculation of the energy spectrum and wave functions to linear algebra.

Jacobian variety and integrable system — after Mumford, Beauville and Vanhaecke

Beauville [A. Beauville, Jacobiennes des courbes spectrales et systèmes hamiltoniens complètement intégrables, Acta. Math. 164 (1990) 211–235] introduced an integrable Hamiltonian system whose general level set is isomorphic to the complement of the theta divisor in the Jacobian of the spectral curve. This can be regarded as a generalization of the Mumford system [D. Mumford, Tata Lectures on Theta II, Birkhäuser, 1984]. In this article, we construct a variant of Beauville’s system whose general level set is isomorphic to the complement of the intersection of the translations of the theta divisor in the Jacobian. A suitable subsystem of our system can be regarded as a generalization of the even Mumford system introduced by Vanhaecke [P. Vanhaecke, Linearising two-dimensional integrable systems and the construction of action-angle variables, Math. Z. 211 (1992) 265–313; P. Vanhaecke, Integrable systems in the realm of algebraic geometry, in: Lecture Notes in Mathematics, vol. 1638, 2001].

Q4: integrable master equation related to an elliptic curve

One of the most fascinating and technically demanding parts of the theory of two-dimensional integrable systems constitutes the models with the spectral parameter on an elliptic curve, including Landau-Lifshitz and Krichever-Novikov equations as well as elliptic Toda and Ruijsenaars-Toda lattices, elliptic Volterra lattice and Shabat-Yamilov lattice. We explain how all these models can be unified on the basis of a single equation Q4 on a quad-graph. This discrete model plays the role of the “master equation” in the (sl2 part of) 2D integrability: most of other models in this area can be obtained from this one by certain limiting procedures. More precisely, discrete versions of all of the above-mentioned models appear from Q4 imposed on a multidimensional cubic lattice. Zero-curvature representations for all models appear as a byproduct of the general construction.

Two-dimensional Krall-Sheffer polynomials and integrable systems

Two-dimensional Krall-Sheffer polynomials are analogues of the classical orthogonal polynomial. They are eigenfunctions of second-order linear partial differential operators and moreover satisfy orthogonality conditions. We show that all Krall-Sheffer polynomials are connected with two-dimensional superintegrable systems on spaces with constant curvature.

SKLYANIN BRACKET AND DEFORMATION OF THE CALOGERO–MOSER SYSTEM

A two-dimensional integrable system being a deformation of the rational Calogero–Moser system is constructed via the symplectic reduction, performed with respect to the Sklyanin algebra action. We explicitly resolve the respective classical equations of motion via the projection method and quantize the system.

New Integrable Two-Dimensional Systems Generated by Deformed Susy Algebras

New solutions of intertwining relations for two-dimensional scalar quantum Hamiltonians by second-order supercharges with Lorentz and degenerate metrics are obtained. The symmetry operators for components of superhamiltonian that lead to integrability of corresponding systems are found. Expressions for the Hamiltonians and the symmetry operators in the classical limit are constructed. A new class of integrable two-dimensional classical systems with integrals of motion of fourth order in momenta is obtained. Bibliography: 23 titles.

Some remarks on complex Hamiltonian systems

The analyticity property of the one-dimensional complex Hamiltonian system H( x, p)= H 1( x 1, x 2, p 1, p 2)+ i H 2( x 1, x 2, p 1, p 2) with p= p 1+i x 2, x= x 1+i p 2 is exploited to obtain a new class of the corresponding two-dimensional integrable Hamiltonian systems where H 1 acts as a new Hamiltonian and H 2 is a second integral of motion. Also a possible connection between H 1 and H 2 is sought in terms of an auto-Bäcklund transformation.