Theory Of Integrable Systems Research Articles

Equal-Time and Equal-Space Poisson Brackets of the N-Component Coupled NLS Equation**Supported by National Natural Science Foundation of China under Grant Nos. 11271168 and 11671177, by the Priority Academic Program Development of Jiangsu Higher Education Institutions and by Innovation Project of the Graduate Students in Jiangsu Normal University

Two Poisson brackets for the N-component coupled nonlinear Schrödinger (NLS) equation are derived by using the variantional principle. The first one is called the equal-time Poisson bracket which does not depend on time but only on the space variable. Actually it is just the usual one describing the time evolution of system in the traditional theory of integrable Hamiltonian systems. The second one is equal-space and new. It is shown that the spatial part of Lax pair with respect to the equal-time Poisson bracket and temporal part of Lax pair with respect to the equal-space Poisson bracket share the same r-matrix formulation. These properties are similar to that of the NLS equation.

Some Open Problems in Random Matrix Theory and the Theory of Integrable Systems. II

We describe a list of open problems in random matrix theory and the theory of integrable systems that was presented at the conference Asymptotics in Integrable Systems, Random Matrices and Random Processes and Universality, Centre de Recherches Mathematiques, Montreal, June 7-11, 2015. We also describe progress that has been made on problems in an earlier list presented by the author on the occasion of his 60th birthday in 2005 (see [Deift P., Contemp. Math., Vol. 458, Amer. Math. Soc., Providence, RI, 2008, 419-430, arXiv:0712.0849]).

Frobenius manifolds and Frobenius algebra-valued integrable systems

The notion of integrability will often extend from systems with scalar-valued fields to systems with algebra-valued fields. In such extensions the properties of, and structures on, the algebra play a central role in ensuring integrability is preserved. In this paper, a new theory of Frobenius algebra-valued integrable systems is developed. This is achieved for systems derived from Frobenius manifolds by utilizing the theory of tensor products for such manifolds, as developed by Kaufmann (Int Math Res Not 19:929–952, 1996), Kontsevich and Manin (Inv Math 124: 313–339, 1996). By specializing this construction, using a fixed Frobenius algebra {mathcal {A}}, one can arrive at such a theory. More generally, one can apply the same idea to construct an {mathcal {A}}-valued topological quantum field theory. The Hamiltonian properties of two classes of integrable evolution equations are then studied: dispersionless and dispersive evolution equations. Application of these ideas are discussed, and as an example, an {mathcal {A}}-valued modified Camassa–Holm equation is constructed.

Variational symmetries and pluri-Lagrangian systems in classical mechanics

We analyze the relation of the notion of a pluri-Lagrangian system, which recently emerged in the theory of integrable systems, to the classical notion of variational symmetry, due to E. Noether. We treat classical mechanical systems and show that, for any Lagrangian system with m commuting variational symmetries, one can construct a pluri-Lagrangian 1-form in the (m+1)-dimensional time, whose multi-time Euler-Lagrange equations coincide with the original system supplied with m commuting evolutionary flows corresponding to the variational symmetries. We also give a Hamiltonian counterpart of this construction, leading, for any system of commuting Hamiltonian flows, to a pluri-Lagrangian 1-form with coefficients depending on functions in the phase space.

New periodic wave solutions of (3+1)-dimensional soliton equation

In this paper, associating with the Hirota bilinear form, the three-wave method, which is applied to construct some periodic wave solutions of (3+1)-dimensional soliton equation, is a powerful approach to obtain periodic solutions for many non-linear evolution equations in the integrable systems theory.

XXIV International Conference on Integrable Systems and Quantum symmetries (ISQS-24)

The XXIV International Conference on Integrable Systems and Quantum Symmetries (ISQS-24), organized by the Department of Mathematics, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University Prague and the Bogoliubov Laboratory of Theoretical Physics of the Joint Institute for Nuclear Research, belongs to the successful series of conferences held at the Czech Technical University which began in 1992 and is devoted to problems of mathematical physics related to the theory of integrable systems, quantum groups and quantum symmetries. During the last 5 years, each of the conferences gathered around 110 scientists from all over the world. 43 papers of plenary lectures and contributions presented at ISQS-24 are published in the present issue of Journal of Physics: Conference Series.

Two-layer interfacial flows beyond the Boussinesq approximation: a Hamiltonian approach

The theory of integrable systems of Hamiltonian PDEs and their near-integrable deformations is used to study evolution equations resulting from vertical-averages of the Euler system for two-layer stratified flows in an infinite two-dimensional channel. The Hamiltonian structure of the averaged equations is obtained directly from that of the Euler equations through the process of Hamiltonian reduction. Long-wave asymptotics together with the Boussinesq approximation of neglecting the fluids’ inertia is then applied to reduce the leading order vertically averaged equations to the shallow-water Airy system, albeit in a non-trivial way. The full non-Boussinesq system for the dispersionless limit can then be viewed as a deformation of this well known equation. In a perturbative study of this deformation, a family of approximate constants of the motion are explicitly constructed and used to find local solutions of the evolution equations by means of hodograph-like formulae.

An introduction to associative geometry with applications to integrable systems

The aim of these notes is to provide a reasonably short and "hands-on" introduction to the differential calculus on associative algebras over a field of characteristic zero. Following a suggestion of Ginzburg's we call the resulting theory associative geometry. We argue that this formalism sheds a new light on some classic solution methods in the theory of finite-dimensional integrable dynamical systems.

A Short Note on a Differential-Difference Gauge Transformation and a New Spectral Problem**Supported by the National Natural Science Foundation of China under Grant No 11371241.

We show that a class of spectral problems are related to the spectral problem of the Volterra lattice through a gauge transformation. The transformation is given. We hope that our discussion can draw attention to the study of gauge transformation theory of differential-difference integrable systems.

Commutator representations and roots of pseudo differential operators

Based on the fundamental commutator representation proposed by Cao [4] we established two explicit expressions for roots of a third order differential operator. By using those expressions we succeeded in clarifying the relationship between two major approaches in theory of integrable systems: the zero curvature and the Lax representations for the KdV and the Boussinesq hierarchies. The proposed procedure could be extended to the general case of higher order of differential operators that leads to the Gel’fand-Dickey hierarchy.

On the Instanton R-matrix

A torus action on a symplectic variety allows one to construct solutions to the quantum Yang-Baxter equations (R-matrices). For a torus action on cotangent bundles over flag varieties the resulting R-matrices are the standard rational solutions of the Yang-Baxter equation, well known in the theory of quantum integrable systems. The torus action on the instanton moduli space leads to more complicated R-matrices, depending additionally on two equivariant parameters t1 and t2. In this paper we derive an explicit expression for the R-matrix associated with the instanton moduli space. We study its matrix elements and its Taylor expansion in the powers of the spectral parameter. Certain matrix elements of this R-matrix give a generating function for the characteristic classes of tautological bundles over the Hilbert schemes in terms of the bosonic cut-and-join operators. In particular we rederive from the R-matrix the well known Lehn’s formula for the first Chern class. We explicitly compute the first several coefficients for the power series expansion of the R-matrix in the spectral parameter. These coefficients are represented by simple contour integrals of some symmetrized bosonic fields.

Circle Complexes and the Discrete CKP Equation

In the spirit of Klein's Erlangen Program, we investigate the geometric and algebraic structure of fundamental line complexes and the underlying privileged discrete integrable system for the minors of a matrix which constitute associated Plucker coordinates. Particular emphasis is put on the restriction to Lie circle geometry which is intimately related to the master dCKP equation of discrete integrable systems theory. The geometric interpretation, construction and integrability of fundamental line complexes in Mobius, Laguerre and hyperbolic geometry are discussed in detail. In the process, we encounter various avatars of classical and novel incidence theorems and associated cross- and multi-ratio identities for particular hypercomplex numbers. This leads to a discrete integrable equation which, in the context of Mobius geometry, governs novel doubly hexagonal circle patterns.

Mode 3 knowledge production: systems and systems theory, clusters and networks

With the comprehensive term of “Mode 3,” we want to draw a conceptual link between systems and systems theory and want to demonstrate further how this can be applied to knowledge in the next steps. Systems can be understood as being composed of “elements”, which are tied together by a “self-rationale”. For innovation, often innovation clusters and innovation networks are being regarded as important. By leveraging systems theory for innovation concepts, one can implement references between the elements of a system and clusters (innovation clusters) and the self-rationale of a system and that of networks (innovation networks). One advantage of this approach is that it makes the tools of systems theory effectively available for research on innovation. Based on original research about the European Union, also the concept of a multi-level hierarchy promises conceptual opportunities. Further integrating systems theory, we can speak of multi-level systems of knowledge (following different levels of aggregation) and multi-level systems of innovation (also following different levels of aggregation). The popular and powerful concept of the national innovation system is being chronically challenged by continuous and ongoing processes of supranational and global integration. Conceptually unlocking the national innovation systems in favor of a broader multi-level logic implies furthermore to accept the existence of national innovation systems but, at the same time, also to emphasize their global embeddedness. Our suggested catch-phrase of “Mode 3”, therefore, integrates several considerations that want to relate systems theory, knowledge, and innovation more directly to each other, and this should be understood as a contribution to a dynamically evolving general discourse on the topics of knowledge and innovation.

Unveiling hidden structure of many-body wave functions of integrable systems via sudden-expansion experiments

In the theory of Bethe-ansatz integrable quantum systems, rapidities play an important role as they are used to specify many-body states, apart from phases. The physical interpretation of rapidities going back to Sutherland is that they are the asymptotic momenta after letting a quantum gas expand into a larger volume making it dilute and noninteracting. We exploit this picture to make a direct connection to quantities that are accessible in sudden-expansion experiments with ultracold quantum gases. By a direct comparison of Bethe-ansatz and time-dependent density matrix renormalization group results, we demonstrate that the expansion velocity of a one-dimensional Fermi-Hubbard model can be predicted from knowing the distribution of occupied rapidities defined by the initial state. Curiously, an approximate Bethe-ansatz solution works well also for the Bose-Hubbard model.

Mini-Workshop: Coideal Subalgebras of Quantum Groups

Coideal subalgebras of quantized enveloping algebras appear naturally if one considers quantum group analogs of Lie subalgebras. Examples appear in the theory of quantum integrable systems with boundary and in harmonic analysis on quantum group analogs of Riemannian symmetric spaces. Recently, much progress has been made to develop a deeper representation theoretic understanding of these examples. On the other hand, coideal subalgebras play a fundamental role in the theory of Nichols algebras. The workshop aimed to discuss these theories in view of the recent developments.

On the Relationship between Two Notions of Compatibility for Bi-Hamiltonian Systems

Bi-Hamiltonian structures are of great importance in the theory of integrable Hamiltonian systems. The notion of compatibility of symplectic structures is a key aspect of bi-Hamiltonian systems. Because of this, a few different notions of compatibility have been introduced. In this paper we show that, under some additional assumptions, compatibility in the sense of Magri implies a notion of compatibility due to Fass\`o and Ratiu, that we dub bi-affine compatibility. We present two proofs of this fact. The first one uses the uniqueness of the connection parallelizing all the Hamiltonian vector fields tangent to the leaves of a Lagrangian foliation. The second proof uses Darboux-Nijenhuis coordinates and symplectic connections.

Realizability of singular levels of morse functions as unions of geodesies

The paper presents a list of special graphs of degree 4 with at most 3 vertices (atoms from the theory of integrable Hamiltonian systems) which could be represented by a union of closed geodesies in one of the following surfaces with a metric of constant curvature: sphere, projective plane, torus, and Klein bottle.

Banach Algebras Associated to Lax Pairs

Lax pairs featuring in the theory of integrable systems are known to be constructed from a commutative algebra of formal pseudodifferential operators known as the Burchnall– Chaundy algebra. Such pairs induce the well known KP flows on a restricted infinite-dimensional Grassmannian. The latter can be exhibited as a Banach homogeneous space constructed from a Banach *-algebra. It is shown that this commutative algebra of operators generating Lax pairs can be associated with a commutative C*-subalgebra in the C*-norm completion of the *-algebra. In relationship to the Bose–Fermi correspondence and the theory of vertex operators, this C*-algebra has an association with the CAR algebra of operators as represented on Fermionic Fock space by the Gelfand–Naimark–Segal construction. Instrumental is the Plucker embedding of the restricted Grassmannian into the projective space of the associated Hilbert space. The related Baker and tau-functions provide a connection between these two C*-algebras, following which their respective state spaces and Jordan–Lie–Banach algebras structures can be compared.

Discrete line complexes and integrable evolution of minors

Based on the classical Plücker correspondence, we present algebraic and geometric properties of discrete integrable line complexes in C P 3 . Algebraically, these are encoded in a discrete integrable system that appears in various guises in the theory of continuous and discrete integrable systems. Geometrically, the existence of these integrable line complexes is shown to be guaranteed by Desargues' classical theorem of projective geometry. A remarkable characterization in terms of correlations of C P 3 is also recorded.

Integrable discretization of hodograph-type systems, hyperelliptic integrals and Whitham equations

Based on the well-established theory of discrete conjugate nets in discrete differential geometry, we propose and examine discrete analogues of important objects and notions in the theory of semi-Hamiltonian systems of hydrodynamic type. In particular, we present discrete counterparts of (generalized) hodograph equations, hyperelliptic integrals and associated cycles, characteristic speeds of Whitham-type and (implicitly) the corresponding Whitham equations. By construction, the intimate relationship with integrable system theory is maintained in the discrete setting.