Lax Operator Research Articles

A modified melting crystal model and the Ablowitz–Ladik hierarchy

This paper addresses the issue of integrable structure in a modified melting crystal model of topological string theory on the resolved conifold. The partition function can be expressed as the vacuum expectation value of an operator on the Fock space of 2D complex free fermion fields. The quantum torus algebra of fermion bilinears behind this expression is shown to have an extended set of ‘shift symmetries’. They are used to prove that the partition function (deformed by external potentials) is essentially a tau function of the 2D Toda hierarchy. This special solution of the 2D Toda hierarchy can also be characterized by a factorization problem of matrices. The associated Lax operators turn out to be quotients of first-order difference operators. This implies that the solution of the 2D Toda hierarchy in question is actually a solution of the Ablowitz–Ladik (equivalently, the relativistic Toda) hierarchy. As a byproduct, the shift symmetries are shown to be related to matrix-valued quantum dilogarithmic functions.

BKP 与CKP 可积系列的递归算子

In this paper, under the constraints of the BKP(CKP) hierarchy, a crucial observation is that the odd dynamical variable $u_{2k+1}$ can be explicitly expressed by the even dynamical variable $u_{2k}$ in the Lax operator $L$ through a new operator $B$. Using operator $B$, the essential differences between the BKP hierarchy and the CKP hierarchy are given by the flow equations and the recursion operators under the $(2n+1)$-reduction. The formal formulas of the recursion operators for the BKP and CKP hierarchy under $(2n+1)$-reduction are given. To illustrate this method, the two recursion operators are constructed explicitly for the 3-reduction of the BKP and CKP hierarchies. The $t_7$ flows of $u_2$ are generated from $t_1$ flows by the above recursion operators, which are consistent with the corresponding flows generated by the flow equations under 3-reduction.

Algebraic curves for factorized string solutions

We show how to construct an algebraic curve for factorized string solution in the context of the AdS/CFT correspondence. We define factorized solutions to be solutions where the flat-connection becomes independent of one of the worldsheet variables by a similarity transformation with a matrix $S$ satisfying $S^{-1}d S=const$. Using the factorization property we construct a well defined Lax operator and an associated algebraic curve. The construction procedure is local and does not require the introduction of a monodromy matrix. The procedure can be applied for string solutions with any boundary conditions. We study the properties of the curve and give several examples for the application of the procedure.

Integrable models for Bose–Einstein condensates in a double-well potential formulated from Holstein–Primakoff transformations

We present three families of integrable models for Bose–Einstein condensates in a double-well potential which are based on the combination, two by two, of three algebraic realizations of the Lax operator: the bosonic realization and the su(2) and su(1, 1) Holstein–Primakoff realizations. The families of Hamiltonians describe Josephson tunneling between two condensates in a double-well potential with boson distribution dependence in the tunneling term. Two families describe models with interacting bosons and the other does not. The integrability is shown by using the quantum inverse scattering method and algebraic Bethe ansatz. The energy eigenvalues and the Bethe ansatz equations are obtained for each model proposed. In the end, we use the energy gap as a testing function to compare the proposed integrable models with the well-known integrable two-site Bose–Hubbard model to show the effect of the boson distribution in the tunneling.

THE GAUGE TRANSFORMATION OF THE CONSTRAINED SEMI-DISCRETE KP HIERARCHY

In this paper, the gauge transformation of the constrained semi-discrete KP(cdKP) hierarchy is constructed explicitly by the suitable choice of the generating functions. Under the m-step successive gauge transformation Tm, we give the transformed (adjoint) eigenfunctions and the τ-function of the transformed Lax operator of the cdKP hierarchy.

Algebraic curves for long folded and circular winding strings in AdS 5 × S 5

For the homogeneous configuration given by the long string limit of the folded string with a spin in AdS3 and a spin and a winding number in S1, we solve the auxiliary linear problem in the finite-gap method and construct the Lax operator to obtain an algebraic curve. We show that the long spiky string in AdS_3 has the same algebraic curve as the null cusp Wilson loop. The algebraic curve for the circular winding string in AdS3xS1 is evaluated. The Virasoro constraint is discussed to characterize the branch points of each curve.

Algebro-geometric Solutions for the Degasperis--Procesi Hierarchy

Though completely integrable Camassa-Holm (CH) equation and Degasperis-Procesi (DP) equation are cast in the same peakon family, they possess the second- and third-order Lax operators, respectively. From the viewpoint of algebro-geometrical study, this difference lies in hyper-elliptic and non-hyper-elliptic curves. The non-hyper-elliptic curves lead to great difficulty in the construction of algebro-geometric solutions of the DP equation. In this paper, we derive the DP hierarchy with the help of Lenard recursion operators. Based on the characteristic polynomial of a Lax matrix for the DP hierarchy, we introduce a third order algebraic curve $\mathcal{K}_{r-2}$ with genus $r-2$, from which the associated Baker-Akhiezer functions, meromorphic function and Dubrovin-type equations are established. Furthermore, the theory of algebraic curve is applied to derive explicit representations of the theta function for the Baker-Akhiezer functions and the meromorphic function. In particular, the algebro-geometric solutions are obtained for all equations in the whole DP hierarchy.

Weak KAM Solutions of a Discrete-Time Hamilton-Jacobi Equation in a Minimax Framework

The purpose of this paper is to study the existence of solutions of a Hamilton-Jacobi equation in a minimax discrete-time case and to show different characterizations for a real number called the critical value, which plays a central role in this work. We study the behavior of solutions of this problem using tools of game theory to obtain a “fixed point” of the Lax operator associated, considering some facts of weak KAM theory to interpret these solutions as discrete viscosity solutions. These solutions represent the optimal payoff of a zero-sum game of two players, with increasingly long time payoffs. The developed techniques allow us to study the behavior of an infinite time game without using discount factors or average actions.

A class of complex solutions to the finite Toda lattice

In this paper, a class of complex-valued solutions to the finite Toda lattice is constructed by using the inverse spectral method. The corresponding Lax operator is a finite complex Jacobi matrix. As the initial values there are taken such complex numbers that the corresponding Jacobi matrix has a simple spectrum. Some examples are given.

Characteristic Classes and Hitchin Systems. General Construction

We consider topologically non-trivial Higgs G-bundles over Riemann surfaces Σg with marked points and the corresponding Hitchin systems. We show that if G is not simply-connected, then there exists a finite number of different sectors of the Higgs bundles endowed with the Hitchin Hamiltonians. They correspond to different characteristic classes of the underlying bundles defined as elements of \({H^{2}(\Sigma_g, \mathcal{Z}(G))}\) , (\({\mathcal{Z}(G)}\) is a center of G). We define the conformal version CG of G - an analog of GL(N) for SL(N), and relate the characteristic classes with degrees of CG-bundles. We describe explicitly bundles in the genus one (g = 1) case. If Σ1 has one marked point and the bundles are trivial then the Hitchin systems coincide with Calogero-Moser (CM) systems. For the nontrivial bundles we call the corresponding systems the modified Calogero-Moser (MCM) systems. Their phase space has the same dimension as the phase space of the CM systems with spin variables, but less number of particles and greater number of spin variables. Starting with these bundles we construct Lax operators, quadratic Hamiltonians, and define the phase spaces and the Poisson structure using dynamical r-matrices. The latter are completion of the classification list of Etingof-Varchenko corresponding to the trivial bundles. To describe the systems we use a special basis in the Lie algebras that generalizes the basis of ’t Hooft matrices for sl(N). We find that the MCM systems contain the standard CM subsystems related to some (unbroken) subalgebras. The configuration space of the CM particles is the moduli space of the stable holomorphic bundles with non-trivial characteristic classes.

Two-dimensional Toda field equations related to the exceptional algebra $\mathfrak{g}_2 $ : Spectral properties of the lax operators

We analyze spectral properties of the Lax operator corresponding to the two-dimensional Toda field equations related to the algebra \(\mathfrak{g}_2 \). We construct two minimal sets of scattering data \(\mathcal{T}_s \), s = 1, 2, understanding the map between the potential and each of the sets \(\mathcal{T}_s \) as a generalized Fourier transformation. We construct explicit recursion operators with special factorization properties.

Dimer models and integrable systems

We explore various aspects of the correspondence between dimer models and integrable systems recently introduced by Goncharov and Kenyon. Dimer models give rise to relativistic integrable systems that match those arising from 5d N=1 gauge theories studied by Nekrasov. We apply the correspondence to dimer models associated to the Y^{p,0} geometries, showing that they give rise to the relativistic generalization of the periodic Toda chain originally studied by Ruijsenaars. The correspondence reduces the calculation of all conserved charges to a straightforward combinatorial problem of enumerating non-intersecting paths in the dimer model. We show how the usual periodic Toda chain emerges in the non-relativistic limit and how the Lax operator corresponds to the Kasteleyn matrix of the dimer model. We discuss how the dimer models for general Y^{p,q} manifolds give rise to other relativistic integrable systems, generalizing the periodic Toda chain and construct the integrable systems for general Y^{p,p} explicitly. The impurities introduced in the construction of Y^{p,q} quivers are identified with impurities in twisted sl(2) XXZ spin chains. Finally we discuss how the physical concept of higgsing a dimer model provides an efficient method for producing new integrable systems starting from known ones. We illustrate this idea by constructing the integrable systems for higgsings of Y^{4,0}.

Calogero–Moser systems for simple Lie groups and characteristic classes of bundles

This paper is a continuation of our paper Levin et al. [1]. We consider Modified Calogero–Moser (CM) systems corresponding to the Higgs bundles with an arbitrary characteristic class over elliptic curves. These systems are generalization of the classical Calogero–Moser systems with spin related to simple Lie groups and contain CM subsystems related to some (unbroken) subalgebras. For all algebras we construct a special basis, corresponding to non-trivial characteristic classes, the explicit forms of Lax operators and quadratic Hamiltonians. As by product, we describe the moduli space of stable holomorphic bundles over elliptic curves with arbitrary characteristic classes.

Integrability of supergravity black holes and new tensor classifiers of regular and nilpotent orbits

In this paper we apply in a systematic way a previously developed integration algorithm of the relevant Lax equation to the construction of spherical symmetric, asymptotically flat black hole solutions of $ \mathcal{N} = 2 $ supergravities with symmetric Special Geometry. Our main goal is the classification of these black-holes according to the H⋆-orbits in which the space of possible Lax operators decomposes. By H⋆ one denotes the isotropy group of the coset UD=3/H⋆ which appears in the time-like dimensional reduction of supergravity from D = 4 to D = 3 dimensions. The main result of our investigation is the construction of three universal tensors, extracted from quadratic and quartic powers of the Lax operator, that are capable of classifying both regular and nilpotent H⋆-orbits of Lax operators. Our tensor based classification is compared, in the case of the simple one-field model S 3, to the algebraic classification of nilpotent orbits and it is shown to provide a simple discriminating method. In particular we present a detailed analysis of the S 3 model, constructing explicitly its solutions and discussing the Liouville integrability of the corresponding dynamical system. By means of the Kostant-representation of a generic Lie algebra element, we were able to develop an algorithm which produces the necessary number of hamiltonians in involution required by Liouville integrability of generic orbits. The degenerate orbits correspond to extremal black-holes and are nilpotent. We present an in depth discussion of their identification and of the construction of the corresponding supergravity solutions. We dwell on the relation between H⋆ orbits and critical points of the geodesic potential showing that there is correspondence yet not one-to-one.

A non-standard Lax formulation of the Harry Dym hierarchy and its supersymmetric extension

For the Harry Dym hierarchy, a non-standard Lax formulation is deduced from that of the Korteweg–de Vries (KdV) equation through a reciprocal transformation. By supersymmetrizing this Lax operator, a new N = 2 supersymmetric extension of the Harry Dym hierarchy is constructed, and is further shown to be linked to one of the N = 2 supersymmetric KdV equations through the superconformal transformation. The bosonic limit of this new N = 2 supersymmetric Harry Dym equation is related to a coupled system of KdV–MKdV equations.

An $${\hbar}$$ -expansion of the Toda hierarchy

A construction of general solutions of the \hbar-dependent Toda hierarchy is presented. The construction is based on a Riemann-Hilbert problem for the pairs (L,M) and (\bar L,\bar M) of Lax and Orlov-Schulman operators. This Riemann-Hilbert problem is translated to the language of the dressing operators W and \bar W. The dressing operators are set in an exponential form as W = e^{X/\hbar} and \bar W = e^{\phi/\hbar}e^{\bar X/\hbar}, and the auxiliary operators X,\bar X and the function \phi are assumed to have \hbar-expansions X = X_0 + \hbar X_1 + ..., \bar X = \bar X_0 + \hbar\bar X_1 + ... and \phi = \phi_0 + \hbar\phi_1 + .... The coefficients of these expansions turn out to satisfy a set of recursion relations. X,\bar X and \phi are recursively determined by these relations. Moreover, the associated wave functions are shown to have the WKB form \Psi = e^{S/\hbar} and \bar\Psi = e^{\bar S/\hbar}, which leads to an \hbar-expansion of the logarithm of the tau function.

Elliptic three-boson system, “two-level three-mode” JCD-type models and non-skew-symmetric classical r-matrices

Using the technique of the classical r-matrices and quantum Lax operators we construct the most general form of quantum integrable multi-boson and spin-multi-boson models associated with linear Lax algebras and sl ( 2 ) ⊗ sl ( 2 ) -valued classical non-dynamical r-matrices with spectral parameters. We consider example of non-skew-symmetric elliptic r-matrix and explicitly obtain one-, two- and three-boson integrable models and the corresponding one-, two- and three-mode two-level Jaynes–Cummings–Dicke-type models. We show that integrable “elliptic” two-level one-mode Jaynes–Cummings–Dicke Hamiltonian is hermitian and contains both rotating and counter-rotating terms.

String equations of the q-KP hierarchy

Based on the Lax operator L and Orlov-Shulman’s M operator, the string equations of the q-KP hierarchy are established from the special additional symmetry flows, and the negative Virasoro constraint generators {L −n , n ≥ 1} of the 2-reduced q-KP hierarchy are also obtained.

General integrable two-level, one-mode Jaynes–Cummings–Dicke models and classicalr-matrices with spectral parameters

Using the technique of classicalr-matrices and quantum Lax operators we construct the most general form of the quantum integrable‘two-level, one-mode’ spin-boson Jaynes–Cummings–Dicke-type model. We consider in detailtwo subclasses of this model and the corresponding examples associated with concrete classicalr-matrices with spectral parameters. We show that the so-called ‘diagonal’ in the root basisr-matrices produces integrable Jaynes–Cummings–Dicke-type models withthe ‘rotating wave approximation’, while all other types of classicalr-matrices produce integrable JCD models without the rotating wave approximation. Weconstruct an explicit example of an integrable Hermitian Jaynes–Cummings–Dicke-typeHamiltonian containing both ‘rotating’ and ‘counter-rotating’ terms. In the case of the diagonalr-matrices we find the spectrum of the corresponding quantum Hamiltonians using thealgebraic Bethe ansatz.

Differential–difference equations associated with the fractional Lax operators

We study integrable hierarchies associated with spectral problems of the form Pψ = λQψ, where P and Q are difference operators. The corresponding nonlinear differential–difference equations can be viewed as inhomogeneous generalizations of the Bogoyavlensky-type lattices. While the latter turn into the Korteweg–de Vries equation under the continuous limit, the lattices under consideration provide discrete analogs of the Sawada–Kotera and Kaup–Kupershmidt equations. The r-matrix formulation and several of the simplest explicit solutions are presented.