On elliptic Lax pairs and isomonodromic deformation systems for elliptic lattice equations
Frank Nijhoff, Neslihan Delice
备注：
38 pages, 2 figures, submitted to the Proceedings of the workshop on Representation Theory, Special Functions and Painleve Equations, RIMS, Kyoto, March 3-6, 2015
In a previous article [N. Delice, F.W. Nijhoff and S. Yoo-Kong, J. Phys. A: Math. Theor. 48(3) (2015), 035206] a novel class of elliptic Lax pairs for integrable lattice equations was introduced. The present article proposes a de-autonomisation of those Lax pairs leading to a class of elliptic discrete isomonodromic deformation problems. We analyse the systems of compatibility conditions using some (possibly novel) higher order elliptic identities.
For all non-equivalent matrix systems of Painlevé-4 type found by authors in arXiv:2107.11680, isomonodromic Lax pairs are presented. Limiting transitions from these systems to matrix Painlevé-2 equations are found.
We present a Lax pair for the sixth Painlevé equation arising as a continuous isomonodromic deformation of a system of linear difference equations with an additional symmetry structure. We call this a symmetric difference-differential Lax pair. We show how the discrete isomonodromic deformations of the associated linear problem gives us a discrete version of the fifth Painlevé equation. By considering degenerations we obtain symmetric difference-differential Lax pairs for the fifth Painlevé equation and the various degenerate versions of the third Painlevé equation.
This is the second article in a suite of articles investigating relations between Stäckel-type systems and Painlevé-type systems. In this article we construct isomonodromic Lax representations for Painlevé-type systems found in the previous paper by Frobenius integrable deformations of Stäckel-type systems. We first construct isomonodromic Lax representations for Painlevé-type systems in the so called magnetic representation and then, using a multitime-dependent canonical transformation, we also construct isomonodromic Lax representations for Painlevé-type systems in the non-magnetic represent…
It is well known that isomonodromic deformations admit a Hamiltonian description. These Hamiltonians appear as coefficients of the characteristic equations of their Lax matrices, which define spectral curves for linear systems of differential and difference systems. The characteristic equations in the case of the associated linear problems for various discrete Painlevé equations is biquadratic in the Painlevé variables. We show that the discrete isomonodromic deformations that define the discrete Painlevé equations may be succinctly described in terms of the characteristic equation of their La…
We show that various models of the elliptic Calogero-Moser systems are accompanied with an isomonodromic system on a torus. The isomonodromic partner is a non-autonomous Hamiltonian system defined by the same Hamiltonian. The role of the time variable is played by the modulus of the base torus. A suitably chosen Lax pair (with an elliptic spectral parameter) of the elliptic Calogero-Moser system turns out to give a Lax representation of the non-autonomous system as well. This Lax representation ensures that the non-autonomous system describes isomonodromic deformations of a linear ordinary dif…
Schlesinger transformations are discrete monodromy preserving symmetry transformations of the classical Schlesinger system. Generalizing well-known results from the Riemann sphere we construct these transformations for isomonodromic deformations on genus one Riemann surfaces. Their action on the system's tau-function is computed and we obtain an explicit expression for the ratio of the old and the transformed tau-function.
Hamiltonian representation of isomonodromic deformations of twisted rational connections: The Painlevé $1$ hierarchy
Université de Lyon（里昂大学）；Université Jean Monnet Saint-Étienne（圣艾蒂安让·莫内大学）；CNRS UMR 5208, Institut Camille Jordan（法国国家科学研究中心 5208 联合实验室，卡米尔·若当研究所）；Institut Universitaire de France（法兰西学院）
This paper presents a new approach to the Hamiltonian structure of isomonodromic deformations of a matrix system of ODE's on a torus. An isomonodromic analogue of the $\rmSU(2)$ Calogero-Gaudin system is used for a case study of this approach. A clue of this approach is a mapping to a finite number of points on the spectral curve of the isomonodromic Lax equation. The coordinates of these moving points give a new set of Darboux coordinates called the spectral Darboux coordinates. The system of isomonodromic deformations is thereby converted to a non-autonomous Hamiltonian system in the spectra…
The Lax pseudo-differential operator plays a key role in studying the general set of KP equations, although it is normally treated in a formal way, without worrying about a complete characterization of its mathematical properties. The aim of the present paper is therefore to investigate the ellipticity condition. For this purpose, after a careful evaluation of the kernel with the associated symbol, the majorization ensuring ellipticity is studied in detail. This leads to non-trivial restrictions on the admissible set of potentials in the Lax operator. When their time evolution is also consider…
A general elliptic $N\times N$ matrix Lax scheme is presented, leading to two classes of elliptic lattice systems, one which we interpret as the higher-rank analogue of the Landau-Lifschitz equations, while the other class we characterize as the higher-rank analogue of the lattice Krichever-Novikov equation (or Adler's lattice). We present the general scheme, but focus mainly of the latter type of models. In the case $N=2$ we obtain a novel Lax representation of Adler's elliptic lattice equation in its so-called 3-leg form. The case of rank $N=3$ is analysed using Cayley's hyperdeterminant of …
We present four classes of nonlinear systems which may be considered discrete analogues of the Garnier system. These systems arise as discrete isomonodromic deformations of systems of linear difference equations in which the associated Lax matrices are presented in a factored form. A system of discrete isomonodromic deformations is completely determined by commutation relations between the factors. We also reparameterize these systems in terms of the image and kernel vectors at singular points to obtain a separate birational form. A distinguishing feature of this study is the presence of a sym…
We build analytic surfaces in $\mathbb{R}cubec$ represented by the most general sixth Painlevé equation $P_{VI}$ in two steps. Firstly, the moving frame of the surfaces built by Bonnet in 1867 is extrapolated to a new, second order, isomonodromic matrix Lax pair of $P_{VI}$, whose elements depend rationally on the dependent variable and quadratically on the monodromy exponents $θ_j$. Secondly, by converting back this Lax pair to a moving frame, we obtain an extrapolation of Bonnet surfaces to surfaces with two more degrees of freedom. Finally, we give a rigorous derivation of the quantum corre…
Hexagonal circle patterns are introduced, and a subclass thereof is studied in detail. It is characterized by the following property: For every circle the multi-ratio of its six intersection points with neighboring circles is equal to -1. The relation of such patterns with an integrable system on the regular triangular lattice is established. A kind of a B"acklund transformation for circle patterns is studied. Further, a class of isomonodromic solutions of the aforementioned integrable system is introduced, including circle patterns analogons to the analytic functions $z^α$ and $\log z$.
In the paper [V. Adler, IMRN {\bf 1} (1998) 1--4] a lattice version of the Krichever-Novikov equation was constructed. We present in this note its Lax pair and discuss its elliptic form.
We investigate the classical limit of the Knizhnik-Zamolodchikov-Bernard equations, considered as a system of non-stationar Schröodinger equations on singular curves, where times are the moduli of curves. It has a form of reduced non-autonomous hamiltonian systems which include as particular examples the Schlesinger equations, Painlevé VI equation and their generalizations. In general case, they are defined as hierarchies of isomonodromic deformations (HID) with respect to changing the moduli of underling curves. HID are accompanying with the Whitham hierarchies. The phase space of HID is the …
QRT maps are translations on smooth biquadratic curves, also known as elliptic curves. Special cases of QRT maps are known to arise as compatibility conditions for an associated system of linear equations, known as a Lax pair. Here, we provide $2\times 2$ Lax pairs explicitly for the whole 18-parameter family of QRT maps.
Hamiltonian representation of isomonodromic deformations of general rational connections on $\mathfrak{gl}_2(\mathbb{C})$
Université Jean Monnet Saint-Étienne, CNRS, Institut Camille Jordan UMR 5208（圣艾蒂安让·莫内大学，法国国家科学研究中心，卡米尔·乔丹研究所）；Institut Universitaire de France（法国高等研究院）；Université de Genève, Section de mathématiques（日内瓦大学数学系）
Nous montrons que les équations du repère mobile des surfaces de Bonnet conduisent à une paire de Lax matricielle isomonodromique d'ordre deux pour la sixième équation de Painlevé. We show that the moving frame equations of Bonnet surfaces can be extrapolated to a second order, isomonodromic matrix Lax pair of the sixth Painlevé equation.
All Painlevé equations can be written as a time-dependent Hamiltonian system, and as such they admit a natural generalization to the case of several particles with an interaction of Calogero type (rational, trigonometric or elliptic). Recently, these systems of interacting particles have been proved to be relevant in the study of $β$-models. An almost two decade old open question by Takasaki asks whether these multi-particle systems can be understood as isomonodromic equations, thus extending the Painlevé correspondence. In this paper we answer in the affirmative by displaying explicitly suita…
The 8-parameter elliptic Sakai difference Painlevé equation admits a Lax formulation. We show that a suitable specialization of the Lax equation gives rise to the time-independent Schrödinger equation for the $BC_1$ 8-parameter 'relativistic' Calogero-Moser Hamiltonian due to van Diejen. This amounts to a generalization of previous results concerning the Painlevé-Calogero correspondence to the highest level in the two hierarchies.
The Darboux transformations for the two dimensional elliptic affine Toda equations corresponding to all seven infinite series of affine Kac-Moody algebras, including $A_l^{(1)}$, $A_{2l}^{(2)}$, $A_{2l-1}^{(2)}$, $B_l^{(1)}$, $C_l^{(1)}$, $D_l^{(1)}$ and $D_{l+1}^{(2)}$, are presented. The Darboux transformation is constructed uniformly for the latter six series of equations with suitable choice of spectral parameters and the solutions of the Lax pairs so that all the reality symmetry, cyclic symmetry and complex orthogonal symmetry of the corresponding Lax pairs are kept invariant. The exact …
In this article we prove that Lax pairs associated with $\hbar$-dependent six Painlevé equations satisfy the topological type property proposed by Bergère, Borot and Eynard for any generic choice of the monodromy parameters. Consequently we show that one can reconstruct the formal $\hbar$-expansion of the isomonodromic $τ$-function and of the determinantal formulas by applying the so-called topological recursion to the spectral curve attached to the Lax pair in all six Painlevé cases. Finally we illustrate the former results with the explicit computations of the first orders of the six $τ$-fun…
In this paper, we construct a new Lax operator for the elliptic Calogero-Moser model with N=2. The nondynamical r-matrix structure of this Lax operator is also studied . The relation between our Lax operator and the Lax operator given by Krichever is also obtained.
This paper presents a construction of isospectral problems on the torus. The construction starts from an SU(n) version of the XYZ Gaudin model recently studied by Kuroki and Takebe in the context of a twisted WZW model. In the classical limit, the quantum Hamiltonians of the generalized Gaudin model turn into classical Hamiltonians with a natural $r$-matrix structure. These Hamiltonians are used to build a non-autonomous multi-time Hamiltonian system, which is eventually shown to be an isomonodromic problem on the torus. This isomonodromic problem can also be reproduced from an elliptic analog…
A generalisation of the Lattice Potential Kadomtsev-Petviashvili (LPKP) equation is presented, using the method of Direct Linearisation based on an elliptic Cauchy kernel. This yields a (3+1)-dimensional lattice system with one of the lattice shifts singled out. The integrability of the lattice system is considered, presenting a Lax representation and soliton solutions. An associated continuous system is also derived, yielding a (3+1)-dimensional generalisation of the potential KP equation associated with an elliptic curve.
New quasilocal recursion and Hamiltonian operators for the Krichever-Novikov and the Landau-Lifshitz equations are found. It is shown that the associative algebra of quasilocal recursion operators for these models is generated by a couple of operators related by an elliptic curve equation. A theoretical explanation of this fact for the Landau-Lifshitz equation is given in terms of multiplicators of the corresponding Lax structure.
We demonstrate that in a certain gauge the elliptic Ruijsenaars-Shneider model with N=2 admits a nondynamical r-matrix structure and the corresponding classical r-matrix is the same as that of its non-relativistic counterpart (Calogero-Moser model) in the same gauge.The relation between our (classical)Lax operator and the Lax operator given by Ruijsenaars is also obtained.
We develop a self-consistent approach to study the spectral properties of a class of quantum mechanical operators by using the knowledge about monodromies of $2\times 2$ linear systems (Riemann-Hilbert correspondence). Our technique applies to a variety of problems, though in this paper we only analyse in detail two examples. First we review the case of the (modified) Mathieu operator, which corresponds to a certain linear system on the sphere and makes contact with the Painlevé $\mathrm{III}_3$ equation. Then we extend the analysis to the 2-particle elliptic Calogero-Moser operator, which cor…
We summarize recent results on the construction of Lax pairs with spectral parameter for the twisted and untwisted elliptic Calogero-Moser systems associated with arbitrary simple Lie algebras, their scaling limits to Toda systems, and their role in Seiberg-Witten theory. We extend part of this work by presenting a new parametrization for the spectral curves for elliptic spin Calogero-Moser systems associated with SL(N).
We present a linear system of difference equations whose entries are expressed in terms of theta functions. This linear system is singular at $4m+12$ points for $m \geq 1$, which appear in pairs due to a symmetry condition. We parameterize this linear system in terms a set of kernels at the singular points. We regard the system of discrete isomonodromic deformations as an elliptic analogue of the Garnier system. We identify the special case in which $m=1$ with the elliptic Painlevé equation, hence, this work provides an explicit form and Lax pair for the elliptic Painlevé equation.
Universal Lax pairs of the root type with spectral parameter and independent coupling constants for twisted non-simply laced Calogero-Moser models are constructed. Together with the Lax pairs for the simply laced models and untwisted non-simply laced models presented in two previous papers, this completes the derivation of universal Lax pairs for all of the Calogero-Moser models based on root systems. As for the twisted models based on B_n, C_n and BC_nroot systems, a new type of potential term with independent coupling constants can be added without destroying integrability. They are called e…
Elliptic N-soliton-type solutions, i.e. solutions emerging from the application of N consecutive Bäcklund transformations to an elliptic seed solution, are constructed for all equations in the ABS list of quadrilateral lattice equations, except for the case of the Q4 equation which is treated elsewhere. The main construction, which is based on an elliptic Cauchy matrix, is performed for the equation Q3, and by coalescence on certain auxiliary parameters, the corresponding solutions of the remaining equations in the list are obtained. Furthermore, the underlying linear structure of the equation…
We construct a Lax pair with spectral parameter for the elliptic Calogero-Moser Hamiltonian systems associated with each of the finite dimensional Lie algebras, of the classical and of the exceptional type. When the spectral parameter equals one of the three half periods of the elliptic curve, our result for the classical Lie algebras reduces to one of the Lax pairs without spectral parameter that were known previously. These Calogero-Moser systems are invariant under the Weyl group of the associated untwisted affine Lie algebra. For non-simply laced Lie algebras, we introduce new integrable s…
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 hie…
A symmetry reduction of the Painlevé IV hierarchy to the Flaschka-Newell Painlevé II hierarchy
Université de Genève（日内瓦大学）；Université Jean Monnet Saint-Étienne, CNRS, Institut Camille Jordan UMR 5208, Institut Universitaire de France（圣艾蒂安让·穆兰大学、法国国家科学研究中心、卡米尔·乔丹研究所UMR 5208、法国高等研究院）
研究偶数阶 Painlevé IV 层级对应的等单值变形问题，通过对称性约化得到 Flaschka-Newell Painlevé II 层级，并显式匹配 Lax 矩阵、Darboux 坐标和哈密顿结构。
We consider a family of classical elliptic integrable systems including (relativistic) tops and their matrix extensions of different types. These models can be obtained from the "off-shell" Lax pairs, which do not satisfy the Lax equations in general case but become true Lax pairs under various conditions (reductions). At the level of the off-shell Lax matrix there is a natural symmetry between the spectral parameter $z$ and relativistic parameter $η$. It is generated by the finite Fourier transformation, which we describe in detail. The symmetry allows to consider $z$ and $η$ on an equal foot…
The main aim of this paper is to discuss recent results on the adaptation of the Fokas-Gel'fand procedure for constructing soliton surfaces in Lie algebras, which was originally derived for PDEs [Grundland, Post 2011], to the case of integrable ODEs admitting Lax representations. We give explicit forms of the $\g$-valued immersion functions based on conformal symmetries involving the spectral parameter, a gauge transformation of the wave function and generalized symmetries of the linear spectral problem. The procedure is applied to a symmetry reduction of the static $ϕ^4$-field equations leadi…
The article is devoted to the study of $R$-matrix-valued Lax pairs for $N$-body (elliptic) Calogero-Moser models. Their matrix elements are given by quantum ${\rm GL}_{\tilde N}$ $R$-matrices of Baxter-Belavin type. For $\tilde N=1$ the widely known Krichever's Lax pair with spectral parameter is reproduced. First, we construct the $R$-matrix-valued Lax pairs for Calogero-Moser models associated with classical root systems. For this purpose we study generalizations of the D'Hoker-Phong Lax pairs. It appeared that in the $R$-matrix-valued case the Lax pairs exist in special cases only. The numb…
We demonstrate that in a certain gauge the elliptic Ruijsenaars--Schneider models admit Lax representation governed by the same dynamical $r$--matrix as their non--relativistic counterparts (Calogero--Moser models). This phenomenon was previously observed for the rational and hyperbolic models.
We derive the Lax connection of the free fermion model on a lattice starting from the fermionic formulation of Bazhanov-Stroganov's three-parameter elliptic parametrization for the R-operator. It results in the Yang-Baxter and decorated Yang-Baxter equations of difference type in one of the spectral parameters, which is the most suitable form to obtain any relativistic model of free fermions in the continuous limit.
Calogero-Moser models and Toda models are well-known integrable multi-particle dynamical systems based on root systems associated with Lie algebras. The relation between these two types of integrable models is investigated at the levels of the Hamiltonians and the Lax pairs. The Lax pairs of Calogero-Moser models are specified by the representations of the reflection groups, which are not the same as those of the corresponding Lie algebras. The latter specify the Lax pairs of Toda models. The Hamiltonians of the elliptic Calogero-Moser models tend to those of Toda models as one of the periods …
The elliptic Korteweg-de Vries (KdV) system is a multi-component generalization of the lattice potential KdV equation, whose soliton solutions are associated with an elliptic Cauchy kernel (i.e., a Cauchy kernel on the torus). In this paper we generalize the class of solutions by using a Sylvester type matrix equation and rederiving the system from the associated Cauchy matrix. Our starting point is the Sylvester equation in the form of $~\boldsymbol{k} \boldsymbol{M}+ \boldsymbol{M} \boldsymbol{k} = \boldsymbol{r} {\boldsymbol{c}}^{T}-g\boldsymbol{K}^{-1} \boldsymbol{r} {\boldsymbol{c}}^{T} \…
We present the Lax pair for the field elliptic Calogero-Moser system and establish a connection between this system and the Kadomtsev-Petviashvili equation. Namely, we consider elliptic families of solutions of the KP equation, such that their poles satisfy a constraint of being balanced. We show that the dynamics of these poles is described by a reduction of the field elliptic CM system. We construct a wide class of solutions to the field elliptic CM system by showing that any N-fold branched cover of an elliptic curve gives rise to an elliptic family of solutions of the KP equation with bala…
We give a new construction of noncommutative surfaces via elliptic difference operators, attaching a 1-parameter noncommutative deformation to any projective rational surface with smooth anticanonical curve. The construction agrees with one implicit in work of Van den Bergh (iterated blowups of noncommutative Hirzebruch surfaces), but the construction enables one to prove a number of new facts about these surfaces. We show that they are noncommutative smooth proper surfaces in the sense of Chan and Nyman, with projective Quot schemes, that moduli spaces of simple sheaves are Poisson and that m…
For the Kowalevski gyrostat change of variables similar to that of the Kowalevski top is done. We establish one to one correspondence between the Kowalevski gyrostat and the Clebsch system and demonstrate that Kowalevski variables for the gyrostat practically coincide with elliptic coordinates on sphere for the Clebsch case. Equivalence of considered integrable systems allows to construct two Lax matrices for the gyrostat using known rational and elliptic Lax matrices for the Clebsch model. Associated with these matrices solutions of the Clebsch system and, therefore, of the Kowalevski gyrosta…
We give the Lax representations for for the elliptic, hyperbolic and homogeneous second order Monge-Ampere equations. The connection between these equations and the equations of hydrodynamical type give us a scalar dispersionless Lax representation. A matrix dispersive Lax representation follows from the correspondence between sigma models, a two parameter equation for minimal surfaces and Monge-Ampere equations. Local as well nonlocal conserved densities are obtained.
Starting from the complete integrable lattice super-KdV equation, two super-mappings are obtained by performing a travelling-wave reduction. The first one is linear and the second is a four dimensional super-QRT mapping containing both Grassmann commuting and anti-commuting dependent variables. Adapting the classical "staircase" method to the Lax super-matrices of the lattice super-KdV equation, we compute the Lax super-matrices of the mapping and the two invariants; the first one is a pure nilpotent commuting quantity and the second one is given by an elliptic curve containing nilpotent commu…
In this paper we suggest generalizations of elliptic integrable tops to matrix-valued variables. Our consideration is based on $R$-matrix description which provides Lax pairs in terms of quantum and classical $R$-matrices. First, we prove that for relativistic (and non-relativistic) tops such Lax pairs with spectral parameter follow from the associative Yang-Baxter equation and its degenerations. Then we proceed to matrix extensions of the models and find out that some additional constraints are required for their construction. We describe a matrix version of ${\mathbb Z}_2$ reduced elliptic t…
In this article, we derive rogue wave (RW) solutions of a fifth-order nonlinear Schrödinger equation over a double-periodic wave background. Choosing the elliptic functions (combinations of $cn$, $dn$ and $sn$) as seed solutions in the first iteration of Darboux transformation and utilizing the nonlinearization of Lax pair procedure, we create the double-periodic wave background for the fifth-order nonlinear Schrödinger equation. By introducing the second linearly independent solution, we generate the RW solutions on the created background for three different eigenvalues. We demonstrate the di…
From the dynamical twisting of the classical r-matrix, we obtain a new Lax operator for the elliptic Ruijsenaars-Schneider model (cf. Ruijsenaars'). The corresponding r-matrix is shown to be the classical $Z_n$-symmetric elliptic r-matrix, which is the same as that obtained in the study of the nonrelativistic version---the $A_{n-1}$ Calogero-Moser model.
We review elliptic solutions to integrable nonlinear partial differential and difference equations (KP, mKP, BKP, Toda) and derive equations of motion for poles of the solutions. The pole dynamics is given by an integrable many-body system (Calogero-Moser, Ruijsenaars-Schneider). The basic tool is the auxiliary linear problems for the wave function which yield equations of motion together with their Lax representation. We also discuss integrals of motion and properties of the spectral curves.
We propose a construction of 1+1 integrable Heisenberg-Landau-Lifshitz type equations in the ${\rm gl}_N$ case. The dynamical variables are matrix elements of $N\times N$ matrix $S$ with the property $S^2={\rm const}\cdot S$. The Lax pair with spectral parameter is constructed by means of a quantum $R$-matrix satisfying the associative Yang-Baxter equation. Equations of motion for ${\rm gl}_N$ Landau-Lifshitz model are derived from the Zakharov-Shabat equations. The model is simplified when ${\rm rank}(S)=1$. In this case the Hamiltonian description is suggested. The described family of models…
