Schedule for: 23w5043 - Lagrangian Multiform Theory and Pluri-Lagrangian Systems

A buffet style dinner is served between 6:00pm and 8:30pm in the Yuxianghu Hotel.

Breakfast is served daily between 7 and 9am in the Xianghu Lake National Tourist Resort

A brief introduction to IASM with important logistical information, technology instruction, and opportunity for participants to ask questions.

I will give a brief introduction to the basic ideas and founding concepts of Lagrangian multiform theory, and present some simple examples to illustrate the ideas. I also aim to touch on the pluri-Lagrangian point of view (which is the one of the Berlin group).

A general framework which performs periodic initial value reductions on discrete 2-dimensional multiforms to derive N-component 1-dimensional multiforms. This is proved generally and then shown for three discrete integrable examples: the discrete potential Korteweg de-Vries equation, the discrete multicomponent Boussinesq equation, and the H3 quad equation.

The pluri-Lagrangian approach emphasises the Multi-Time Euler-Lagrange equations, which characterise critical points of the action integral over arbitrary submanifolds. The Lagrangian multiform approach focuses on the exterior derivative of the Lagrangian k-form, which should be zero on solutions. These two approaches are bridged by the fact that the Multi-Time Euler-Lagrange equations can also be obtained by taking variations of the exterior derivative of the Lagrangian k-form. In more pragmatic terms, if the coefficients of the exterior derivative can be written as a product of factors that vanish on a set of equations (or a sum of such products), then this set of equations is sufficient for criticality, i.e. it implies the Multi-Time Euler-Lagrange equations. This property has been central (though sometimes implicit) in the construction of Lagrangian k-forms. I will give an updated presentation, which explicitly makes use of this property, of the construction of some key examples of continuous and semi-discrete Lagrangian 2-forms. In addition, I will highlight some cases where the equivalence between the Multi-Time Euler-Lagrange equations and the factors of the exterior derivative is nontrivial. This showcases the use of Lagrangian k-forms as a tool to establish surprising connections.

Lunch is served daily between 12:00am and 1:30pm at the Academic Island.

The pluri-Lagrangian theory is very satisfying from the aesthtical point of view. But there is a frequently asked question: what is the added value of knowing the pluri-Lagrangian structure for a given system? In this talk, I will give a couple examples of a possible answer.

I will present how integrable lattice models of statistical mechanics are related to Lagrangian multiform theory/pluri-Lagrangian systems for Adler-Bobenko-Suris (ABS) quad equations. Integrable lattice models of statistical mechanics satisfy a special form of the Yang-Baxter equation called star-triangle relation. A well-known example of such a model is the two-dimensional Ising model. One may take a quasi-classical expansion of the star-triangle relation, and in the leading order one obtains a classical relation that is precisely the closure relation for Lagrangian multiforms. There is also an analogue of invariance of the action functional for Lagrangian multiforms, which is the property of Z-invariance, for which the partition function of the lattice model of statistical mechanics is invariant under certain deformations of a lattice. For such models, the quasi-classical expansion of the partition function leads to an action functional for systems of discrete Laplace-type equations that are related to ABS equations. The lattice models of statistical mechanics thus provide a quantization of these discrete systems in the sense of the path-integral formulation.

It has been proved by Ishizaki that the autonomous Schwarzian differential equations, which admit transcendental meromorphic solutions, have six canonical forms. In this talk, we will present all of these transcendental meromorphic solutions. This talk is based on joint works with Liangwen Liao, Jie Zhang and Donghai Zhao.

The multi-time propagator in the continuous case for quadratic Lagrangian 1-forms will be presented. The integrability condition, path-independent feature as a result of the critical condition of the propagator, on the space independent variables is given. The two-time harmonic oscillators shall be given as a toy example.

I will report on recent joint work with Daniel Riccombeni and Frank Nijhoff on the construction of a Lagrange multiform structure for the Darboux-KP system. The Lagrange multiform is derived from a certain matrix-valued Chern-Simons theory in infinite dimensional space, by judiciously choosing the fields that are allowed. We will explain the required elementary Chern-Simons theory used in the construction. References: JFM, Frank W Nijhoff, Daniel Riccombeni, The Darboux-KP system as an integrable Chern-Simons multiform theory in infinite dimensional space, arXiv:2305.03182 [math-ph].

A buffet dinner is served between 6:00pm and 8:30pm in the Yuxianghu Hotel.

Breakfast is served daily between 7 and 9am in the Xianghu Lake National Tourist Resort

Diffeomorphism symmetry plays an important role for general relativity and in particular for quantum gravity. It can be also introduced into other mechanic (then known as reparametrization invariance) or field theoretic systems.  The fate of diffeomophism symmetry under discretizations has led to a considerable amount of debate. I will explain that diffeomorphism symmetry in the discrete is a very powerful symmetry. However discretizing diffeomorphism symmetric systems typically breaks this symmetry. The symmetry can be restored, e.g. via constructing perfect discretizations which often amounts to solving the continuum dynamics of the system.

Four-dimensional Wess-Zumino-Witten (4dWZW) models are analogous to the two dimensional WZW models. Equation of motion of the 4dWZW model is the Yang equation which is equivalent to the anti-self-dual Yang-Mills (ASDYM) equation. It is well known as the Ward conjecture that the ASDYM equations can be reduced to many classical integrable systems, such as the KdV eq. and Toda eq. [Ward, Mason-Woodhouse,...]. On the other hand, 4d Chern-Simons (CS) theory has connections to many quantum integrable systems such as spin chains and principal chiral models [Costello-Yamazaki-Witten, ...]. Furthermore, these two master equations have been derived from a 6dCS theory on a twistor space like a double fibration [Bittleston-Skinner]. This suggests a nontrivial duality between the 4dWZW model and the 4dCS theory. In this talk, I would like to discuss integrability aspects of the 4dWZW model and construct soliton solutions of it by the Darboux technique. We calculate the action density of the solutions and found that the soliton solutions behaves as the KP-type solitons, that is, the one-soliton solution has localized action (energy) density on a 3d hyperplane in 4-dimensions (soliton wall) and the N-soliton solution describes N intersecting soliton walls with phase shifts. We note that the Ward conjecture holds mostly in the split signature (+,+,-,-) and then the 4dWZW model describes the open N=2 string theory in the four-dimensional space-time. Hence a unified theory of integrable systems can be proposed in this context with the split signature from Lagrangian viewpoints. This talk is partially based on the collaboration with Shan-Chi Huang, Hiroaki Kanno (Nagoya) and Claire Gilson, Jon Nimmo (Glasgow): [arXiv:2212.11800, 2106.01353, 2004.09248, 2004.01718]

Tetrahedron and 3D reflection equations are natural generalizations of the Yang-Baxter and reflection equations into three dimensions. We construct a new solution to them associated with the quantum cluster algebra defined on the Fock-Goncharov quivers. The key is to realize a cluster transformation of quantum Y-variables as an adjoint action by using q-Weyl algebras. (Joint work with Rei Inoue and Yuji Terashima.)

In this talk, the notion of boundary conditions for integrable quad-graph systems will first be given. The boundary conditions are characterized by equations on triangles that are the elementary patterns supporting a boundary for a quad-graph system. The integrability criterion is defined by the so-called boundary consistency. Then, I will provide a Lagrangian formalism for integrable boundary conditions. This includes a set of Euler-Lagrange equations and a closure relation associated with the boundary consistency. Explicit examples will be given for equations in the ABS classification.

In this talk, I will first present a Bäcklund transformation (BT) which connects the continuous to discrete Boussinesq system. This BT is obtained by using elliptic function solutions of the continuous Boussinesq equation. We apply this BT to establish the Lax pair and N-times Darboux transformation for the continuous Boussinesq equation. Starting from an elliptic seed solution, the Darboux transformation is used to construct elliptic soliton solutions. The BT can also be used to construct discrete elliptic seed solution for the lattice Boussinesq system. Furthermore, we establish an infinite family of solutions in terms of elliptic functions of the lattice KP system by setting up an elliptic direct linearisation scheme. Through a dimensional reduction of this elliptic direct linearisation scheme, we obtain the elliptic multi-soliton solutions of the lattice Boussinesq system.

Motivated by some observations and the study of well known examples, in this talk I will present some ideas about the Lagrangian formulation of hierarchies of integrable differential-difference equations along with their connection to canonical conservation laws and local symplectic operators. I will introduce the notion of a master Lagrangian which can used to generate the Lagrangian for every member in the hierarchy in the same way a master symmetry can be used to generate the whole hierarchy of differential-difference equations. Several hierarchies will be considered as examples to demonstrate these ideas.

I will review some recent developments on the study of the gauge-theoretic origin of both finite-dimensional integrable models and (1+1)-dimensional integrable field theories, as proposed by Costello, Witten and Yamazaki. In particular, I will explain how the Zakharov-Mikhailov action naturally arises from 4d Chern-Simons theory in the presence of suitable surface defects. Similarly, I will show how a 1-dimensional action describing the Gaudin model realised on a product of coadjoint orbits naturally arises from 3d mixed BF theory in the presence of suitable line defects. The talk will be mainly based on the joint works [2201.07300] with J. Winstone and [2012.04431] with V. Caudrelier and M. Stoppato.

A round table banquet dinner will be served from 18:00 - 20:30 at Yuxianghu Hotel.

In the first part, I start by recalling the two well-known classes of partial differential equations that admit infinite hierarchies of higher order generalized symmetries: 1) linear and linearizable systems that admit a nontrivial point symmetry group; 2) integrable nonlinear equations such as Korteweg--de Vries, nonlinear Schrödinger, and Burgers'.  I will then introduce a new general class: 3) underdetermined systems of partial differential equations that admit an infinite dimensional symmetry algebra depending on one or more arbitrary functions of the independent variables.  An important subclass of the latter are the underdetermined Euler--Lagrange equations arising from a variational principle that admits an infinite-dimensional variational symmetry algebra depending on one or more arbitrary functions of the independent variables.  According to Noether's Second Theorem, the associated Euler--Lagrange equations satisfy Noether dependencies; examples include general relativity, electromagnetism, and parameter-independent variational principles. $\\$ Noether's First Theorem relates strictly invariant variational problems and conservation laws of their Euler--Lagrange equations.  The Noether correspondence was extended by her student Bessel-Hagen to divergence invariant variational problems. In the second part of this talk, I highlight the role of Lie algebra cohomology in the classification of the latter, and conclude with some provocative remarks on the role of invariant variational problems in fundamental physics.

In this talk, I’ll talk about some recent results in matrix-valued orthogonal polynomials and non-commutative integrable lattices by using the technique of quasi-determinants. Backlund transformation of non-commutative integrable systems will also be discussed in the talk from the perspective of orthogonal polynomials theory.

After introducing the prolongation structure for finite difference equations, we define the difference variational bicomplex and study its exactness. Similar to its differential counterpart, the difference variational bicomplex offers a convenient framework for exploring discrete variational calculus, inverse problems, symmetry analysis, and more. In particular, we will introduce its connection with discrete integrable systems that admit Lagrangian multiforms. This is based on joint works with Peter Hydon (Kent) and Frank Nijhoff (Leeds).

After reviewing the main ideas and ingredients of Lagrangian multiform theory pioneered by Lobb and Nijhoff, I will focus on methods to construct Lagrangian multiforms efficiently for field theories in 1+1 dimensions and for finite-dimensional systems. These methods involve 3 main sources of inspiration: the generating function formalism for hierarchies advocated by Flaschka-Newell-Ratiu and Nijhoff, the insightful construction by Zakharov-Mikhailov of an action for zero-curvature equations of Zakharov-Shabat type and the classical r-matrix/Lie dialgebras theory developed by Semenov-Tian-Shansky. In the case of field theories, the resulting generating Lagrangian multiform contains a huge class of hierarchies and I will show how many old (AKNS, sine-Gordon) and new (trigonometric Zakharov-Mikhailov, coupled hierarchies) examples can be constructed by fixing a small amount of data (marked points on the Riemann sphere, a Lie algebra and an r-matrix). Based on these multiforms, I will also explain how some aspects of Lagrangian multiform theory are related to the classical r-matrix theory familiar in Hamiltonian aspects of integrable systems and to the classical Yang-Baxter equation. In particular, I will show the exact relation between the closure relation and the involutivity of Hamiltonians in the case of finite-dimensional systems and I will comment on the situation in infinite dimensions. This result supports the proposal of using Lagrangian multiforms as a variational criterion for integrability. This is based on joint works with M. Dell'Atti, A. Singh, M. Stoppato and B. Vicedo.

Algebro-geometric solutions of the lattice potential modified Kadomtsev-Petviashvili (lpmKP) equation are constructed. A Darboux transformation of the Kaup-Newell spectral problem is employed to generate a Lax triad for the lpmKP equation, as well as to define commutative integrable symplectic maps which generate discrete flows of eigenfunctions. These maps share the same integrals with the finite-dimensional Hamiltonian system associated to the Kaup-Newell spectral problem. We investigate asymptotic behaviors of the Baker-Akhiezer functions and obtain their expression in terms of Riemann theta function. Finally, algebro-geometric solutions for the lpmKP equation are reconstructed from these Baker-Akhiezer functions.

Gaudin models are a general class of integrable systems associated with quadratic Lie algebras. In this talk, I will describe the construction of the Lagrangian 1-form for the case of the rational Gaudin model, based on a joint work with V. Caudrelier and M. Dell’Atti. We use the theory of Lie dialgebras, due to Semenov-Tian-Shansky, to construct a general Lagrangian 1-form living on a coadjoint orbit. Lie dialgebras are related to Lie bialgebras, but are more flexible in that they incorporate the case of non-skew-symmetric r-matrices. I will illustrate how this construction can be employed in the setting of loop algebras, needed when dealing with Lax matrices with spectral parameters. I will also briefly discuss some natural next steps and some possible connections to other recent works with gauge-theoretic flavours.

In this talk I will give a short review on the integrability of the semi-discrete and discrete Burgers equations, which are featured as integrable equations that are linearisable. The continuous and semi-discrete Burgers hierarchies are related to the mKP system via squared eigenfunction symmetry constraints. The semi-discrete Burgers equation acts as a Bäcklund transformation for the continuous Burgers hierarchy. The fully discrete Burgers equation is a simple 3-point lattice equation that are consistent around the cube. It is also the Bianchi identity of the Bäcklund transformation. The Lagrangian of the discrete Burgers equation is not known.

A buffet dinner is served between 6:00pm and 8:30pm in the Yuxianghu Hotel.

We show that the regular soliton solutions of the KP equation can be explicitly expressed by the Riemann theta functions on singular curves in the sense of Mumford. These solitons can be also obtained by applying vertex operators. In particular, applying the vertex operator to quasi-periodic solutions, we have KP solitons on quasi-periodic background. We also discuss possible models of soliton gases as physical applications of this formulation.

In 1967, Japanese physicist Morikazu Toda proposed an integrable lattice model to describe motions of a chain of particles with exponential interactions between nearest neighbors. Since then, Toda lattice and its generalizations have become the test models for various techniques and philosophies in integrable systems and wide connections are built with many other branches of mathematics. In this talk, I will characterize singular structure of solutions of the so-called full Kostant-Toda (f-KT) lattices de ned on simple Lie algebras in two different ways: through the $\tau$-functions and through the Kowalevski-Painlevé analysis. Fixing the spectral parameters which are invariant under the f-KT flows, we build a one to one correspondence between solutions of the f-KT lattices and points in the corresponding flag varieties. This talk is based on preprint arXiv:2212.03679.

There are quantum integrable systems which ride on the background of classical integrable systems. In this talk I will introduce the notion of such systems, and will give some examples. Also the connection with well known structures will be explained.

The constrained KP hierarchy is introduced by Yi Cheng. In this talk, we extend the matrix-resolvent approach for studying tau-functions for the constrained KP hierarchy and the bigraded Toda hierarchy. In particular, we obtain expressions of n-point functions of both hierarchies. We show that the tau-function of an arbitrary solution to the bigraded Toda hierarchy is a tau-function for the constrained KP hierarchy, which generalizes the Carlet--Dubrovin--Zhang theorem. This is a joint work with Ang Fu and Dafeng Zuo (arXiv:2306.09115).

In this lecture, I will review some typical bosonizations for KP, BKP, CKP and DKP hierarchies. Then give some recent results on the applications of bosonizations in symmetries, Darboux transformations and reductions for KP integrable hierarchy.