A novel approach to integrability of semi-discrete systems
Lead Research Organisation:
University of Kent
Department Name: Sch of Maths Statistics & Actuarial Sci
Abstract
Physical phenomena are often described in terms of nonlinear differential/difference equations.In general it is impossible to obtain exact analytic solutions to most nonlinear systems and the best one can do is to use approximate, asymptotic or numerical methods. Meanwhile, there exists a remarkable class of nonlinear systems called integrable systems. Integrable systems can be analysed meticulously, possess rich hidden algebraic structures and often have geometric realisations. The interest in integrable systems is surging. The range of their applications and unexpected connections with other areas of Mathematics is growing fast. Therefore, the central problems are to determine whether a given equation is integrable, and ultimately to make a complete classification of integrable systems.
In this project we propose to test new methods in the theory of integrable systems inspired by recent developments. We propose to re-formulate the problem of integrability in rigorous terms of difference algebra. Our novel idea is to study non-local extensions of the corresponding difference field in order to achieve a better flexibility in the application of formal pseudo--difference series and aiming to find universal necessary integrability conditions. We are going to develop a symbolic representation of the objects involved to re-cast the problem in terms of symmetric Laurent polynomials (similar ideas proved to be successful in the differential case, but have not been developed in the difference setting). We aim to make a progress in a long standing problem, whose solution would have a lasting impact on the development of Mathematics, Mathematical Physics, Numerical Analysis and far beyond.
We also will attempt to extend these new methods to non-commutative (free associative and quantum) settings to explore uncharted terrain of non-commutative integrable systems, their Poisson structures and quanisations. A recently emerged new approach to quantisation is based on the study of dynamical systems for functions with values in a free associative algebra and certain invariant differential ideals of the algebra. We aim to extend the theory semi-discrete integrable systems to free associative and quantised algebra domains and to link it with non-commutative algebraic geometry, quantum and statistical mechanics.
This small research project is a spring-board or feasibility studies for the future full scale research projects suitable for standard mode applications.
In this project we propose to test new methods in the theory of integrable systems inspired by recent developments. We propose to re-formulate the problem of integrability in rigorous terms of difference algebra. Our novel idea is to study non-local extensions of the corresponding difference field in order to achieve a better flexibility in the application of formal pseudo--difference series and aiming to find universal necessary integrability conditions. We are going to develop a symbolic representation of the objects involved to re-cast the problem in terms of symmetric Laurent polynomials (similar ideas proved to be successful in the differential case, but have not been developed in the difference setting). We aim to make a progress in a long standing problem, whose solution would have a lasting impact on the development of Mathematics, Mathematical Physics, Numerical Analysis and far beyond.
We also will attempt to extend these new methods to non-commutative (free associative and quantum) settings to explore uncharted terrain of non-commutative integrable systems, their Poisson structures and quanisations. A recently emerged new approach to quantisation is based on the study of dynamical systems for functions with values in a free associative algebra and certain invariant differential ideals of the algebra. We aim to extend the theory semi-discrete integrable systems to free associative and quantised algebra domains and to link it with non-commutative algebraic geometry, quantum and statistical mechanics.
This small research project is a spring-board or feasibility studies for the future full scale research projects suitable for standard mode applications.
Publications
Berjawi S
(2021)
Second-Order PDEs in 3D with Einstein-Weyl Conformal Structure
in Annales Henri Poincaré
Buchstaber V
(2021)
KdV hierarchies and quantum Novikov's equations
Buchstaber V
(2023)
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Buchstaber V
(2021)
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Buchstaber V
(2023)
Cyclic Frobenius algebras
in Russian Mathematical Surveys
Buchstaber V
(2021)
Integrable polynomial Hamiltonian systems and symmetric powers of plane algebraic curves
in Russian Mathematical Surveys
Carpentier S
(2022)
Quantisations of the Volterra hierarchy
Carpentier S
(2022)
Quantisations of the Volterra hierarchy
in Letters in Mathematical Physics
Casati M
(2022)
Hamiltonian Structures for Integrable Nonabelian Difference Equations
in Communications in Mathematical Physics
| Description | 1. We explicitly prove that the nonabelian Volterra together with the whole hierarchy of its symmetries admits a deformation quantisation. We show that all odd-degree symmetries of the Volterra hierarchy admit also a non-deformation quantisation. We discuss the quantisation problem for periodic Volterra hierarchy including their quantum Hamiltonians, central elements of the quantised algebras, and demonstrate super-integrability of the quantum systems obtained. We show that the Volterra system with period 3 admits a bi-quantum structure, which can be regarded as a quantum deformation of its classical bi-Hamiltonian structure. 2. We studied a non-commutative version of the N-th Novikov's equation defined on a finitely generated free associative algebra with generators. For N=1,2,3,4, we found two-sided homogeneous ideals (quantisation ideals) which are invariant with respect to the N-th Novikov's equation and such that the quotient algebra has a well defined Poincare-Birkhoff-Witt basis. It enables us to define the quantum N-th Novikov's equation on the quotient algebra . We also showed that the quantum N-th Novikov's equation and its finite hierarchy can be written in the standard Heisenberg form. We introduced the notion of cyclic Frobenius algebras (CF-algebras), which leads to explicit expressions for first integrals of the N-th Novikov hierarchy, suitable for classical, non-Abelian and quantum cases. 3. We proposed a new method to tackle the integrability problem for evolutionary differential-difference equations of arbitrary order. We first defined and developed symbolic representation for the difference polynomial ring, difference operators and formal series. We then introduced a novel quasi-local extension of the difference ring to formulate necessary integrability conditions. We applied our new approach to solve the classification problem of integrable equations for anti-symmetric quasi-linear equations of order (-3, 3) and produced a list of 17 equations satisfying the necessary integrability conditions. For every equation from the list we presented an infinite family of integrable higher order relatives. Some of the equations obtained are new. 4. We extensively studied the notion of Hamiltonian structure for nonabelian differential-difference systems, exploring the link between the different algebraic (in terms of double Poisson algebras and vertex algebras) and geometric (in terms of nonabelian Poisson bivectors) definitions. We introduced multiplicative double Poisson vertex algebras (PVAs) as the suitable noncommutative counterpart to multiplicative PVAs, used to describe Hamiltonian differential-difference equations in the commutative setting, and proved that these algebras are in one-to-one correspondence with the Poisson structures defined by difference operators, providing a sufficient condition for the fulfillment of the Jacobi identity. As an application we obtain some results towards the classification of local scalar Hamiltonian difference structures and construct the Hamiltonian structures for the nonabelian Kaup, Ablowitz-Ladik and Chen-Lee-Liu integrable lattices. 5. Based on the method of hydrodynamic reductions, the integrability conditions are derived (in the form of an involutive PDE system for the Hamiltonian density) for a class of Hamiltonian equations in 2+1 dimensions. We show that the generic integrable density is expressed in terms of the Weierstrass s-function. Dispersionless Lax pairs, commuting flows and dispersive deformations of the resulting equations are also discussed. 6. Three-dimensional Einstein-Weyl structures naturally arise on solutions of second-order dispersionless integrable PDEs in 3D. We demonstrate that, for generic second-order PDEs (for in- stance, for all equations not of Monge-Amp`ere type), not only the conformal structure but also the covector is also expressible in terms of the equation, thus providing an efficient dispersionless integrability test for generic second order PDEs in 3D. They provides a dispersionless Lax pair by an explicit formula which is apparently new. Some partial classification results of PDEs with Einstein-Weyl characteristic conformal structure are obtained. A rigidity conjecture is proposed according to which for any generic second-order PDE with Einstein-Weyl property, all dependence on the 1-jet variables can be eliminated via a suitable contact transformation. |
| Exploitation Route | Part of the findings had been included in MAGIC course "Integrable Systems'' in 2022. |
| Sectors | Education |
| URL | http://arxiv.org |
| Description | Part of finding has been delivered in a number of international conferences, and had been included in MAGIC course ''Integrable Systems" 2022 |
| First Year Of Impact | 2022 |
| Sector | Education |
| Description | QSMS |
| Organisation | Seoul National University |
| Country | Korea, Republic of |
| Sector | Academic/University |
| PI Contribution | Alexander Mikhailov and Jing Ping Wang continued their collaboration with Sylvain Carpentier, who is currently a senior researcher at the center for quantum structures in modules and spaces (QSMS). We both have research expertise in non-commutative integrable systems such as Lax representations, Hamiltonian structures and recursion operators. |
| Collaborator Contribution | Sylvain Carpentier has a strong background in non-commutative algebras, which is crucial for our collaboration. |
| Impact | Research output: Carpentier, S., Mikhailov, A.V. & Wang, J.P. Quantisations of the Volterra hierarchy. Lett Math Phys 112, 94 (2022). https://doi.org/10.1007/s11005-022-01588-1 |
| Start Year | 2017 |