Exact solutions for discrete and continuous nonlinear systems
Lead Research Organisation:
University of Leeds
Department Name: Applied Mathematics
Abstract
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People |
ORCID iD |
| A Mikhailov (Principal Investigator) |
Publications
Adamopoulou P
(2020)
Drinfel'd-Sokolov construction and exact solutions of vector modified KdV hierarchy
in Nuclear Physics B
Adamopoulou P
(2019)
Drinfel'd-Sokolov construction and exact solutions of vector modified KdV hierarchy
Buchstaber V
(2018)
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Buchstaber V
(2018)
Polynomial Hamiltonian integrable systems on symmetric powers of plane curves
in Russian Mathematical Surveys
Buchstaber V
(2017)
Infinite-dimensional Lie algebras determined by the space of symmetric squares of hyperelliptic curves
in Functional Analysis and Its Applications
Bury R
(2021)
Automorphic Lie algebras and corresponding integrable systems
in Differential Geometry and its Applications
Bury R
(2017)
Wave fronts and cascades of soliton interactions in the periodic two dimensional Volterra system
in Physica D: Nonlinear Phenomena
Bychkov B
(2019)
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Bychkov B
(2019)
Polynomial graph invariants and linear hierarchies
in Russian Mathematical Surveys
| Description | 1, For two dimensional Volterra system we derived explicit formulas for soliton solutions of arbitrary rank: • A new class of exact solutions corresponding to wave fronts is presented. • A full classification of rank 1 solutions is given. • Soliton solutions similar to breathers resemble soliton webs in the KP theory. • The full classification is associated with the Schubert decomposition of the Grassmannians. 2. We construct Lie algebras of vector fields on universal bundles of symmetric squares of hyperelliptic curves. As an application, we obtain integrable polynomial dynamical systems. Furthermore, we have constructed a new a new class of integrable systems on symmetric powers of general plane algebraic curves. A generalisation of the theory of Abelian functions is required for integration of these systems. 3. We have improved the result of Chmutov, Kazarian, and Lando who proved that the generating series for polynomial graph invariants are formal t -functions of the Kadomtsev-Petviashvili integrable hierarchy. We have shown that these generating series together with the t -function are solutions of the linear hierarchy 4. A new conservative finite element discretisations for the vectorial modified KdV equation and a consistent Galerkin scheme have been designed and applied to the problems with periodic boundary conditions. 5. A hierarchy of a multi-component generalisation of modified KdV equation and exact solutions to its associated members have been found. 6. We have formulated the concept of a graded isomorphism and classified SL_2(C) - based automorphic Lie algebras corresponding to all finite reduction groups. We have shown that hierarchies of integrable systems, their Lax representations and master symmetries can be naturally formulated in terms of automorphic Lie algebras. 7. We have developed a rigorous theory of pseudo difference (rational) recursion and Hamiltonian operators. A new concept of pre-Hamiltonian operators enables us to give a constructive definition of Hamiltonian operators in the pseudo-difference case. 8. A new approach to quantisation of dynamical and differential difference systems have been formulated. It starts from systems on free associative algebras and its two-sided invariant differential ideals. |
| Exploitation Route | The outcome was used for the development of the MAster/PhD graduate cursed given within the MAGIC consortium. |
| Sectors | Education |
| Description | The findings have been used for the development of the MAGIC graduate course "Integrable Systems" . |
| First Year Of Impact | 2018 |
| Sector | Education |
| Description | Differential Algebra and Related Topics IX, LMS Scheme 1 |
| Amount | £6,000 (GBP) |
| Funding ID | 11721 |
| Organisation | London Mathematical Society |
| Sector | Academic/University |
| Country | United Kingdom |
| Start | 07/2018 |
| End | 08/2018 |
| Description | Formal series and integrable hierarchies, Research in pairs, Scheme 4. |
| Amount | £1,000 (GBP) |
| Funding ID | 41824 |
| Organisation | London Mathematical Society |
| Sector | Academic/University |
| Country | United Kingdom |
| Start | 04/2019 |
| End | 05/2019 |
| Description | Poisson Structures and Noncommutative Integrability |
| Amount | £4,500 (GBP) |
| Funding ID | Ref 11944 |
| Organisation | London Mathematical Society |
| Sector | Academic/University |
| Country | United Kingdom |
| Start | 04/2020 |
| End | 07/2022 |
| Description | The research team Jing Ping Wang and Alexander Mikhailov started the collaboration with Dr Sylvain Carpentier in July 2017. |
| Organisation | Columbia University |
| Country | United States |
| Sector | Academic/University |
| PI Contribution | We provide our knowledge in Integrable Systems, in particular, their Lax representations, recursion operators and Hamiltonian structures. |
| Collaborator Contribution | Dr Sylvain Carpentier brought his expertise in noncommutative algebras to the project. |
| Impact | We have published three high-quality research papers: 1. PreHamiltonian and Hamiltonian operators for differential-difference equations. Nonlinearity 33 (3), 915, 2020. 2. Rational recursion operators for integrable differential-difference equations. Communications in Mathematical Physics 370 (3), 807-851, 2019 3. Quantisations of the Volterra hierarchy Letters in Mathematical Physics 112 (5), 94 |
| Start Year | 2017 |