Mini-symposium
Nonlinear Dynamics and Integrability
in the frame of the International Conference “Mathematics Days in Sofia – 2026”
The proposed mini-symposium aims to bring together experts in nonlinear dynamics and nonlinear modeling working on problems related to integrability, soliton theory and propagation of nonlinear waves.
The equations of interest are nonlinear PDEs. These equations are rich of underlying algebraic and geometric structures which necessitates a variety of mathematical methods to be used in their analysis. These methods include spectral theory of operators, functional analysis, geometry, inverse scattering of discrete and continuous integrable systems, symmetry groups and theory of Lie groups and algebras.
The focus of the presented results will be on particular solution techniques as well as qualitative results and further developments of the mathematical methods. An important aim of this mini symposium is also to establish and emphasize the connection of the nonlinear PDE to modeling of various physical phenomena in areas such as classical and quantum mechanics, fluid mechanics, nonlinear optics and geophysics.
Organizing Committee
Georgi Boyadjiev, Institute of Mathematics and Informatics, Bulgaria
Georgi Grahovski, University of Essex, UK
Rossen Ivanov, Technological University Dublin, Ireland
Participants
Aleksander Stefanov, Institute of Mathematics and Informatics, Bulgaria
Alexander Mikhailov, University of Leeds, UK
Andrew Hone, University of Kent, UK
Georgi Boyadjiev, Institute of Mathematics and Informatics, Bulgaria
Georgi Grahovski, University of Essex, UK
Georgios Papamikos, University of Essex, UK
Jing Ping Wang, Ningbo University, China
Lubomir Markov, Barry University, USA
Nikola Stoilov, Universite de Bourgogné, France
- Pavlos Kassotakis, University of Patras, Greece
- Rossen Ivanov, Technological University Dublin, Ireland
Stoyan Mishev, New Bulgarian University, Bulgaria
Tihomir Valchev, Institute of Mathematics and Informatics, Bulgaria
- Vesselin Vatchev, University of Texas at Rio Grande Valley, USA
Vladimir Gerdjikov, Institute of Mathematics and Informatics, Bulgaria
Program and Abstracts
We review some of the common methods of constructing and solving higher dimensional integrable systems. Starting from Zakharov and Manakov’s non-local
Riemann-Hilbert problem formulation, we restrict our attention to models admitting a zero curvature representation with explicit dependence on the spectral parameter. This form is suitable for the use of reductions, which leads to new integrable models.
Known integrable systems with noncommutative dependent variables are typically formulated over free associative algebras, quantum algebras, or Grassmann algebras. For differential–difference integrable equations, we identify a new class of noncommutative algebras compatible with the dynamics, which may be viewed as lying between free and quantum algebras.
In this talk, I discuss reductions of the nonabelian Volterra hierarchy to these new algebras, the properties of the resulting systems, and connections with quantum systems. This represents a new direction in the theory of integrable systems, with many open questions yet to be answered. The approach extends to a broad class of integrable systems, including the Toda lattice, the Ablowitz–Ladik system, and many others.
Joint work with S. Carpentier and J. P. Wang.
We find an exact expression for the unique positive solution of a discrete Painlevé equation that arises in the context of static membranes, and as an example of a quantum minimal surface considered by Arnlind, Hoppe and Kontsevich. It transpires that this is generated by a special combination of Backlund transformations for Painlevé V, admitting a sequence of classical solutions which we are able to express explicitly using Wronskians of modified Bessel functions. Extensions to other quantum curves and higher order discrete Painlevé equations will briefly be mentioned, as well as the connection with very recent results by Felder and Hoppe which relate these solutions to orthogonal polynomials.
This is joint work with Peter Clarkson, Anton Dzhamay and Ben Mitchell.
We generalise a family of quadrirational parametric Yang–Baxter (YB) maps with Lax matrices by introducing additional essential parameters. These maps preserve a prescribed Poisson structure which originates from the Sklyanin bracket. We investigate various low-dimensional reductions of this family, as well as degenerate limits with respect to the parameters that were introduced. As a result, we derive several birational YB maps, and we discuss some of their integrability properties. This work is part of a more general classification of YB maps admitting a strong Lax matrix with a linear dependence on the spectral parameter.
The classification of integrable equations is one of the central problems in the theory of integrable systems. Among the various methods developed to address this problem, the symmetry approach has proven to be particularly effective and powerful.
In this talk, we will discuss recent advances in the symmetry approach to classification of two-component integrable evolutionary systems. These developments have been achieved primarily through the symbolic representation of the ring of differential polynomials, which enables the application of tools from algebraic geometry and number theory and allows one to derive necessary and sufficient integrability conditions.
We will then explain how these recently established integrability conditions can be used to classify two-component homogeneous integrable evolutionary systems. In particular, we will present classification results for integrable systems of orders 3 and 5.
Finally, we will illustrate the algebraic properties of the resulting systems by providing either their Lax representations or transformations that relate them to previously known integrable systems.
This is joint work with A. V. Mikhailov and V. Novikov.
We provide necessary and sufficient conditions for maps that satisfy associative-like conditions on families of n-ary magmas to be pentagon maps. We obtain parametric-pentagon maps and we propose a procedure that generates families of multicomponent pentagon and entwining pentagon maps from a given pentagon map.
We analyze the effect of imposing constant boundary conditions (CBC) on integrable nonlinear evolution equations (NLEE). We consider two types of NLEE:
a) N-wave type equations related to the algebra sp(4)
and
b) multicomponent NLS (MNLS) equations.
They correspond to two basic gradings of sp(4):
a) to the C.I symmetric space
and
b) to the homogeneous space related to sp(4).
Our conclusion is that the CBC may not be compatible with the initial grading of the algebra.
This is a joint work with Prof. Vladimir Gerdjikov.
We consider multicomponent nonlinear equations in two independent variables that admit a zero curvature representation. Their Lax pairs are related to simple complex Lie algebras and are subject to some constraints. Simple examples that belong to this class of equations include the well-known Heisenberg ferromagnet equation for an arbitrary simple Lie algebra and the Golubchik-Sokolov equation.
Using Lagrange’s method, we introduce a stream function and derive a second-order approximation of the system. For the resulting equations in a traveling-wave setting, we develop an algorithm to construct approximate solutions on the entire domain. We establish convergence and stability results that allow approximation to arbitrary precision. Finally, we present simple trigonometric approximate solutions, including a traveling wave that resembles a hexagonal pattern on the circle.
The Event is Supported by:
