Mini-symposium

Integrability and Mathematical Physics

#### in the frame of the International Conference “Mathematics Days in Sofia”

**T**

**he proposed mini-symposium**aims to bring together experts in Mathematical physics working on problems, related to integrability /exact solvability of nonlinear PDEs. The equations of interest are rich of underlying algebraic and geometric structures which predetermines the variety of mathematical methods used in the area. 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 on 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 PDE models under consideration to real life applications in fluid mechanics, nonlinear optics and geophysics.

#### Organizing Committee

George Papamikos (University of Essex, UK)

Georgi Boyadjiev (Institute of Mathematics and Informatics, Bulgaria)

Georgi Grahovski (University of Essex, UK)

Rossen Ivanov (Technological University Dublin, Ireland)

### Participants

**Alexander Stefanov**, Sofia University “St. Kliment Ohridski”, Bulgaria**George Papamikos**, University of Essex, UK**Georgi Boyadjiev**, Institute of Mathematics and Informatics, Bulgaria**Georgi Grahovski**, University of Essex, UK**Michail D. Todorov**, Technical University of Sofia, Bulgaria**Nikola Stoilov**, University of Burgundy, France

**Panagiota Adamopoulou**, Heriot Watt University, UK**Pavlos Kassotakis**, University of Warsaw, Poland**Rossen Ivanov**, Technological University Dublin, Ireland**Sotiris Konstantinou-Rizos**, Yaroslavl University, Russia**Stoyan Mishev**, New Bulgarian University, Bulgaria**Tihomir Valchev**, Institute of Mathematics and Informatics, Bulgaria**Vladimir Gerdjikov**, Institute of Mathematics and Informatics, Bulgaria

### Program and Abstracts

We consider the extended bivariate Pólya-Aeppli process (BPAP) and the bivariate risk model with BPAP counting process. Our interest is in the properties of the ruin and non-ruin probabilities of the given model. For this risk model we present the solution of the derived partial differential equation for the non-ruin probability in the case of exponentially distributed claims using the technique of the double Laplace transform.

Acknowledgments. The research work of Meglena Lazarova is supported by the Bulgarian National Science Fund under Project KP-06-M62/1 “Numerical deterministic, stochastic, machine and deep learning methods with applications in computational, quantitative, algorithmic finance, biomathematics, ecology and algebra” from 2022.

Boussinesq’s equation (BE) was the first model for the propagation of surface waves over shallow inviscid fluid layer. He found an analytical solution of his equation and thus proved that the balance between the steepening effect of the nonlinearity and the flattening effect of the dispersion maintains the shape of the wave. This discovery can be properly termed ‘Boussinesq Paradigm.’ Apart from the significance for the shallow water flows, this paradigm is very important for understanding the particle-like behavior of nonlinear localized waves. As it should have been expected, most of the physical systems are not fully integrable (even in one spatial dimension) and only a numerical approach can lead to unearthing the pertinent physical mechanisms of the interactions. A different approach to removing the incorrectness is by changing the spatial fourth derivative to a mixed fourth derivative, which resulted into an equation know nowadays as the Regularized Long Wave Equation (RLWE) or Benjamin–Bona–Mahony equation (BBME) In fact, the mixed derivative occurs naturally in Boussinesq derivation and was changed by Boussinesq to a fourth spatial derivative under an assumption that second temporal derivative is approximately equal to the second partial derivative,which is currently known as the ‘Linear Impedance Relation’ (or LIA). The LIA has produced innumerable instances of unphysical results.

In this work, we study the behavior of Zakharov Kuznetsov (ZK) equations using advanced numerical and analytical tools, many of the latter inspired by integrability theory. As a nonlinear dispersive PDE, ZK possesses solutions that develop a singularity in finite time from smooth initial data. It has a rich behavior and phenomena including, inelastic ”soliton” interaction, soliton resolution and finite time blow up. We propose several conjectures for its long-term behavior . Based on joint work with Christian Klein, Svetlana Roudenko and Jean-Claude Saut.

The soliton hierarchies are characterised by a pair of Lax operators (L and M), which depend on a spectral parameter, usually involving finitely many positive and negative powers of the spectral parameter.

We present recent results on integrable systems arising from quadratic pencil of Lax operator L, with values in a Hermitian symmetric space. The counterpart operator M in the Lax pair defines positive, negative and rational flows, according to the dependence on the spectral parameter. The examples include extensions of the integrable Kaup-Newell, Gerdjikov-Ivanov, Fokas-Lenells equations, as well as new systems. The results are illustrated with examples from the A.III symmetric space. Our approach originates from the classical papers such as [1,2], see for example [3,4]

References:

[1] Fordy AP, Kulish PP, Nonlinear Schrödinger equations and simple Lie algebras, Commun. Math. Phys. 89 (1983) 427. https://doi.org/10.1007/BF01214664 [2] Athorne C, Fordy A, Generalised KdV and MKdV equations associated with symmetric spaces, J Phys A: Math Gen. 20 (1987) 1377–86. https://doi.org/10.1088/0305-4470/20/6/021. [3] Ivanov R, NLS-type equations from quadratic pencil of Lax operators: Negative flows, Chaos, Solitons & Fractals 161 (2022) 112299, https://doi.org/10.1016/j.chaos.2022.112299. [4] Gerdjikov VS and Ivanov R, Multicomponent Fokas–Lenells equations on Hermitian symmetric spaces, Nonlinearity 34 (2021) 939; ttps://doi.org/10.1088/1361-6544/abcc4b, arXiv:2104.00154 [nlin.SI]Set-theoretical solutions of the Pentagon relation, also referred to as Pentagon maps, turn to be essentially three-dimensional discrete integrable systems. In this talk we focus into a connection of the Pentagon maps with discrete integrability, via Lax pair formulation. In detail, we propose a specific class of matrices that participate to factorization problems that turn to be equivalent to constant and entwining (non-constant) Pentagon maps, expressed in totally non-commutative variables. We show that factorizations of order N = 2 matrices of this specific class, turn equivalent to the homogeneous normalization map, while from order N = 3 matrices we obtain an extension of the homogeneous normalization map, as well as novel entwining Pentagon, reverse-Pentagon and Yang-Baxter maps.