Research seminars

(Landau Institute of Theoretical and Theoretical Physics of the Russian Academy of Sciences, Moscow)

Alexei Shabat, PhD, is an author of more than 120 research works, Laureate of the State Award of the Russian Federation in the field of Science and Technology. Dr.Shabat’s research works are dedicated to general theory of integrable systems and its applications. Dr.A.Shabat developed a theory of integrating non-linear equations of the mathematical physics. Under Dr.Shabat’s supervision, the research for creating the soliton’s theory has started.

Since September 2018, Dr.Alexei Shabat teaches a seminar for undergraduate and graduate students at the Faculty of Mathematics and Computer Sciences at Adyghe State University.

**II semester of 2018-2019 academic year**

**7.02 The seminar was dedicated to R.Miura and was organized with R.Kulaev, PhD (North Ossetian State University).**

The name of R. Miura is connected to the transformation (differential substitution) which unites a pair of partial differential equations that have solutions of the solitary wave type called solitons. The peculiarity of these waves is that, in the interaction, they retain their individuality and model particle-like solutions. The connection between the solutions of these equations, generally speaking, is irreversible and is one-sided. Robert M. Miura in 1967 showed that for the KdV equation, the problem of constructing a differential substitution is correct.

Although the transformation (0.1) does not in itself simplify the solution of the Korteweg-de Vries equation since it connects the solutions of two nonlinear equations, nevertheless, this transformation was the key to the discovery of the inverse scattering method and the development of a symmetry approach to the integrability problem.

**I semester of 2018-2019 academic year**

**21.09 The theory of dispersion, the Korteweg equation (by Ruslan Kulayev)**

**28.09 Riccati equation (by A.E.Artisevich)**

The Riccati equation is one of the most interesting nonlinear first order differential equations. The Riccati equation is found in various areas of mathematics (for example, in algebraic geometry and in the theory of conformal mappings) and physics. It also often arises in applied mathematical problems.

The talk includes the discussion of reversible changes of variables applicable to the general Riccati equation that do not violate its structure and reduce the equations to a simpler.

**12.10 Analytical continuations (by A.B.Shabat)**

**19.10 Vronskian (by Alina Allakhverdyan)**

**26.10 Bessel equation (by A.A.Panesh)**

A linear second-order ordinary differential equation of the form y^”+xy^’+(x^2-v^2 )y=0 is called the Bessel equation. The number is called the order of the Bessel equation.

This differential equation was named after the German mathematician and astronomer Friedrich Wilhelm Bessel, who studied it in detail and showed (in 1824) that the solutions of the equation are expressed through a special class of functions called the Bessel cylindrical functions.

The seminar considers the solution of the Bessel equation in cases where:

- The order v is non-integer;
- The order v is an integer.

**2.11 Matrix Exhibitor (by Alina Eremina)**

Matrix entry of linear systems of differential equations is considered during the talk. The iteration method produces a solution representing a convergent matrix series. The case of an arbitrary differentiable matrix-valued function with a nonzero determinant is also considered.

**9.11 Numbers Catalan (by A.A.Panesh)**

Catalan numbers are a numerical sequence found in a number of combinatorial problems. This sequence is named after the Belgian mathematician Catalan who lived in the 19th century.

Catalan numbers are found in a large number of combinatorial problems. The seminar will consider the problem of finding the number of ways to completely divide the n + 1 multiplier (nth Catalan number) with brackets.

**16.11 Symmetric polynomials (by Nadezhda Prozorova)**

Symmetric polynomial – a polynomial in n variables for all derangement of its variables. The seminar discusses the representation of symmetric polynomials in terms of elementary symmetric polynomials and generating functions.

**23.11 Elliptic functions (by N.A. Loboda)**

There are various ways to define elliptic functions. The seminar considers elliptic functions as solutions of a system of differential equations {y1^’=y2*y3, y2^’=-y1*y3, y3^’=-k^2*y2*y1. where 0 < k < 1. All the basic properties of elliptic functions are obtained directly from this system.