Vol. 5, No. 3 (2021): Regular Issue

Two dates coincided in 2019: the centenary of the full member of Russian Academy of Sciences Lev Vasilievich Ovsyannikov (April 22, 1919 -- May 23, 2014) and the eightieth anniversary of his follower, professor Nail Hairullovich Ibragimov (January 18, 1939 -- November 04, 2018). Through the efforts of them and their numerous students and followers, Sophus Lie's theory of local transformation groups was revived and received promising development.

Louise Petrén (Hedvig Louise Beata Petrén-Overton, 1880-1977) was the first Swedish woman to defend her doctoral thesis in mathematics in 1911. It took nearly a hundred years for her thesis to be duly appreciated.

In study of the structure of shock waves in bubbly liquids we need to find integral curve connecting two singular points of the system of differential equations. We linearized the system of basic equations around equilibrium state. The condition of existence of nontrivial solution of the linear equations leads to the characteristic transcendental equation with respect to exponential index. The structure of roots of such equation is studied by using argument principle [1]. It is shown that for compression waves such equation has one real root.

Medical ultrasound imaging provides images of the internal body for diagnosis. Speckle noise is a major problem degrading image quality. In this paper, a wide class of noise reduction models is considered from the group analysis point of view. Group classification is performed, and conservation laws and invariant solutions of the studied equation are presented.

In this paper we obtain Lie symmetries and travelling wave solutions for a generalized Bogoyavlensky-Konopelchenko equation in $(2+1)$-dimensions. Moreover, we determine some low-order conservation laws which are invariant under the translation symmetry; consequently they are inherited by the reduced differential equations.

The paper is devoted to preliminary group classification of the spatially homogeneous and isotropic Boltzmann equation with a source term. In fact, the Fourier transform of the Boltzmann equation with respect to the molecular velocity variable is considered. The equivalence transformations considered earlier were extended by the change of time. This extends the set of source functions. The analysis of the extended equivalence group is given. Using optimal systems of finite-dimensional subalgebras of these extended equivalence set of transformations, preliminary classification of the source function is obtained.

The complete classification of inequivalent realizations of the Virasoro algebra by Lie vector fields over the three-dimensional field of real numbers is obtained. Based on this result is our group classification of second-order differential equations admitting infinite-dimensional Virasoro algebras. In particular, we derive all inequivalent second-order partial differential equations, which admit the direct sum of the Witt algebras. The two well known examples of equations belonging to this class are the wave and hyperbolic Liouville equations. We prove that there is one more nonlinear differential equation enjoying the same group properties as the wave equation.

In this paper we show some short notes concerned the nonlinear selfadjointess of a class of wave equations. These results, obtained together with prof. Ibragimov between the end of 2017 and January 2018, were the beginning of a wide study that remained incomplete.

In this paper, analytical dynamics of a discontinuous system with a hyperbolic control boundary is studied. The analytical conditions for flow switchability of the discontinuous system are developed from a flow passability theory at boundaries, which are for a better understanding of dynamics of such a discontinuous dynamical system. With the analytical switchability conditions, the sliding and non-sliding motions are described through generic mappings. Periodic motions in the discontinuous dynamical system with hyperbolic boundary are studied through mapping structures. The bifurcation trees of periodic motions are presented, and the corresponding stability and bifurcation of periodic motions are analyzed. The grazing bifurcation for a flow to the boundary is discussed as periodic motion switching. Numerical simulations are completed for illustration of complex periodic motions and flow switchability at the hyperbolic boundary. This paper is dedicated to Nail Ibragimov as a good friend and colleague for 20 years friendship with Albert Luo.