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Background. In this work, a new representation is constructed for equations of the Korteweg – de Vries (KdV) type. The proposed approach allows to obtain a universal Lax representation for a set of nonlinear partial differential equations, for which such a representation was not previously known. Materials and methods. The construction of the Lax representation for the new equations is based on the reduction of the general compatibility condition for two nonlinear first-order equations with a polynomial dependence on the unknown function. Results. A new general scheme for calculating the Lax representations in the form of two linear operators of the first order with a spectral parameter for the set of 1 + 1 equations integrable using the inverse problem method is obtained in this work. Infinite series of differential conservation laws for these equations are calculated and a special type of Backlund transformations for them is indicated. Conclusions. For a whole class of equations of the KdV-type, there is a general form of Lax representations that allows the inverse problem method to be applied to them.

For the Riemann problem $$ \left\{\begin{array}{l}u_t+(\Phi(u,x))_x=0,\\ u|_{t=0}=u_-+[u]\theta(x) \end{array}\right. $$ a new definition of the solution is proposed. We associate a Hamiltonian system with initial conservation law, and define the geometric solution as the result of the action of the phase flow on the initial curve. In the second part of this paper, we construct the equalization procedure, which allows us to juxtapose a geometric solution with a unique entropy solution under the condition that $\Phi$ does not depend on $x$. If $\Phi$ depends on $x$, then the equalization procedure allows us to construct a generalized solution of the original Riemann problem.

The paper addresses an extended (1+1)-dimensional Sawada-Kotera (SK) equation. The Lie symmetry analysis leads to many plethora of solutions to the equation. The non-linear self-adjointness condition for the SK equation established and subsequently used to construct simplified independent conserved vectors. In particular, we also get conservation laws of the equation with the corresponding Lie symmetry.

In this study, we consider a 4-th order (1+1)-dimensional PDE called Zoomeron equation. Some conservation laws are derived based on direct method. We also derived some similarity solutions using the symmetries.

This paper discusses the structure of a geologic medium represented by accessible lithified rocks and provides an overview of methods used to describe its movements. Two basic opinions are considered in the framework of the discussion: (1) an initially homogeneous and continuous geologic medium acquires the structure composed of blocks in the process of the geologic medium’s deformation/destruction/degradation, and (2) a geologic medium is composed of blocks (and often has hierarchic, active, energy-saturated features), and the continuity model is thus not valid for describing the geologic medium’s deformation. Proponents of the first point of view actively apply the standard or modified continuum model of a solid deformed body (SDB) in estimations of the stress-strain state, but the input parameters of this model do not contain any information on discreteness in principle. Authors who support the second opinion, either explicitly or implicitly assume that the block structure of the geologic medium, which is detectable by geological methods, makes a direct and unambiguous impact on all other mechanical properties of the geologic medium and, above all, on the nature of its movements.Based on results obtained by interpreting the data collected in our long-term field studies of rock fracturing, mathematical processing of GPS-measurements, and theoretical models, we agree with the concept of the geologic medium’s block structure, but argue that the geologic block-structure property is not acquired but congenital. Regarding sedimentary rocks, it means that the discrete structure has been already embodied in the rock before sediment lithification, regardless of the intensity of macroscopic deformations. A discrete structure is the form of the geologic medium existence and a cause of the congenital anisotropy of the geologic medium’s strength characteristics. Due to subsequent deformation of the geologic medium, some elements of the structure can be manifested more clearly, and the structure itself can become more complex due to secondary effects. At the same time, the structure of geologic blocks is not directly manifested in the spatial and temporal features of the recent movements in the geologic medium. However, it is not an obstacle to developing continuum models of such movements in the same way as, for instance, the adequacy of the continuum general theory of relativity is not denied in view of the discrete-hierarchical structure of the Universe.The key requirements to a model include its applicability, testability, confirmability/deniability of its predictions in the investigated space-time scale, and compliance with conservation laws. This paper briefly discusses the most important aspects of the continuum approach based on the concept of an effective continuous medium and, above all, the Cauchy continuum model, envisaging that the dynamic response of the medium in spatial descriptions is given only by the Cauchy symmetric stress tensor (T). In more general continuum models of the medium (such as moment, micropolar, micromorphic and other models), the dynamics of the medium may be characterized by asymmetric tensors of force and couple stresses.This paper refutes the unjustified criticism of the continuum model as such criticism is rooted in the mistaken identification of quite special assumptions or ways of setting the problems with the general principles of the continuity model. Special attention is given to critical comments received from supporters of the active geologic medium concept. The paper considers actual difficulties encountered in studies using the continuity model, specifically in coordinated descriptions of the medium containing mobile defects, as well as the medium that is subject to deformation due to movements on its structure, while this structure is hierarchical at any scale level, down to the zero level (which, in particular, concerns the fractal structure). Discussed are causes of some widespread misunderstandings and mistakes in the geoscience literature, as well as the occurrence of conditions facilitating the revival of Aristotle ideas and preNewton concepts in geology, which repeatedly gain the upper hand over the modern ideas of classical physics. The paper considers the problem of reconstruction of stresses in the geologic medium from in-situ kinematic indicators, specifically from irreversible slips. Attention is drawn to the fact that the currently dominating approach ‘imposes’ a priori speculative rules on the geologic medium, such as a relationship between stresses (to be estimated) and the slip directions. Under this approach, the conservation laws are inevitably ignored, which makes it impossible to interpret the obtained results in terms of stress. Under the alternative approach proposed earlier by the author of this paper, the conservation laws being taken into account allows not only to reconstruct the stress tensor field, but also to judge on the geologic medium rheology. It is concluded that rejecting the continuum approach a priori, with a reference to the geologic medium discreteness, is at least unconstructive.

This work is aimed at demonstrating the possibility of the inclusion of religious elements contained in Holy Scripture and Holy Tradition in the general natural scientific courses based on the principle of symmetry.The method used in the work is confined to a comparison of perceptions formed in modern science and is closely related to the forms of symmetry and invariance principles (symmetry principles) and, in particular, space-time concepts with those of the Book of Genesis. Such a comparison reveals the following unexpected feature: most profound presentation of modern natural sciences is closer to the provisions of Holy Scripture and Holy Tradition than a look at the same things existed in the earlier stages of the development of science. This allows the authors to formulate the hypothesis that in the process of development of scientific knowledge, it gradually becomes closer to the religious worldview. This process is slow, so its results have become visible only within 3500 years after the establishment of the truth of the Old Testament and 2000 years after the New Testament.Results and scientific novelty. The «firmament of heaven» and «water under the firmament» concepts are explained in the terms of the model of the Kleinert – Planck World crystal and understanding of the properties of matter and fields which are related with the conservation law of the wave-function parity. The relational nature of phenomena such as «life» and «death» in the course of universe evolution as a general trend is considered as the process of lowering the degree of symmetry of matter after the Big Bang wherein the Universe was created. The concepts used by E. Wigner for the description of the structure of the scientific knowledge are analysed. Its structure is determined by shapes and specific principles of the symmetry of exact sciences. The analysis of the concept «natural phenomenon» has shown that they are different in the degree of space-time localization. As the nonlocality of nature phenomenon becomes intensive, the limits of the scientific knowledge are approached. Understanding of creatures with the utmost degree of nonlocality is beyond the scientific knowledge. There is a tendency in modern science to study the behavior of objects in frames of nonlocal spacetime description. This trend is reflected, for example, in a study of the phenomenon of quantum entanglement. It can be stated that in this respect the position of science closes in the positions of the religious worldview.Practical significance. In this paper the authors present a few examples of selection of content of the course based on the Principle of symmetry.

The development of the non-equilibrium thermodynamics of the viscous fluid not using the local thermodynamic equilibrium hypothesis is considered. The theory is based on the causal mechanics of the heat conducting continuum, which includes the 1st law of thermodynamics as a theorem. The conditions of applicability of the 2nd law of thermodynamics and the dissipation of the kinetic energy problem are discussed. Main conclusions are illustrated using the example from the numerical analysis.

The problem has been considered, to what extent the kinetic models of cellular metabolism fit the matter which they describe. Foundations of stoichiometry of the whole metabolism and its large regions have been stated. A bioenergetic representation of stoichiometry based on a universal unit of chemical compound reductivity, viz., redoxon, has been described. Equations of mass-energy balance (bioenergetic variant of stoichiometry) have been derived for metabolic flows including those of protons possessing high electrochemical potential H+, and high-energy compounds. Interrelations have been obtained which determine the biomass yield, rate of uptake of energy source for cell growth and other important physiological quantities as functions of biochemical characteristics of cellular energetics. The maximum biomass energy yield values have been calculated for different energy sources utilized by cells. These values coincide with those measured experimentally.

At the initial level information physics is studied. The assumption of the primacy of information naturally leads to the idea of the existence of God. The total differential of information and the laws of conservation of information are obtained. It is pointed out that the physics of information is a quantum mechanic that studies the motion of information quantum in the channels of information transmission. This subject is well studied science and is known as Coding Theory. [ABSTRACT FROM AUTHOR]