Now exist several alternative cosmological models that describe observable properties of our universe. In particular, it is such models as F (R) and F (T ) gravity. We consider properties of their generalization as F (R, T ) model of gravity with kessence. We obtained some exact solutions of particular cass of scale factor a for general formof the F (R, T ) functions with scalar field. These solutions describe the accelerated/deccelerated periods of the universe.

In mathematics and physics, one of the main tasks is to relate differential geometry and non-linear differential equations, which means that the study of particular cases of subvarieties, curves, and surfaces are of great importance. Soliton surfaces associated with the integrable system play an essential role in many problems with the physical application. In this paper, we study the complex modi ed Korteweg de Vries (cmKdV) equation. It is well known that the cmKdV equation is a very important integrable equation. We present the relationship between an integrable system and soliton surfaces and namely Lax representation of the cmKdV equation was used to obtain the rst and second fundamental forms, surface area and curvature.

It is shown that the inflationary model is the result of the symmetry of the generalized F(R,T,X,φ)-cosmological model using the Noether symmetry. It leads to a solution, a particular case of which is Starobinsky’s cosmological model. It is shown that even in the more particular case of cosmological models F(R,X,φ) and F(T,X,φ) the Monge–Ampère equation is still obtained, one of the solutions including the Starobinsky model. For these models, it is shown that one can obtain both power-law and exponential solutions for the scale factor from the Euler–Lagrange equations. In this case, the scalar field φ has similar time dependences, exponential and exponential. The resulting form of the Lagrangian of the model allows us to consider it as a model with R2 or X2. However, it is also shown that previously less studied models with a non-minimal relationship between R and X are important, as one of the possible models. It is shown that in this case the power-law model can have a limited evolutionary period with a negative value of the kinetic term.

We investigated the gravity model F (R, T), which interacts with a fermion field in a uniform and isotropic at spacetime FLRW. The main idea and purpose of the work donewas to create a mathematical model and find a particular solution for the scale factor a, since it describes the dynamics of the evolution of the Universe. The solutions for this universe are obtained using the Noether symmetry method. With its help, a specific form of the Lagrangian is obtained. And the possible types of the scale factor were found. The evolution of the resulting cosmological model has been investigated.

Physical processes are described using mathematical models. Many of them are nonlinear in nature. For this reason, the theory of a nonlinear medium is relevant and very extensive. From a mathematical point of view, the subject of physics of nonlinear phenomena is systems described by nonlinear partial differential equations that have partial solutions - solitons. A traveling wave that rapidly decreases at infinity is called a solitary wave or soliton. Soliton theory has many fundamental methods for detailed analysis of processes. One of these methods is the geometric interpretation of the physical process. This paper is devoted to the study of the Lax pair of isomonodromic deformation. The isomonodromy condition is equivalent to the existence of a compatible pair of linear equations, the Lax pair. In this pair, one of the equations undergoes deformation, and the other describes the deformation. Isomonodromic deformation is the theory of isomonodromy (that is, monodromy conservation) of the deformation of ordinary differential equations. This method was used to obtain expressions for the coordinate angle. It is proved that the deformation of a system is isomonodromic if and only if the first and second fundamental forms that define this deformation satisfy the integrability condition. It is shown that, similarly, the area of a soliton surface is represented as a semi-saddle graph.