Zaur Khukhunashvili

We investigate an algebraic structure of the space of solutions of autonomous nonlinear differential equations of certain type. It is shown that for these equations infinitely many binary algebraic laws of addition of solutions exist. We extract commutative and conjugate commutative groups which lead to the conjugate differential equations. Besides one is being able to write down particular form of extended Fourier series for these equations. It is shown that in a space with a moving field, there always exist metrics geodesics of which are the solutions of a given differential equation and its conjugate equation. Connection between the invariant group and the algebraic structure of solution space has also been studied.

In the proposed work which continues [Z. Z. Khukhunashvili, V. Z. Khukhunashvili, Alternative Analysis Generated by a Differential Equation, E. J. Qualitative Theory of Diff. Equ., No. 2. (2003), pp. 1-31], we study the algebraic properties of processes described by autonomous differential equations. We have found that a wide class of differential equations contains an algebraic object isomorphic to the object consisting of superposed two alternatively acting numerical fields with common neutral elements. Using its own algebraic field, each process constructs its own (differential and integral) calculus with a simultaneous definition of its own frame of reference. It appears that in its own calculus the differential equation of this process takes the linear form, while the arisen system of reference becomes inertial. Along with this, because of the existence of a double algebraic field an alternative antiprocess is assigned to each process. The developed theory makes it possible to describe one process from the standpoint of the other process. It should be said that the inertial systems of one process do not necessarily coincide with the inertial systems of the other process. All the results and conclusions follow exclusively from the algebraic properties of differential equations without using any other postulates and assumptions. These studies enable us to get an idea of the algebraic structure of the Fourier method in the case of nonlinear equations. We succeeded in writing out the exact solution of equations of hydrodynamics in implicit form.

In this paper the geometry of a space is investigated using not the logic of motion of a classical particle, but the properties of motion of a field. This appears to be sufficient for the algebraic theory of differential equations to bring us unambiguously to a qualitatively new mathematical space and field theory. It turns out that each differential equation describing some process constructs its own geometry – field geometry. With that, the arising corresponding space represents a union of three: space-time, inner space and dynamical space. The principles of relativity are qualitatively broadened, an explanation is found for the existence of unitary symmetry that commutes with the Lorentz group but is generated by its representation.

Algebraic theory of a differential equation describing some process, generates proper geometry, field geometry. The localization of group parameters is performed. From the scalar curvature, a single Lagrangian is derived for Maxwell, Yang-Mills, Dirac and Einstein equations for strong gravitation. In this case, in the first approximation there arise standard interaction terms and even mass terms. In subsequent approximations the equations of Maxwell and equations of Dirac become non-linear. As to usual gravitation, though it is involved in the field theory developed in the paper, it has an absolutely different nature than all other fields. The alternative properties of the algebraic theory of differential equations allow us to conclude immediately that all fields must be quantized. An exception is a gravitation field whose quantization is meaningless. The developed theory suggests the existence of the double world. There exists only a gravitational interaction between these worlds, all other interactions are absent.