Discrete Quantum Transitions, Duality: Emergence of Physical Structures and Occurrence of Observed Formations (Hidden Properties of Mathematical Physics Equations)

With the help of skew-symmetric differential forms, the hidden properties of the mathematical physics equations that describe discrete quantum transitions and emergence the physical structures are investigated. It is shown that the mathematical physics equations possess a unique property. They can describe discrete quantum transitions, emergence of physical structures and occurrence observed formations. However, such a property possesses only equations on which no additional conditions, namely, the conditions of integrability, are imposed. The intergrability conditions are realized from the equations themselves. Just under realization of integrability conditions double solutions to the mathematical physics equations, which describe discrete transitions and so on, are obtained. The peculiarity consists in the fact that the integrability conditions do not directly follow from the mathematical physics equations;they are realized under the description of evolutionary process. The hidden properties of differential equations were discovered when studying the integrability of differential equations of mathematical physics that depends on the consistence between the derivatives in differential equations along different directions and on the consistence of equations in the set of equations. The results of this work were obtained with the help of skew-symmetric differential forms that possess a nontraditional mathematical apparatus such as nonidentical relations, degenerate transformations and the transition from nonintegrable manifolds to integrable structures. Such results show that mathematical physics equations can describe quantum processes.

Some Remarks to Numerical Solutions of the Equations of Mathematical Physics

The equations of mathematical physics, which describe some actual processes, are defined on manifolds (tangent, a companying or others) that are not integrable. The derivatives on such manifolds turn out to be inconsistent, i.e. they don’t form a differential. Therefore, the solutions to equations obtained in numerical modelling the derivatives on such manifolds are not functions. They will depend on the commutator made up by noncommutative mixed derivatives, and this fact relates to inconsistence of derivatives. (As it will be shown, such solutions have a physical meaning). The exact solutions (functions) to the equations of mathematical physics are obtained only in the case when the integrable structures are realized. So called generalized solutions are solutions on integrable structures. They are functions (depend only on variables) but are defined only on integrable structure, and, hence, the derivatives of functions or the functions themselves have discontinuities in the direction normal to integrable structure. In numerical simulation of the derivatives of differential equations, one cannot obtain such generalized solutions by continuous way, since this is connected with going from initial nonintegrable manifold to integrable structures. In numerical solving the equations of mathematical physics, it is possible to obtain exact solutions to differential equations only with the help of additional methods. The analysis of the solutions to differential equations with the help of skew-symmetric forms [1,2] can give certain recommendations for numerical solving the differential equations.

The Peculiarity of Numerical Solving the Euler and Navier-Stokes Equations

The analysis of integrability of the Euler and Navier-Stokes equations shows that these equations have the solutions of two types: 1) solutions that are defined on the tangent nonintegrable manifold and 2) solutions that are defined on integrable structures (that are realized discretely under the conditions related to some degrees of freedom). Since such solutions are defined on different spatial objects, they cannot be obtained by a continuous numerical simulation of derivatives. To obtain a complete solution of the Euler and Navier-Stokes equations by numerical simulation, it is necessary to use two different frames of reference.