Abstract
With the help of skew-symmetric differential forms the hidden properties of the mathematical physics equations are revealed. It is shown that the equations of mathematical physics can describe the emergence of various structures and formations such as waves, vortices, turbulent pulsations and others. Such properties of the mathematical physics equations, which are hidden (they appear only in the process of solving these equations), depend on the consistency of derivatives in partial differential equations and on the consistency of equations, if the equations of mathematical physics are a set of equations. This is due to the integrability of mathematical physics equations. It is shown that the equations of mathematical physics can have double solutions, namely, the solutions on the original coordinate space and the solutions on integrable structures that are realized discretely (due to any degrees of freedom). The transition from the solutions of the first type to one of the second type describes discrete transitions and the processes of origin of various structures and observable formations. Only mathematical physics equations, on what no additional conditions such as the integrability conditions are imposed, can possess such properties. The results of the present paper were obtained with the help of skew-symmetric differential forms.
Highlights
We studied the equations of mathematical physics that are a set of equations and on which no additional conditions are imposed
From the nonidentity of evolutionary relation it follows that the mathematical physics equations have double solutions, namely, the solutions that are not functions and solutions that are discrete functions
From the evolutionary relation it follows that the equations of mathematical physics can have double solutions, namely, the solutions, which are not functions and generalized solutions, i.e. solutions that are discrete functions
Summary
We studied the equations of mathematical physics that are a set of equations and on which no additional conditions are imposed. When studying the consistency of equations in system of equations, the evolutionary relation in skew-symmetric differential forms is obtained. It is known that the mathematical physics equations for material media consist of the equations of conservation laws for energy, linear momentum, angular momentum, and mass [2, 3], that is, they are a set of equations. The integrability of such a set of differential equations depends on the consistency of equations in the set of equations. Such an evolutionary relation describes processes of emergence of various structures
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
