Symmetry in Nonlinear Mathematical Physics - 2009
Ludmila Petrova (Moscow State University, Russia)
Specific features of differential equations of mathematical physics
Investigation of the equations of mathematical physics has been carried out in terms of
the skew-symmetric differential forms. In addition
to exterior forms, the skew-symmetric forms, which, in contrast to exterior forms, are defined on nonintegrable manifolds (such as tangent manifolds of
differential equations, Lagrangian manifolds and so on), were used.
It has been found that the differential equations describing any processes, in particular, the equations of mechanics and physics of continuous medium, are nonintegrable equations if the additional conditions are not realized.
This follows from the analysis of the functional relation that is derived from these equations and have the form of the relations in differential forms. This functional relation connects the differential
of state functional and the skew-symmetric form.
Such a relation proves to nonidentical since the skew-symmetric form for real processes appears to be unclosed and cannot be a differential. From this nonidentical relation one cannot obtain the differential of state functional that may point to the integrability of original equations.
However, if there are any degrees of freedom, it can be realized the conditions of symmetry, namely, the conditions of degenerate transformation (which doesn't conserve a differential), under which
from unclosed skew-symmetric form it is obtained the exterior inexact form (being closed on a certain pseudostructure), and from nonidentical relation it is obtained the identical relation from which one can obtain the interior (only on pseudostructure) differential. This points to the realization of local integrability and the availability of generalized solution.
The degenerate transformations (to which the vanishing of such functional expressions as determinants, Jacobians, Poisson's brackets, residues and others is correspond) are realized mathematically as the transition from nonintegrable manifold to pseudostructures (for example, the transition from
tangent manifold to sections of cotangent bundles).
Thus, it turns out that only the generalized solutions can be solutions to the differential equations of mathematical physics.
This investigation is of principal significance for the qualitative theory of differential equations.