Skew-symmetric forms: On integrability of equations of mathematical physics

TL;DR

This paper explores the integrability of mathematical physics equations using skew-symmetric forms, revealing conditions under which solutions are locally integrable and how unclosed forms relate to nonintegrability.

Contribution

It introduces the use of skew-symmetric forms on nonintegrable manifolds to analyze the integrability of PDEs in physics and mechanics, providing new insights into solution structures.

Findings

Nonidentical relations indicate nonintegrability of equations.

Degenerate transformations lead to local integrability.

Generalized solutions depend on unclosed form commutators.

Abstract

The study of integrability of the mathematical physics equations showed that the differential equations describing real processes are not integrable without additional conditions. This follows from the functional relation that is derived from these equations. Such a relation connects the differential of state functional and the skew-symmetric form. This relation proves to be nonidentical, and this fact points to the nonintegrability of the equations. In this case a solution to the equations is a functional, which depends on the commutator of skew-symmetric form that appears to be unclosed. However, under realization of the conditions of degenerate transformations, from the nonidentical relation it follows the identical one on some structure. This points out to the local integrability and realization of a generalized solution. In doing so, in addition to the 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. In the present paper, the partial differential equations, which describe any processes, the systems of differential equations of mechanics and physics of continuous medium and field theory equations are analyzed.