On Integrability of Mathematical Physics Equations

Abstract

A study of differential equations has shown that the integrability of mathematical physics equations depends on the consistency of the derivatives of described functions. The study of the consistency of derivatives and equations that form the equations of mathematical physics showed that on the original coordinate space the differential equation turns out to be non-integrable. That is, the solution is not a function. However, if there are any degrees of freedom, then integrable structures, on which the derivatives of a differential equation form a differential, are realized. That is, discrete functions are the solution to the equations of mathematical physics on integrable structures. This indicates to the differential equation integrability. Double solutions of the equations of mathematical physics describes the emergence of various formations such as waves, vortex,  and so on.