L.I. Petrova “Specific features of differential equations of mathematical physics.”

1. L.I. Petrova “Specific features of differential equations of mathematical physics.” Investigation of the equations of mathematical physics with the help the skew-symmetric differential forms has shown that all equations describing any processes, in particular the equations of mechanics and physics of continuous medium, are not integrable. It follows from the functional relation that is derived from these equations. This relation connects the differential of state functional and the skew-symmetric form. Such a relation proves to be nonidentical , and this fact points to the nonintegrability of the equations. In this case the solution the equations is a functional, which depends on the commutator of skew- symmetric form. However, under realization of the conditions of degenerate transformations (symmetry, due any degrees of freedom) from a nonidentical relationit follows the identical one on some structure, more exactly, on pseudostructure. This points out to the local integrability and realisation of the generalised solution, namely, a solution that is a function, i.e. it depends only on independent variables.

2. Analysis of the first-order partial differential equation Let us take the simplest case: the first-order partial differential equation • (1) • Let us consider the functional relation • (2) where is a skew-symmetric differential form of the first degree. (the summation over repeated indices is implied). In the general case, when differential equation (1) describes any physical processes, the functional relation (2) is nonidentical one. If totake the differential of this relation, we will have in the left-hand side whereas in the right-hand side : where the differential form commutator constructed of the mixed derivatives is nonzero. : from equation (1)it does not follow (explicitly)that the derivatives which obey to the equation(and given boundary or initial conditions) are consistent, their mixed derivatives are commutative, that is, the commutator is nonzero.

3. The nonidentity of functional relation (2) means that the equation (1) is nonintegrable: the derivatives of equation do not make up a differential. The solution to the equation , obtained from such derivativesis not be a function of only variables . This solutionwill depend on the commutator , that is, it isa functional. To obtain a solution that is a function (i.e., the derivatives of this solution make up a differential), it is necessary to add the closure condition for the form and for relevant dual form (the functional plays a role of a form dual) :

4. If to expand the differentials, one gets a set of homogeneous equations with respect to and (in the -dimensional tangent space): (4) From the solvability condition of these equations – vanishing the determinant (composed of coefficients at , ) it follows the relation (5) This relation specify the integrating direction - a pseudostructure, on which the form turns out to be closed one, i.e. is a differential. On thispseudostructure(in present case this is a characteristics of the equation) the derivatives of differential equation (1) constitute a differential and this meansthat the solution of equation (1) becomes a function.It is generalizedsolutions.It is evident that this solution is obtained only under degenerate transformation, that is, when the determinant vanishes. Similar functional properties have the solutions to all differentialequations describing physical processes.

5. Integrabilityof theequations of mechanics andphysics Under description ofmechanics andphysics of continuous medium it is necessary to investigate the conjugacy of not only derivatives in different directions but also the conjugacy (consistency) of the equations. In this case from set of equations one also obtains nonidentical relation that allows to study the integrabilityof equations and features of their solutions. • Equations of mechanics and physics of continuous media are • equations that describe the conservation laws for energy, linear • momentum, angular momentum and mass. • Let us analyze the equations of energy and linear momentum.

6. Evolutionary relation • In the accompanying reference system, which is tied to the manifold built by the trajectories of particles(elementsofmaterial system)the equations of energy and linear momentum are written in the form • here - the functional of the state (action (1) functional, entropy,wave function are • examples of the functional) • -coordinate along thetrajectory (2) - coordinates normal to the trajectory -is an expression that depends on the external actions and the system characteristics Eqs. (1), (2) can be convoluted into the relation • where • Since the equations are evolutionary ones, • this relation is also an evolutionary relation

7. Properties ofevolutionary relation Evolutionary relationisanonidentical relation as it involves an unclosed differential form : the commutator of the form is nonzero: The coefficients of the form depend on energetic action and on force action which have different nature, and hence, the commutator constructed from derivatives of such coefficients cannot be equal to zero. The evolutionary nonidentical relation is a selfvarying one (it is an evolutionary relation and it contains two objects one of which appears to be unmeasurable and cannot be compared with another one, and therefore the process of mutual variation cannot terminate). • The selfvarying evolutionary relation leads torealization of the conditions of degenerate transformation. Underdegenerate transformationfromnonidenticalrelation the relation that isidentical onpseudostructure is obtained.

8. Obtaining an identical relation from anonidentical one Under degenerate transformation the closure conditionsof the exterior and dual form are realized The condition is an equation of the pseudostructure on which the differential of evolutionary form vanishes and on which the closed (inexact) exterior form is obtained. On the pseudostructure from evolutionary relation it is obtained an identical relation (closed exterior formis differential: , thereforerelation is identical one)

9. The identity of the equation obtained from the evolutionary relation means that on pseudostructures the original equations for material systems (the equations of conservation laws) become consistent and integrable. Pseudostructures constitute the integral surfaces (such as the characteristics, integral surfaces) on which the quantities of material system desired (such as the temperature, pressure, density) become functions of only independent variables and do not depend on the commutator (and on the path of integrating). This are generalizedsolutions. One can see that the integral surfaces are obtained from the condition of degenerate transformation of evolutionary relation.

10. The conditions of degenerate transformation • The conditions of degenerate transformation are a vanishing of such functional expressions as determinants, Jacobians, Poisson's brackets, residues and others. • They are connected with symmetries, which can be due to the degrees of freedom (for example, translation, rotation, oscillation freedom of the of material system). The degenerate transformation is realized as a transition from the noninertial frame of reference to the locally inertial system, i.e. a transition from nonintegrablemanifold (for example, tangent one) tothe integrable structures and surfaces (such as the characteristics, potential surfaces, eikonal surfaces, singular points).

11. Conclusion • Thus, the equations describing any processes can have two types of solutions, namely, the solutions that depends on the commutator (a functional) and a generalized solution, which is obtained only under degenerate transformation. • The dependence of the solution on the commutator maylead to instability. The instability develops when theintegrability conditions are not realized and exact (generalized)solutions are not formatted. (Thus, the solutions to the equationsof the elliptic type may be unstable.) • One can see that the qualitative theory of differential equations that solves the problem of unstable solutions and integrability is based on the properties of nonidentical functional relation.

12. Many thanks for yourattention