## On Vessiot's Theory of Partial Differential Equations

##### Abstract

The object of research presented here is Vessiot's theory of partial

differential equations: for a given differential equation one

constructs a distribution both tangential to the differential

equation and contained within the contact distribution of the jet

bundle. Then within it, one seeks n-dimensional subdistributions

which are transversal to the base manifold, the integral

distributions. These consist of integral elements, and these again

shall be adapted so that they make a subdistribution which closes

under the Lie-bracket. This then is called a flat Vessiot connection.

Solutions to the differential equation may be regarded as integral

manifolds of these distributions.

In the first part of the thesis, I give a survey of the present state

of the formal theory of partial differential equations: one regards

differential equations as fibred submanifolds in a suitable jet

bundle and considers formal integrability and the stronger notion of

involutivity of differential equations for analyzing their

solvability.

An arbitrary system may (locally) be represented in reduced Cartan

normal form. This leads to a natural description of its geometric

symbol. The Vessiot distribution now can be split into the direct sum

of the symbol and a horizontal complement (which is not unique). The

n-dimensional subdistributions which close under the Lie bracket

and are transversal to the base manifold are the sought tangential

approximations for the solutions of the differential equation. It is

now possible to show their existence by analyzing the structure

equations.

Vessiot's theory is now based on a rigorous foundation. Furthermore,

the relation between Vessiot's approach and the crucial notions of

the formal theory (like formal integrability and involutivity of

differential equations) is clarified. The possible obstructions to

involution of a differential equation are deduced explicitly.

In the second part of the thesis it is shown that Vessiot's approach

for the construction of the wanted distributions step by step

succeeds if, and only if, the given system is involutive. Firstly,

an existence theorem for integral distributions is proven. Then an

existence theorem for flat Vessiot connections is shown. The

differential-geometric structure of the basic systems is analyzed and

simplified, as compared to those of other approaches, in particular

the structure equations which are considered for the proofs of the

existence theorems: here, they are a set of linear equations and an

involutive system of differential equations. The definition of

integral elements given here links Vessiot theory and the dual

Cartan-Kähler theory of exterior systems.

The analysis of the structure equations not only yields theoretical

insight but also produces an algorithm which can be used to derive

the coefficients of the vector fields, which span the integral

distributions, explicitly. Therefore implementing the algorithm in

the computer algebra system MuPAD now is possible.

differential equations: for a given differential equation one

constructs a distribution both tangential to the differential

equation and contained within the contact distribution of the jet

bundle. Then within it, one seeks n-dimensional subdistributions

which are transversal to the base manifold, the integral

distributions. These consist of integral elements, and these again

shall be adapted so that they make a subdistribution which closes

under the Lie-bracket. This then is called a flat Vessiot connection.

Solutions to the differential equation may be regarded as integral

manifolds of these distributions.

In the first part of the thesis, I give a survey of the present state

of the formal theory of partial differential equations: one regards

differential equations as fibred submanifolds in a suitable jet

bundle and considers formal integrability and the stronger notion of

involutivity of differential equations for analyzing their

solvability.

An arbitrary system may (locally) be represented in reduced Cartan

normal form. This leads to a natural description of its geometric

symbol. The Vessiot distribution now can be split into the direct sum

of the symbol and a horizontal complement (which is not unique). The

n-dimensional subdistributions which close under the Lie bracket

and are transversal to the base manifold are the sought tangential

approximations for the solutions of the differential equation. It is

now possible to show their existence by analyzing the structure

equations.

Vessiot's theory is now based on a rigorous foundation. Furthermore,

the relation between Vessiot's approach and the crucial notions of

the formal theory (like formal integrability and involutivity of

differential equations) is clarified. The possible obstructions to

involution of a differential equation are deduced explicitly.

In the second part of the thesis it is shown that Vessiot's approach

for the construction of the wanted distributions step by step

succeeds if, and only if, the given system is involutive. Firstly,

an existence theorem for integral distributions is proven. Then an

existence theorem for flat Vessiot connections is shown. The

differential-geometric structure of the basic systems is analyzed and

simplified, as compared to those of other approaches, in particular

the structure equations which are considered for the proofs of the

existence theorems: here, they are a set of linear equations and an

involutive system of differential equations. The definition of

integral elements given here links Vessiot theory and the dual

Cartan-Kähler theory of exterior systems.

The analysis of the structure equations not only yields theoretical

insight but also produces an algorithm which can be used to derive

the coefficients of the vector fields, which span the integral

distributions, explicitly. Therefore implementing the algorithm in

the computer algebra system MuPAD now is possible.