Vessiot distribution

[Tehseen 2014]
For finite $k$ , Cartan distribution $C$ is generated by

D_{x^{i}}^{(k)} := \partial x_{i} + \sum_{α = 1}^{q} \sum_{0 \leq | J | < k} u_{J, i}^{α} \partial_{u_{J}^{α}}, 1 \leq i \leq p

V_{α}^{J} := \partial_{u_{J}^{α}}, | J | = k, 1 \leq α \leq q .

Cartan distribution restricted to the submanifold $S$ of $J^{k} (E)$ given by a system of DEs is known as the Vessiot distribution. See also [Vitagliano 2017] page 22, where it defines the distribution:

C (S) : z \mapsto C (S)_{z} := C_{z} \cap T_{z} S

In general, is not Frobenius integrable, but Vessiot gave a method to construct all the integrable subdistributions for a given distribution.

The integral manifolds of this distribution are the solutions of system of DEs (provided they have the proper dimension, i.e. equal to the number of independent variables $p$ ).

Excess dimension. For a PDE with $p$ independent variables, a solution is a $p$ -dimensional integral manifold of $V$ . When $\dim V > p$ , there is a continuous family of valid tangent planes at each point: one must choose a $p$ -dimensional sub-plane at every point of the domain, and a smooth field of such choices is a function. Each extra dimension beyond $p$ therefore corresponds to an arbitrary function in the general solution. See differential constraint method for a detailed discussion and the relation to Cartan–Kähler theory.