Finite type system (PDEs)

Definition

A system of PDEs, viewed as a submanifold of the jet bundle $\mathcal{E}\subset J^k$, is of finite type if its Vessiot distribution $\mathcal{V}$ on $\mathcal{E}$ has rank exactly $n$ (the number of independent variables) and is involutive (Frobenius integrable).

Equivalently: the vector fields ${\mathbf{v}}_1,\dots ,{\mathbf{v}}_n$ spanning $\mathcal{V}$ satisfy

Consequence

A finite type system has a finite-dimensional solution space: solutions are determined by a finite set of constants (initial data at a point), rather than by arbitrary functions.

Sketch of why

Why rank $>n$ means arbitrary functions.
If $\dim (\mathcal{V})>n$, at each point $p\in \mathcal{E}$ one must choose which $n$-plane inside $\mathcal{V}(p)$ to continue along. This choice is made continuously at every point of the domain — that is exactly what a function is. Each extra dimension of $\mathcal{V}$ above $n$ contributes one arbitrary function to the general solution.

What rank $=n$ with involutivity buys.
When $\dim (\mathcal{V})=n$ and $\mathcal{V}$ is involutive, there is no choice at each point: the $n$-plane is uniquely determined by $\mathcal{V}(p)$ itself. The Frobenius theorem then guarantees that through each point $p\in \mathcal{E}$ there passes a unique integral manifold.

Why that means finite-dimensional.
Fix a base point $x_0$ in the domain. A solution through $x_0$ is determined by specifying where in the fiber ${\mathcal{E}}_{x_0}$ one starts — i.e., the values of $u$ and its derivatives up to order $k$ at $x_0$. That fiber has dimension $\dim (\mathcal{E})-n$, which is finite. Once the starting point is fixed, Frobenius uniquely propagates the integral manifold everywhere.

Summary. Involutivity kills the pointwise freedom of choosing directions; Frobenius turns the unique distribution into a unique leaf; and a leaf is pinned by finitely many numbers (its starting point in the fiber ${\mathcal{E}}_{x_0}$).

Context

Most PDE systems are not of finite type — their Vessiot distribution has rank $>n$ and the general solution depends on arbitrary functions. The differential constraint method is one systematic way to produce finite type systems: by appending compatible constraints one shrinks $\mathcal{E}$ until $\dim (\mathcal{V})=n$, and finite type is the condition that guarantees a solution submanifold actually exists.