The Cauchy-Kowalevski Theorem

AKA Cauchy-Kovalevskaya theorem
Given a Cauchy problem, the Cauchy–Kowalevski theorem states:
If all the functions $F_{i}$ are analytic in some neighborhood of the point

(t^{0}, x_{1}^{0}, x_{2}^{0}, \dots, ϕ_{j, k_{0}, k_{1}, \dots, k_{n}}^{0}, \dots),

and if all the functions $ϕ_{j}^{(k)}$ are analytic in some neighborhood of the point

(x_{1}^{0}, x_{2}^{0}, \dots, x_{n}^{0}),

then the Cauchy problem has a unique analytic solution in some neighborhood of the point

(t^{0}, x_{1}^{0}, x_{2}^{0}, \dots, x_{n}^{0}) .

For better understanding, let's focus on the case of first order PDEs (see @olver1995equivalence):

Definition 15.1. A system of first order partial differential equations for functions $u (x) = (u^{1} (x), \dots, u^{q} (x))$ , $x = (x^{1}, \dots, x^{p})$ is said to be in Kovalevskaya form if it has been solved for the derivatives of the $u$ 's with respect to one of the $x$ 's, say $x^{p}$ , so that

\begin{matrix} (15.1) & \frac{\partial u^{α}}{\partial x^{p}} = Δ^{α} (x, \tilde{u^{(1)}}), α = 1, \dots, q, \end{matrix}

where the right hand side depends on $x^{1}, \dots, x^{p}$ (denoted as $x$ ) and $u^{1}, \dots, u^{q}$ , and the first order partial derivatives $u_{j}^{β} = \partial u^{β} / \partial x^{j}$ for $β = 1, \dots, q, j = 1, \dots, p - 1$ (denoted as $\tilde{u^{(1)}}$ ).

The Cauchy-Kovalevskaya Theorem concerns the existence and uniqueness of solutions to the Cauchy problem for a system in Kovalevskaya form (15.1), with initial Cauchy data

\begin{matrix} (15.2) & u^{α} (x^{1}, \dots, x^{p - 1}, x_{0}^{p}) = h^{α} (x^{1}, \dots, x^{p - 1}), i = 1, \dots, q, \end{matrix}

prescribed on an open subset of the initial hypersurface ${x ∣ x^{p} = x_{0}^{p}}$ . The fact that the system can be solved for all the $x^{p}$ -derivatives means that this hypersurface is noncharacteristic (in a sense, it means that the hypersurface is transversal to the natural flow given by the PDE). The Cauchy problem for the system (15.1) is to find a (local) solution $u (x)$ to the system satisfying the given Cauchy data.

Theorem 15.2. Let (15.1) be an analytic system of partial differential equations in Kovalevskaya form, and let $h^{α} (x^{1}, \dots, x^{p - 1})$ be analytic functions defined in a neighborhood of $(x_{0}^{1}, \dots, x_{0}^{p - 1})$ . Then, in a neighborhood of $x_{0} = (x_{0}^{1}, \dots, x_{0}^{p})$ , there exists a unique analytic solution to the Cauchy problem consisting of (15.1) and initial data (15.2).