Cauchy–Kovalevskaya on a noncharacteristic analytic hypersurface

Use F1 to choose analytic coordinates carrying the surface to t=0. For Euclidean normal jets use the normal-line coordinates of F1, so the prescribed functions are exactly the coordinate t-jets. At a selected compatible nonlinear jet F1 gives the analytic implicit branch; otherwise the equation is already solved. Every right-hand derivative now has total order at most m and t-order below m, and the lower/mixed initial jets lie in its analytic neighborhood by the data-compatibility hypothesis.

F2 subtracts the polynomial of the prescribed coordinate jets. F3 constructs the finite first-order jet system with zero analytic data. Its right side is analytic near the zero initial value and spatial jet, because F2 makes the scalar right side analytic at its zero-data centre and F3 replaces each allowed highest jet by a first spatial derivative. If there is at least one spatial variable, F4 supplies an analytic vector solution; with no spatial variables, the system is a finite analytic ODE system and F5 supplies it. F3 then recovers derivative compatibility and proves that the zero-order component solves the scalar zero-data equation. Adding the polynomial restores every prescribed jet. All arguments are local, so shrink finitely until the solution jet stays in the selected branch neighborhood; continuity and its prescribed centre jet ensure this. Composing with the analytic inverse coordinates gives a solution to the original PDE with its original normal data.

Two analytic solutions with these data and jets in the selected branch pull back to solutions of the same solved coordinate equation. Subtracting the same polynomial and taking derivative vectors gives two solutions of the same analytic jet system supplied by F3. If there is at least one spatial variable, F4 makes those vector solutions equal; with no spatial variables, F5 does so. Their zero-order components are therefore equal. Adding the polynomial and composing back proves equality of the original germs. No comparison with a different nonlinear branch or with nonanalytic solutions is used.