Reduction of higher-order normal form with jet compatibility

For $\mathrm{\mid }\beta \mathrm{\mid }\le m-2$ prescribe ${\mathrm{\partial }}_t{U}_{\beta }=U_{\beta +e_t}$. For $\mathrm{\mid }\beta \mathrm{\mid }=m-1$ other than $(m-1)e_t$, choose the least spatial i with ${\beta }_i>0$ and prescribe ${\mathrm{\partial }}_t{U}_{\beta }={\mathrm{\partial }}_i{U}_{\beta -e_i+e_t}$. For $\beta =(m-1)e_t$, prescribe ${\mathrm{\partial }}_t{U}_{\beta }=F$, replacing jets of order at most m-1 by U, and each allowed order-m jet $\gamma$ by ${\mathrm{\partial }}_i{U}_{\gamma -e_i}$ using the least spatial i with ${\gamma }_i>0$. Such i exists because the pure m-time jet is excluded. Thus every right side uses only U and first spatial derivatives of U.

The prescribed data are analytic derivatives of the g_j, and the finite right-hand side is analytic near their compatible initial jet. For $d\ge 1$, F2 supplies a unique analytic vector U. If $d=0$, there are no spatial derivatives or mixed jets, and F4 supplies that vector as an analytic ODE solution. Independently F1 supplies the scalar formal solution u for the original data. Formal differentiation shows that $D^{\beta }u$ solves every equation in step 1.1 and has initial trace ${\mathrm{\partial }}_x^{\alpha }{g}_j$. The first-order formal uniqueness clause of F1 therefore identifies the Taylor series of each analytic U_beta with $D^{\beta }u$.

In particular the Taylor series of U_0 is u. Differentiating its convergent series shows $D^{\beta }{U}_0$ and U_beta have identical Taylor series; hence they coincide on a smaller neighborhood. Substituting in the pure-time equation in step 1.1 yields the scalar equation, and its first m data follow from the traces of U_{je_t}. Conversely for any analytic scalar solution, F3 makes its actual derivative vector satisfy every equation and datum in step 1.1. These constructions are inverse, including m=1 when the sole unknown is U_0 and the pure-time equation is the original equation.