Manuscript VII-A@1 · VII-A.tex · lemma · 11da3fe6

Lemma: Uniqueness of the switched scalar jet

The specified switched scalar action › The switched solution with fixed free input · lines 702–721 · proof included or linked

VII-A@1#lem:scalarunique

Equations (eq:phi1)–(eq:phi2) give the unique scalar jet satisfying the switched equation and the specified Cauchy data.

Proof. At order $κ$ the equation is

P_{m} ϕ_{1, g} = - V_{g} [h] ϕ_{0} .

At order $κ^{2}$ it is

P_{m} ϕ_{2, g} = - V_{g} [h] ϕ_{1, g} - W_{g} [h, k] ϕ_{0} .

Existence follows from the Cauchy solution operator, and uniqueness follows successively at the two orders.

Displayed formulas in this passage

unlabeled · line 710
$P_{m} ϕ_{1, g} = - V_{g} [h] ϕ_{0} .$
unlabeled · line 714
$P_{m} ϕ_{2, g} = - V_{g} [h] ϕ_{1, g} - W_{g} [h, k] ϕ_{0} .$