Manuscript X-A-ST@1 · X-A-ST.tex · proposition · d9ccbde1

Proposition: Unique formal coordinate coefficients

Flat Cauchy construction and coefficient equations › Coordinate equations, with every ramp term · lines 755–798 · proof included or linked

X-A-ST@1#prop:coordeq, X-A-ST@1#eq:coordinatesolutions, X-A-ST@1#eq:coordinateequations, X-A-ST@1#eq:affinek

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The coordinate problem has the unique solution

w^{μ} v^{μ} = - E_{0} \partial_{α} (σ a_{1}^{α μ} [h]), = - E_{0} (V_{ε} [h] w^{μ} + W_{ε} [h, k] x^{μ}) .

Equivalently, with zero data at each positive order,

□ w^{μ} □ v^{μ} = σ C^{μ} [h] - σ^{'} a_{1}^{0 μ} [h], = σ (C^{μ} [k] - \partial_{α} a_{2}^{α μ} [h] - \partial_{α} (a_{1}^{α β} [h] \partial_{β} w^{μ})) - σ^{'} (a_{1}^{0 μ} [k] + a_{2}^{0 μ} [h] + a_{1}^{0 β} [h] \partial_{β} w^{μ}) .

Also,

v [h, k] = w [k] + v [h, 0] .

Proof. Insert (eq:coordinateoperators) and $Co = x + κ w + κ^{2} v$ into (eq:coordinateproblem). Since $□ x^{μ} = 0$ , coefficient comparison gives the displayed equations. Lemma lem:cauchy proves existence and uniqueness successively. Only the source $- \partial_{α} (σ a_{1}^{α μ} [k])$ contains $k$ , proving the final identity.

Displayed formulas in this passage

X-A-ST@1#eq:coordinatesolutions · line 758
$w^{μ} v^{μ} = - E_{0} \partial_{α} (σ a_{1}^{α μ} [h]), = - E_{0} (V_{ε} [h] w^{μ} + W_{ε} [h, k] x^{μ}) .$
X-A-ST@1#eq:coordinateequations · line 769
$□ w^{μ} □ v^{μ} = σ C^{μ} [h] - σ^{'} a_{1}^{0 μ} [h], = σ (C^{μ} [k] - \partial_{α} a_{2}^{α μ} [h] - \partial_{α} (a_{1}^{α β} [h] \partial_{β} w^{μ})) - σ^{'} (a_{1}^{0 μ} [k] + a_{2}^{0 μ} [h] + a_{1}^{0 β} [h] \partial_{β} w^{μ}) .$
X-A-ST@1#eq:affinek · line 783
$v [h, k] = w [k] + v [h, 0] .$

Proposition: Unique formal coordinate coefficients

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