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 Incoming links: Quadratic oscillator phase-modulation condition

Use the fast time

T_{0} = t

T = ϵ^{2} t

. A possible intermediate scale

ϵ t

has no leading modulation: the absence of first-order resonance imposes independence of that scale. Write

U_{0} = A (T) e^{i T_{0}} + \overline{A} (T) e^{- i T_{0}} .

At first order choose the nonresonant particular solution

U_{1} = 2∣ A ∣^{2} - \frac{1}{3} A^{2} e^{2 i T_{0}} - \frac{1}{3} \overline{A}^{, 2} e^{- 2 i T_{0}},

with any needed fundamental homogeneous correction assigned consistently when matching the initial data. At second order,

(D_{0}^{2} + 1) U_{2} = 2 U_{0} U_{1} - 2 D_{0} D_{T} U_{0} .

e^{i T_{0}}

on the right is

(10/3) ∣ A ∣^{2} A - 2 i A_{T}

. A bounded oscillatory correction requires this resonance coefficient, and its conjugate, to vanish. The quadratic oscillator phase-modulation condition is therefore

A_{T} = - \frac{5 i}{3} ∣ A ∣^{2} A .

It resums the secular phase accumulation into a slow modulation instead of leaving it as a growing term in the correction.