i u_{t} + Δ u + ∣ u ∣^{2} u = 0.

The imaginary terms proportional to

P_{η} + y \cdot \nabla P_{η}

cancel because

b = - λ λ_{t}

. After multiplying the remaining real equation by

λ^{3}

, one obtains

Δ P_{η} + P_{η}^{3} - λ^{2} γ_{t} P_{η} + \frac{λ ^{2} b _{t} + b ^{2}}{4} ∣ y ∣^{2} P_{η} = 0.

λ^{2} b_{t} + b^{2} = - η

. Comparison with the equation for

P_{η}

therefore requires

γ_{η}^{'} (t) = \frac{1}{λ _{η} ( t ) ^{2}} = \frac{1}{t ^{2} + η} .

Thus, up to an arbitrary constant phase,

γ_{η} (t) = γ_{η} (- 1) + \frac{1}{η} (arctan \frac{t}{η} + arctan \frac{1}{η}) .

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