Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 331 / 3 / a / i

Table of contents
- Solution i

Solution

Take the physical coefficients

U, μ, c_{d}

real. Substitution of

e^{i (k x - ω t)}

into the linear complex Ginzburg-Landau equation gives

- iω + ik U - μ + (1 + i c_{d}) k^{2} = 0

, hence

ω = U k + (c_{d} - i) k^{2} + i μ .

Let

A = c_{d} - i

. Completing the square yields

ω = A (k + U / (2 A))^{2} + i μ - U^{2} / (4 A)

. Therefore the absolute wavenumber and absolute frequency are

k_{0} = - \frac{U}{2 ( c _{d} - i )}, ω_{0} = i μ - \frac{U ^{2}}{4 ( c _{d} - i )} .

Their explicitly separated parts are

k_{0} = - \frac{U ( c _{d} + i )}{2 ( 1 + c _{d}^{2} )}, ω_{0} = - \frac{U ^{2} c _{d}}{4 ( 1 + c _{d}^{2} )} + i [μ - \frac{U ^{2}}{4 ( 1 + c _{d}^{2} )}] .

Thus

d ω / d k = U + 2 (c_{d} - i) k

vanishes at

k_{0}

. The complex saddle describes the localized Green function response; it is different from selecting a real wavenumber for a Fourier mode for temporal amplification.

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