Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 4 / 14A / b

Table of contents
- Solution b

Solution

For

q = k^{2}

, the mode matrix is

J - q diag (D_{1}, D_{2})

. Its trace remains negative. Its determinant is

D_{1} D_{2} q^{2} - \frac{D _{2} β ( 1 - λ β )}{α - β} q + \frac{β ^{2}}{α - β}, λ = \frac{D _{1}}{D _{2}} .

It is negative for some

q > 0

exactly when

1 - λ β > 0, (1 - λ β)^{2} > 4 λ (α - β) .

Together with homogeneous stability, the spatial-instability region is

β > 1, 0 < λ < \frac{1}{β}, β < α < β + \frac{( 1 - λ β ) ^{2}}{4 λ} .

For fixed

β

, this is the region below the last curve, above

α = β

, and left of

λ = 1/ β

.

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