Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 355 / 2 / b

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- Solution b

Solution

To first order about the fixed point,

u_{t} = D_{n} u_{xx} - χ n_{*} v_{xx} - δ u, v_{t} = D_{c} v_{xx} + αu - β v .

For a mode proportional to

e^{σ t + ik x}

, the linear matrix is

M (k) = (- δ - D_{n} k^{2} α χ n_{*} k^{2} - β - D_{c} k^{2}) .

Its trace is always negative, so instability occurs exactly when its determinant is negative. Writing

q = k^{2}

,

det M = D_{n} D_{c} q^{2} + (δ D_{c} + β D_{n} - \frac{α χ γ}{δ}) q + δ β .

At onset this quadratic touches zero, requiring

χ_{c} = \frac{δ}{α γ} (δ D_{c} + β D_{n})^{2} .

The repeated positive root is

q_{c} = \frac{δ β}{D _{n} D _{c}},

and therefore

k_{c} = (\frac{δ β}{D _{n} D _{c}})^{1/4} .

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