 Solution

ID: past-exam-of-the-topics-in-applied-mathematics-course-of-the-university-of-oxford/2007/a/4/iii/solution

The cosine modes satisfy both zero-flux endpoint conditions and

\partial_{x}^{2} cos (k_{n} x) = - k_{n}^{2} cos (k_{n} x)

. Linearizing the reaction terms gives

f_{a} δ a + f_{b} δ b

and

g_{a} δ a + g_{b} δ b

. Cancelling the common mode factor yields

λ (b _{n} a _{n}) = [J - (k_{n}^{2} 0 0 D k_{n}^{2})] (b _{n} a _{n}), k_{n} = nπ / L .

The derivatives are exactly the entries of the matrix computed in (ii). The zero mode reproduces the stable homogeneous perturbation; positive integers give the possible spatial instabilities.

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