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

For $q=k^2$, the mode matrix is $J-q\mathrm{diag}(D_1,D_2)$. Its trace remains negative. Its determinant isIt is negative for some $q>0$ exactly whenTogether with homogeneous stability, the spatial-instability region isFor fixed $\beta$, this is the region below the last curve, above $\alpha =\beta$, and left of $\lambda =1/\beta$.