Time dependence of perturbations

One possible result of our perturbation is that all the modes associated with every possible integer value of n in the above equations will decay. Then A - 0, Tn - 0 for all n as t - oo. In this case, the uniform state will be stable to all perturbations. 

The decay or possible growth of a given mode will be determined by the eigenvalues of the Jacobian matrix associated with the perturbation equations above these in turn are determined by the trace and determinant. For a given n the trace has the form 

We can recognize the first term as the trace of the matrix for the well-stirred system of chapter 4 (let us call this tr(U)) multiplied by the positive quantity y. We have specified that we are to consider here systems which have a stable stationary state when well stirred, i.e. for which tr(U) is negative. The additional term associated with diffusion in eqn (10.47) can only make tr(J) more negative, apparently enhancing the stability. There are no Hopf bifurcations (where tr(J) = 0) induced by choosing a spatial perturbation with non-zero n. 

Again the first term on the right-hand side can be recognized as being proportional to the determinant of the uniform system (now multiplied by y2), which was necessarily positive. Equation (10.48) has, however, extra terms which can cause the sign of det(J) to change, particularly if the ratio of the diffusivities P is large. 

If det(J) is positive for all n, then the amplitudes of all the components of any perturbation will decay back to the spatially uniform stationary state. As mentioned above, det(J) is positive for n = 0, and clearly will always be positive for sufficiently large n when the last term dominates. However, eqn (10.48) is a quadratic in n2 a completely stable uniform state arises if there are no real solutions to the condition det(J) = 0- We can write the 

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The first term in the right-hand side of Equation 6.128 is responsible for the propagation of the perturbation with the velocity V(1 — 2po(t,x)) along the axis x (ifpo(t.x) < 1/2) or in the opposite direction (if po(t.x) > 1/2). The second term is important for understanding the thermomigration-driven roughening. This means that the time dependence of perturbation amphtude should be exponential,... 

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