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Unformatted text preview: Secondary Instabilities and Stability Balloons To determine the physical relevance of the steady, nonlinear, spatially-periodic states formed above the instability of the spatially uniform state we must in turn test their stability against small perturbations. For a translationally invariant state the ansatz for the stability analysis was a simple sinusoidal mode. The generalization to the case of a (real) base state that is spatially periodic in the extended coordinates is a perturbation of the form where the function u 1 , that depends on both the wave vector q of the base pattern and the wave vector Q of the perturbation, has the same spatial periodicity with respect to as the base solution u q . This form of the perturbation about a periodic state was introduced by Felix Bloch in the context of quantum mechanics of electrons in a lattice, and is analogous to Floquet theory for the stability of time-periodic states. For a stripe state, if we take the x axis along the wave vector of the original pattern (i.e., perpendicular to the stripes), we may restrict the range of consideration for Q x to since any perturbation wave vector outside of this range is equivalent to some wave vector in this range by redefining u 1 . (For example, a wave vector with Q x = 2 q/ 3 can be expressed instead as a perturbation with wave vector Q x = q/ 3 with the function u 1 changed to which has the same periodicity as u 1 ). In signal processing, a similar restriction of the range of frequencies in the power spectrum analysis is known as the Nyquist range , and in the study of waves in periodic structures in solid state physics the range is known as the first Brillouin zone . ) , ( z x u q x , ) , ( ) , ( 1 t i e e z Q q x Q x u u = x 2 / 2 / q Q q x &lt; &lt; iqx e 1 1 u u = We can then identify different classes of instabilities according to the values of the wave vector Q :- Q may be strictly zero, when the perturbation does not change the basic spatial periodicity of the pattern.- Q x may be q/ 2 (i.e., Q is on the zone boundary in the language of solid state physics), so that the spatial period in the x direction is doubled at the instability, and Q y may be zero (no structure in the y direction) or have a nonzero value.- The instability may occur at long wavelength- Q may take on a general value, with Q x incommensurate with the base wave number. In addition there may be different instability types characterized by different behavior under discrete symmetries . For example, if the dynamical equations and base state are unchanged by the parity transformation then perturbations with Q x = 0 or Q x = q/ 2 can be classified according to the signature of u 1 (even or odd) under this transformation. (A general Q x eliminates the symmetry, and u 1 does not reflect the parity symmetry for these values.) Finally, the instability may be stationary or oscillatory ....
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- Spring '11