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Unformatted text preview: Secondary Instabilities and Stability Balloons To determine the physical relevance of the steady, nonlinear, spatiallyperiodic 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 timeperiodic 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
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