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Unformatted text preview: Phys. 7268 Assignment 9 Due: 4/7/11 Problem 1 Long wavelength instabilities of stripe state in the amplitude equation formalism. It turns out that many calculations that could, in principle, be performed using the Galerkin expansion, simplify considerably in the amplitude equation formalism. One particular example is the stability calculations for the stripe state, which can be performed using the scaled version of the amplitude equation ∂ T ¯ A = ¯ A + ( ∂ X- i 2 ∂ 2 Y ) 2 ¯ A- | ¯ A | 2 ¯ A. (1) (a) Is it possible to analyze the cross-roll instability using the amplitude equation (1)? Explain why. (b) Find the amplitude of the nonlinear saturated stripe solution with dimensional wave number q = q ˆ x . Express the answer in terms of the scaled wave number deviation K =- 1 / 2 ξ ( q- q c ) and scaled coordinates X . (c) The stability of such stripe states can be tested by writing the amplitude in the form ¯ A ( X,Y,T ) = ¯ A K ( X ) + δ ¯ A ( X,Y,T ) . By linearizing (1) about ¯ A K derive the equation satisfied by small disturbances δ ¯ A . By careful, the amplitude is complex!...
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- Spring '11
- Wavelength, Fundamental physics concepts, Wavenumber, Wave vector, growth rate σK, amplitude equation formalism