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Unformatted text preview: Phys. 7268 Assignment 6 Due: 3/10/11 Problem 1 Nonlinear saturation in the Swift-Hohenberg equation. Consider the Swift-Hohenberg equation ∂ t u = ru- (1 + ∂ 2 x ) 2 u- u 3 (1) (a) Repeat the calculation for the nonlinear saturated stripe state using Galerkin expansion truncated to one mode for a general wave vector, i.e., calculate a q in the expression u = a q cos qx + ··· (Note: it is helpful to understand how large q- q c is compared to r .) (b) Repeat the calculation for the nonlinear saturated stripe state using Galerkin expansion truncated to three modes for the critical wave vector q = q c = 1, i.e., calculate a 1 , a 3 and a 5 in the expression u = a 1 cos x + a 3 cos3 x + a 5 cos5 x + ··· In principle we could immediately substitute this expansion into the Swift-Hohenberg equation with ∂ t u = 0. Setting the coefficients of three lowest order harmonics to zero would then provide the equations that can be solved for a 1 through a 5 . However, analytical solution of this system is difficult due to the unknown relative....
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- Spring '11
- Calculation, Expression, Bifurcation theory, Pitchfork bifurcation, Fmk