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Unformatted text preview: 7224 Homework Solution 7 Problem 1 We need to find the saturated nonlinear states of the system described by the following PDE: ∂ t u = ru (1 + ∇ 2 ) 2 u + ∇ · [( ∇ u ) 2 ∇ u ] . (a) Linearizing about the uniform state u = 0 (we can drop the last term since it is cubic in u = δu ) we get ∂ t δu = rδu (1 + ∇ 2 ) 2 δu. Subsitituting an infinitesimal perturbation in the usual form δu = e σ q t e i q · x we obtain the same growth rate as in the SwiftHohenberg equation σ q = r (1 q 2 ) 2 . The neutral stability curve defined by setting σ q = 0 is r = (1 q 2 ) 2 . Its minimum r c = 0 is achieved at q c = 1. (b) The saturated stipe state with the periodicity we have just found is u = a cos x + ··· (we could have chosen to orient the stripes in any other direction as well). Plugging this Galerkin expansion into the original evolution equation we see that the last term gives us ∇ · [( ∇ u ) 2 ∇ u ] = ∂ 2 x u [( ∂ x u ) 2 + ∂ y u ) 2 ] + ∂ x u∂ x [( ∂ x u ) 2 + ∂ y u ) 2 ] + ∂ 2 y u [( ∂ x u ) 2 + ∂ y u ) 2 ] + ∂ y u∂ y [( ∂ x u ) 2 + ∂ y u ) 2 ] = ∂ 2 x u ( ∂ x u ) 2 + ∂ x u∂ x ( ∂ x u ) 2 = 3 a 3 sin 2 x cos x + ··· = 3 a 3 1 4 (cos x cos3 x ) + ··· . Collecting all terms proportional to cos x and ignoring all higher harmonics we get 0 = ra cos x 3 4 a 3 cos x + ··· (the lefthandside is zero because we are looking for a steady state). Setting the coefficient of cos x to zero ra 3 4 a 3 = 0 we get solutions for the amplitude of the saturated stripe state a = r 4 3 r. (c) Now we repeat this analysis for the square state u = a (cos x + cos y ) + ··· . Various terms in the original equation give 1 (1 + ∇ 2 ) 2 u = (1 + ∂ 2 x + ∂ 2 y ) 2 a (cos x + cos y ) + ··· = 0 + ··· , ∇ · [( ∇ u ) 2 ∇ u ] = a (2cos x + 2cos y ) a 2 (sin 2 x + sin 2 y ) + ··· = ( 3 4 a 3 1 2 a 3 )(cos x + cos y ) + ··· = 5 4 a 2 (cos x + cos y ) + ··· Collecting terms proportional to cos x and to cos y we see that they have the same coefficient (that shouldn’t surprise us – the original equation is invariant with respect to rotations, so x and y...
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This note was uploaded on 08/25/2011 for the course PHYS 7268 taught by Professor Staff during the Spring '11 term at Georgia Tech.
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