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mdtrm_sln

# mdtrm_sln - Problem 1(a Linearization is trivial(just like...

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Problem 1 (a) Linearization is trivial (just like that of the standard Swift-Hohenberg equation) and we get so we immediately get the growth rate of an infinitesimal solution δ u = e iqx e σ t . For u b = 0 the real part of the growth rate is the same as for the standard SHE: so expanding it about the bifurcation point r c = 0, q c = 1 (i.e., it’s a type-I instability) we get Since the bifurcation occurs at r c = 0 we can just take ε = r , which immediately determines the characteristic time scale τ = 1 and correlation length ξ c = 2 by comparing with the canonical expression for a type-I instability. For nonzero s , the instability is oscillatory, i.e., type-I o with critical frequency ω c = Im σ q | q = 1 = s . (b) Substituting the growth rate into the solution of the linearized equation we get which by a change of variables reduces to the corresponding expression for a standard SHE. This change is equivalent to a translation of the growing pattern with speed s ! It is trivial to check that this change of variables reduces the modified SHE to the standard form. , 3 ) 1 ( 2 2 2 u u u u s u r u b x x t δ δ δ δ δ + + = isq q u r b q + = 2 2 2 ) 1 ( ) 3 ( σ , ) 1 ( Re 2 2 q r q q = = σ γ . ) 1 ( 2 ) 1 ( ) 1 ( 2 2 2 2 L + = + = q r q q r q γ ] ) ) 1 ( exp[( )] ( exp[ 2 2 t q r st x iq e e u t iqx + = = σ δ st x y x + =

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