Problem 1(a) Linearization is trivial (just like that of the standard Swift-Hohenberg equation) and we getso we immediately get the growth rateof an infinitesimal solution δu= eiqxeσt. For ub= 0 the real part of the growth rate is the same as for the standard SHE:so expanding it about the bifurcation point rc= 0, qc= 1 (i.e., it’s a type-I instability) we getSince the bifurcation occurs at rc= 0 we can just take ε= r, which immediately determines the characteristic time scale τ= 1 and correlation length ξc= 2 by comparing with the canonical expression for a type-I instability. For nonzero s, the instability is oscillatory, i.e., type-Iowith critical frequency ωc= Imσq|q = 1= s.(b) Substituting the growth rate into the solution of the linearized equation we getwhich 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(222uuuusurubxxtδδδδδ−∂+−∂+=∂isqqurbq+−−−=222)1()3(σ,)1(Re22qrqq−−==σγ.)1(2)1()1(2222L+−−=+−−=qrqqrqγ]))1(exp[()](exp[22tqrstxiqeeutiqx−−+==σδstxyx+=→
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