170mm x 244mm Collerton c10. tex V3-11/21/2014 5:28 P. M. Page 233 10. 3 CONDITIONS FOR THE LOSS OF STABILITY OF THE HOMOGENEOUS STATE 233 +𝛽∫R2𝑤lat(x-x′,𝜙)u(x′,𝜙)dx′] (10. 10) Because of translation symmetry the solutions of equation 10. 10 can be written in the form u(x,𝜙)=𝑣(𝜙-𝜑)eik⋅x+𝑣(𝜙-𝜑)e-ik⋅x(10. 11) where k=q(cos𝜑,sin𝜑)and 𝜆𝑣(𝜙)=-𝛼𝑣(𝜙)+f′𝜇[ ∫𝜋 0𝑤loc(𝜙-𝜙′)𝑣(𝜙′)d𝜙′ 𝜋+𝛽̃𝑤lat(k,𝜙+𝜑)𝑣(𝜙)] (10. 12) and̃𝑤(k,𝜙)isthe Fouriertransformof 𝑤lat(x,𝜙). Themainresultthatcanbederived from these formulae is that solutions take the form 𝑣(𝜙)⋅cos(k⋅x) in which (a) 𝑣(𝜙)=cos2𝜙,( b )𝑣(𝜙)=sin2𝜙,( c )𝑣(𝜙)=1. The first solution cos2𝜙⋅cos(k⋅x)generatesevencontouredplanforms,becausecos2 𝜙=cos(-2𝜙) is even, the second sin2 𝜙⋅cos(k⋅x)generatesoddcontoured planforms, because sin2𝜙=-sin(-2𝜙)is odd, and the third generates non-contoured planforms cos(k⋅x). Figure 10. 13 shows the dispersion curves (the eigenvalue 𝜆plotted as a function of the wave number q), derived from equation 10. 12. It will be seen that theinstabilityfirstoccursatthelowestvalueof 𝜆,forevenplanforms. [Thisoccurs in case there is some scatterin the lateral connectivity 𝑤lat(x,𝜙). ] 0. 98 odd even qcλ 0. 96 0. 94 0. 92 0. 90 12345q Figure 10. 13 Dispersion curves 𝜆(q)obtained from equation 10. 12 giving the conditions for marginal stability. The critical wavenumber at which spontaneous pattern formation occurs is q=qc.