A modified pattern-forming equation with a stronger nonlinear saturation is _t u = ru…

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Thermal, Statistical & Complex Systems · Fourier analysis and integral transforms

Many Parts, One Behaviour, Ch Pattern formation: instabilities that choose a wavelength

Problem

A modified pattern-forming equation with a stronger nonlinear saturation is ∂tu=ru−(1+∂x2)2u−2u3,\partial_t u = ru - (1+\partial_x^2)^2u - 2u^3, with u(x,t)u(x,t) the pattern field and rr the control parameter (dimensionless units). ((a)) Following the same projection method used for the ordinary Swift-Hohenberg equation, substitute u≈A(t)cos⁡xu \approx A(t)\cos x (the critical mode, qc=1q_{\rm c}=1), use cos⁡3x=34cos⁡x+14cos⁡3x\cos^3x = \tfrac34\cos x + \tfrac14\cos3x, and match the coefficient of cos⁡x\cos x to derive the amplitude equation for A(t)A(t). ((b)) Find the saturated amplitude A∗(r)A^*(r) and evaluate it at r=0.12r=0.12.

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