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# hwk8 - Phys 7268 Assignment 8 Due You are highly encouraged...

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Phys. 7268 Assignment 8 Due: 3/31/11 – You are highly encouraged to use Maple or Mathematica to do these problems. – You are welcome (and encouraged) to use and modify the Maple program(s) posted on the class homepage. Problem 1 Quintic nonlinearity. Consider a variation of the Swift-Hohenberg equation with a quintic nonlinearity t u = ru - (1 + 2 ) 2 u - u 5 , where u ( x, y, t ) is a real field in a two-dimensional space. (a) Conduct the linear stability analysis and determine the type of the primary instability. (b) The next several questions consider base patterns with wave numbers equal to the critical one, q = q c . Find the nonlinear saturated stripe solution. You will need to determine which harmonics will be present in the Galerkin expansion and find the coefficients of the three leading terms. (c) Find the nonlinear saturated square solution. You may stop after you find the leading order term in the Galerkin expansion. (d) Find the nonlinear saturated hexagon solution. You may stop after you find the leading order term in the Galerkin expansion. (e) Determine the linear stability of the stripe state with respect to a cross-roll instability where the new set of stripes can have arbitrary orientation and wave number.

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