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midterm - Phys 7268 Midterm exam Due Problem 1...

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Phys 7268 Midterm exam Due: 3/10/11 Problem 1 Generalizations of Swift-Hohenberg equation. The form of the Swift-Hohenberg equation is determined by the symmetries of the system, such as translational and rotational symmetries. For instance, breaking the parity symmetry leads to an extra term s∂ x u with s an arbitrary constant: t u = ru + s∂ x u - (1 + 2 x ) 2 u - u 3 . (a) Compute the growth rate and determine the type of instability (i.e., I s , III o , etc.) of the trivial solution u = 0 which arises for different values of parameter s . Furthermore, determine the coherence length ξ c , the characteristic timescale τ and, if the instability is oscillatory, the characteristic frequency ω c . (b) By substituting the growth rate into the linear solution determine the meaning of the new term. Can you show that by a suitable change of variables the above equation can be reduced to a standard Swift-Hohenberg equation? (b) Determine the stability of the two nontrivial uniform solutions to the above equation. If it happens that these
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