Phys 7268Midterm examDue:3/10/11Problem 1Generalizations of Swift-Hohenberg equation.The form of the Swift-Hohenberg equation is determined bythe symmetries of the system, such as translational and rotational symmetries.For instance, breaking the paritysymmetry leads to an extra terms∂xuwithsan arbitrary constant:∂tu=ru+s∂xu-(1 +∂2x)2u-u3.(a) Compute the growth rate and determine the type of instability (i.e., Is, IIIo, etc.)of the trivial solutionu= 0 which arises for different values of parameters.Furthermore, determine the coherence lengthξc, thecharacteristic timescaleτand, if the instability is oscillatory, the characteristic frequencyωc.(b) By substituting the growth rate into the linear solution determine the meaning of the new term.Can youshow that by a suitable change of variables the above equation can be reduced to a standard Swift-Hohenbergequation?(b) Determine the stability of the two nontrivial uniform solutions to the above equation. If it happens that these
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