Phys. 7224 Midterm exam 3/11/03
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∂xuwith san arbitrary constant:
∂tu=ru +s∂xu−(1 + ∂2
x)2u−u3.
(a) Compute the growth rate and determine the type of instability (i.e., Is, IIIo, etc.) of the trivial solution
u= 0 which arises for different values of parameter s. Furthermore, determine the coherence length ξc, the
characteristic timescale τand, if the instability is oscillatory, the characteristic frequency ωc.
(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
two solutions are stable when the trivial solution is destabilized, explain why they will or will not be selected
as opposed to a nonuniform pattern.
Problem 2
Slime mold aggregation. The amoebae of the slime mold Dictyostelium discoideum secrete a chemical attractant,
Cyclic-AMP, and spatial aggregations of amoebae start to form. One of the models for this process gives rise to
the system of equations for the densities of amoebae and C-AMP, n(x, t) and a(x, t), respectively, which in their
one-dimensional form, are
∂tn=−χ∂x(n∂xa) + Dn∂2
xn,
∂ta=hn −ka +Da∂2
xa,
where χ,h,k, and the diffusion coefficients Dnand Daare all positive constants.
(a) Nondimensionalize the equations to reduce the number of parameters to a minimum.
(b) Examine the linear stability about the steady state (which introduces a further parameter into the problem) by
calculating the growth rate σqin a laterally infinite system. What types of linear instabilities are possible in
this system for different choices of parameters?
(c) If one considers a laterally finite system with no-flux boundary conditions, what is the minimal size of the system
for which the heterogeneous pattern becomes possible for given nondimensional parameters?
(d) Compare the critical wavelength of the patterns which arise in a laterally infinite system in the process of
natural evolution characterized by gradual increase in χ(with all other parameters fixed) with those that arise
in a system whose size is gradually increased (again holding all other parameters fixed).
(a) Now go back to the dimensional equations and explain the meaning of all the terms in the equations and all
parameters. Explain intuitively how spatial aggregation takes place.