In class we derived via Taylor expansion the following approximation for the exponential growth rate of a sinusoidal perturbation of wavenumber for a Type I-s instability, near the critical wavenumber (), and close to onset of instability ().
Here and are system-dependent constants in units of time and length respectively.
(a) Determine these constants for the 1d Swift-Hohenberg equation.
(b) Cross and Greenside describe and as characteristic time and length scales of the instability. What observable characteristics of the Swift-Hohenberg equation do these time and length scales govern?
(c) Adapt the time-integration code for the Kuramoto-Sivashisky equation to Swift-Hohenberg, and use it to verify your answers to (b) with numerics.
Derive the reduced-order ODE model for the Swift-Hohenberg equation just above threshhold and at critical wavenumber and compare its behavior to numerical simulations of the PDE, in the following steps:
(a) Starting from the Swift-Hohenberg PDE on the periodic domain and with , expand the unknown field in Fourier modes
Substitute into the PDE, exploit orthogonality of the Fourier basis, and truncate to four modes to obtain a system of four ODEs in the four coefficients (class notes 2012-02-08). You can fix the phase to be even in and use a cosine Fourier expansion, as we did in class, or use a complex Fourier basis as written above to represent at arbitrary phase. In the latter case you will need to include the complex conjugates of in the expansion.
(b) Show that the equations for and decouple, leaving a 2d system of ODEs in just and .
(c) Use Center Manifold Reduction to derive an algebraic model for in terms of , and then use that result to form a reduced-order nonlinear evolution equation for alone. What is the long-term stable equilibrium state predicted by the reduced-order model?
(d) Use a numerical ODE integration routine to integrate your ODE models from (b) and (c) and the time-integration code from problem 1 for the PDE, for . For each model and the PDE simulation, produce phase plots of versus for a handful of initial conditions scattered in the plane. (If you used the complex Fourier representation, plot versus and choose real-valued initial conditions.) Plot the approximation of the center manifold on the phase plane as well. You should see rapid approach to the center manifold followed by slow evolution on it.
(e) How accurate are the ODE models and the reduced-order equilibrium in the long term, as a function of ? Assume that the PDE simulation gives an accurate numerical solution of the Swift-Hohenberg equation. Using the fixed initial condition , produce a log-log plot of asymptotic error
versus where is the asymptotic state of the PDE simulation and is first the the ODE model from (b), second the reduced-order model from (c), and third the reduced-order equilibrium. Plot these as three lines in log-log plot of error versus . I suggest using and .
Note that the ODE systems for (b) and (c) will be stiff, in that the high-order coefficients evolve very rapidly until the system equilibrates to and moves slowly on the center manifold. You might need to use a stiff ODE integrator instead of the classic explicit schemes like RK4.