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====== IAM 950 HW1 ======

=====Problem 1=====
 In class we derived via Taylor expansion the following approximation 
for the exponential growth rate $\sigma$ of a sinusoidal perturbation of wavenumber 
$q$ for a Type I-s instability, near the critical wavenumber ($q \approx q_c$), and close 
to onset of instability ($|\epsilon| = |p - p_c| << 1$).

<latex>
\sigma_{q}(\epsilon) \approx \frac{1}{\tau_0} \left[ \epsilon - \xi_0^2 (q -q_c)^2 \right]
</latex>

Here $\tau_0$ and $\xi_0$ 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 $\tau_0$ and $\xi_0$ 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.

=====Problem 2=====

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 $[0, 2\pi]$ and with $0<r<<1$,
expand the unknown field $w(x,t)$ in Fourier modes

<latex>
w(x,t) = \sum_k a_k(t) e^{ikx}
</latex>

Substitute into the PDE, exploit orthogonality of the Fourier basis, and truncate to four modes $a_0, a_1, a_2, a_3$
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 $x$ and use a cosine Fourier expansion, as we did in class, or use a complex Fourier basis 
as written above to represent $w(x,t)$ at arbitrary phase. In the latter case you will need to include the
complex conjugates of $a_0, a_1, a_2, a_3$ in the expansion.

**(b)** Show that the equations for $a_0$ and $a_2$ decouple, leaving a 2d system
of ODEs in just $a_1$ and $a_3$.

**%%(c)%%** Use Center Manifold Reduction to derive an algebraic model for $a_3$ in terms of $a_1$, and 
then use that result to form a reduced-order nonlinear evolution equation for $a_1$ 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 $r=1/8$. For each model and the PDE simulation,
produce phase plots of $a_3(t)$ versus $a_1(t)$ for a handful of initial conditions scattered in the 
$a_1, a_3$ plane. (If you used the complex Fourier representation, plot $Re a_3$ versus $Re a_1$ 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 $r$? Assume that the PDE simulation gives an accurate numerical solution of the Swift-Hohenberg 
equation. Using the fixed initial condition $w(x,0) = 0.1 cos x + 0.1 cos 3x$, produce a log-log plot 
of asymptotic error 

<latex>
err = \lim_{t \rightarrow \infty} \sqrt{ \int_0^{2\pi} |\hat{w}(x,t) - w(x)|^2 dx}
</latex>

versus $r$ where $w(x)$ is the asymptotic state of the PDE simulation and $\hat{w}$ 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 $r$. I suggest 
using $r = 1/16, 1/8, 1/4, 1/2,$ and $1$. 

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.
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