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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 $a_1$
to the first, and then use that result to form a reduced-order nonlinear evolution equation
for $a_1$ alone. 

**(d)** Use a numerical ODE integration routine to integrate your ODE models from (a), (b), and %%(c)%% 
for the initial condition $w(x,0) = 0.1 cos x$ and compare them to the numerical solution produced 
by the time-integration code for the PDE from problem 1. We are interested in the long-term behavior, 
so for the comparison, define the error of an ODE model as 

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

Produce a log-log plot of this error measure versus $r$ for each of the three ODE models. 
I suggest using $r = 1/32, 1/16, 1/8, 1/4, 1/2,$ and $1$.
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