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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.

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