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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.
