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1. Consider ut = uxxx and let the initial conditions consist of two narrow wavepackets. The first packet starts at x1 = 0 and has wavenumber k1 = 2 and the second starts at x2 = 10 and has wavenumber k2 = 1. When and where will their paths cross? 2. Let f(x) be a smooth function with Fourier transform ˆf(k). What is the Fourier transform of the scaled function f(x/σ), where σ > 0 is a real scaling parameter? 3. Let f(x) be a smooth function and consider the solution u(x, t) = 1 2π Z +∞ −∞ exp(i[kx − ω(k)t]) ˆu0(k)dk for some dispersion function ω(k) to the initial-value problem in which uˆ0(k) = F T{exp(ik0x)f(x/σ)} with k0σ  1. We did this in class with f(x) being a Gaussian, so we had a wavepacket. Replicate the argument in class for this general f(x) and derive that u(x, t) ≈ exp(i[k0x − ω0t])f([x − cgt]/σ) where the constants ω0 and cg are to be related to the dispersion function ω(k). What is the interpretation of previous expression? 4. Consider a dispersion relation with two real roots for the frequency that are equal and opposite, i.e. ω1,2 = ±ω(k) for some function ω(k). Show that the solution to the initial-value problem with u(x, 0) = A(x) and ut(x, 0) = B(x) can be written as u(x, t) = 1 2π Z +∞ −∞ exp(ikx) " Aˆ(k) cos ω(k)t + Bˆ(k) ω(k) sin ω(k)t # dk.

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Applied Differential Equations
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