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Movies of Linear Dispersion on a Periodic Domain

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Linear evolution equation: u_{t} = L[u] with dispersion relation ω(k).

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Solution to the periodic initial-boundary value problem with step function initial data at t = 0.

For detailed explanation and discussion, see:

Chen, G., and Olver, P.J., Dispersion of discontinuous periodic waves, Proc. Roy. Soc. London A 469 (2012), 20120407.
pdf

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Linearized RLW/BBM equation: ω = k/(1 + k^{3}/6)

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Starting at t = 0

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Starting at t = 1000 π

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Starting at t = 10000 π

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Linearized regularized Boussinesq equation: ω = k /
√ 1 + k2/3

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Starting at t = 0

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Starting at t = 1000 π

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Starting at t = 10000 π

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Water waves: ω =
√ k tanh k

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Starting at t = 0

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Starting at t = 1000 π

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Square root dispersion: ω =
√ |k|

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Asymptotically linear dispersion: ω =
√ |k(1 + k)|

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Starting at t = 0

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Starting at t = 1000 π

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Eleven Tenths dispersion: ω = |k|^{11/10}

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Starting at t = 0

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Starting at t = 1000 π

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Three halves dispersion: ω = |k|^{3/2}

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Linearized Boussinesq equation: ω = k
√ 1 + k2/3

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Rational time step

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Irrational time step

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Linearized Benjamin-Ono equation: ω = k^{2} sign k

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Rational time step

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Irrational time step

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Five halves dispersion: ω = |k|^{5/2}

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Rational time step

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Irrational time step

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Linearized Korteweg--deVries equation: ω = k - k^{3}/6

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Rational time step

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Irrational time step