Movies of Linear Dispersion on a Periodic Domain


Linear evolution equation:     ut = L[u]    with dispersion relation   ω(k).

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

    Linearized RLW/BBM equation:      ω = k/(1 + k3/6)

    Starting at t = 0

     

    Starting at t = 1000 π

     

    Starting at t = 10000 π

     

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

    Starting at t = 0

     

    Starting at t = 1000 π

     

    Starting at t = 10000 π

     

    Water waves:      ω = √ k tanh k  

    Starting at t = 0

     

    Starting at t = 1000 π

     

    Square root dispersion:      ω = √ |k| 

     

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

    Starting at t = 0

     

    Starting at t = 1000 π

     

    Eleven Tenths dispersion:      ω = |k|11/10

    Starting at t = 0

     

    Starting at t = 1000 π

     

    Three halves dispersion:      ω = |k|3/2

     

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

    Rational time step

     

    Irrational time step

     

    Linearized Benjamin-Ono equation:      ω = k2 sign k

    Rational time step

     

    Irrational time step

     

    Five halves dispersion:      ω = |k|5/2

    Rational time step

     

    Irrational time step

     

    Linearized Korteweg--deVries equation:      ω = k - k3/6

    Rational time step

     

    Irrational time step