KdV equation

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This is the Korteweg de Vries equation for a quantity A(x,t) in physical form
A_t = -c A_x + \alpha A A_x + \beta A_{xxx}\,
where subscripts denote partial derivatives.

The parameter c denotes the linear advective wave speed, the parameter \alpha denotes the nonlinear effects while the parameter \beta denotes the dispersive effect. In practice these parameters depend on the physical situation considered.

To understand the meaning of the nonlinear and dispersive terms consider them one at a time.

  • The nonlinear term: A_t = -(c-\alpha A) A_x.\,
    • You can see that if A>0 and \alpha is negative then larger waves have faster propagation speeds.
  • The Fourier transform of the dispersive term: \bar{A}_t = -ik(c-\beta k^2) \bar{A}. (where k is the wavenumber)
    • Thus when \beta is positive shorter waves (with larger k) propagate slower.

The Wave Envelope page uses the KdV equation to illustrate the difference between group velocity and phase speed via wave envelopes.