KdV equation

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.