KdV equation

This is the Korteweg de Vries equation in physical form

$A_{t}=-cA_{x}+\alpha AA_{x}+\beta A_{xxx}$

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. first the nonlinear term

$A_{t}=-(c-\alpha A)A_{x}.$

You can see that if alpha is negative then larger waves have faster propagation speeds. For the dispersive we need to Fourier transform to see that

${\bar {A}}_{t}=-ik(c-\beta k^{2}){\bar {A}}.$

where k is the wavenumber. Thus when beta is positive shorter waves (with larger k) propagate slower.