KdV Equation - weakly nonlinear weakly disspersive

[quid]

Hi guys,

I have been reading up on the KdV equation and everywhere it states that the waves need to be 'weakly nonlinear and weakly disspersive.' From a physical point of view does this mean that the wave amplitude must be small?

If that is the case how would one model large solitons numerically? Or would that be impossible?

Thanks in advance.

 

[Andy Resnick]

The blurb in Arnold's "Mathematical Methods of Classical Mechanics" notes that the equation first arose by considering waves in shallow water, but doesn't state any restrictions on the general solution.

Boyd's "nonlinear optics" has a section on solitions- solitions occur when the self-phase modulation cancels group-velocity dispersion-but again does not require anything to be 'small'.

What's your book?

 

[hunt_mat]

The KdV equation equation is derived under the assumption of shallow water and small amplitude.

 

[Bobbywhy]

The correct term for a wave which is localized and retains its form over a long period of time is: solitary wave. However, a soliton is a solitary wave having the additional property that other solitons can pass through it without changing its shape. But, in the literature it is customary to refer to the solitary wave as a soliton, although this is strictly incorrect. Solitons are stable, nonlinear pulses which exhibit a fine balance between non-linearity and dispersion.

The KdV equation is the most famous solitary wave equation which describes small amplitude shallow water waves in a channel. It is interesting to note that, a KdV solitary wave in water that experiences a change in depth will retain its general shape. However, on encountering shallower water its velocity and height will increase and its width decrease; whereas, on encountering deeper water its velocity and height will decrease and its width increases.

 

[hunt_mat]

It is possible to derive a KdV equation in 3D as well, this is usually called the KP equation.