KdV and mKdV Equations of Higher-orders

Abstract

It is well known that the third order KdV equation is the generic model for studying weakly nonlinear waves. The equation models surface waves with small amplitude and long wavelength on shallow water. The KdV equation involves a balance between weak nonlinearity and linear dispersion. The KdV equation is completely integrable and the collision between solitary waves is elastic, which means that the solitons retain original identity after collision.