The Family of the KdV Equations

Abstract

The ubiquitous Korteweg de-Vries (KdV) equation [14] in dimensionless variables reads

$$ u_t+ auu_x+ u_{xxx}= 0, $$

(13.1)

where subscripts denote partial derivatives. The parameter a can be scaled to any real number, where the commonly used values are a=±1 or a=±6. The KdV equation molels a variety of nonlinear phenomena, including ion acoustic waves in plasmas, and shallow water waves. The derivative u _t characterizes the time evolution of the wave propagating in one direction, the nonlinear term uu _x describes the steepening of the wave, and the linear term u _xxx accounts for the spreading or dispersion of the wave. The KdV equation was derived by Korteweg and de Vries to describe shallow water waves of long wavelength and small amplitude. The KdV equation is a nonlinear evolution equation that models a diversity of important finite amplitude dispersive wave phenomena. It has also been used to describe a number of important physical phenomena such as acoustic waves in a harmonic crystal and ion-acoustic waves in plasmas. As stated before, this equation is the simplest nonlinear equation embodying two effects: nonlinearity represented by uu _x, and linear dispersion represented by u _xxx. Nonlinearity of uu _x tends to localize the wave whereas dispersion spreads the wave out. The delicate balance between the weak nonlinearity of uu _x and the linear dispersion of u _xxx defines the formulation of solitons that consist of single humped waves. The stability of solitons is a result of the delicate equilibrium between the two effects of nonlinearity and dispersion.