# The Family of the KdV Equations

• Abdul-Majid Wazwaz
Chapter
Part of the Nonlinear Physical Science book series (NPS)

## Abstract

The ubiquitous Korteweg de-Dries (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.

## Keywords

Solitary Wave Soliton Solution Travel Wave Solution mKdV Equation Kink Solution
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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