The Generalized Equal Width Equation

The equal width (EW) equation for long waves propagating in the positive x-direction, has the form 0    xxt x t u uu u   (6.1.1) where  and  are positive constants, which require the boundary conditions 0  u as   x .The EW equation is a model nonlinear partial differential equation for the simulation of one-dimensional wave propagation in nonlinear media with dispersion process. Solitary waves are wave packets or pulses which propagate in nonlinear dispersive media. Due to balance between the nonlinear and dispersive effects these waves retain a stable waveform. A soliton is a very special type of solitary wave which also keeps its wave form after collision with other solitons. Peregrine [74] was the first to derive the regularized long wave (RLW) equation to model the development of an undular bore. Later on, Benjamin et al. [9] proposed the use of RLW equation as preferred alternative to the more classical Korteweg-de Vries (KdV) equation to model a larger class of physical phenomena. The EW equation, which is less well-known and proposed by Morrison et al. [71], is an alternative description to the more usual KdV equation and RLW equation. It has been shown to have solitary wave solution and govern a large number of important physical phenomena such as the nonlinear transverse waves in shallow water, ion-acoustic and magneto-hydrodynamic waves in plasma and phonon packets in nonlinear crystals. Indeed, the EW equation is a special case of the generalized equal width (GEW) equation of the form 0    xxt x p t u u u u   (6.1.2)