Dispersion Effect on Traveling Wave Solution of K-dV Equation

Abstract

All solutions of the Korteweg-de Vries(K-dV) equation that are bounded on the real line are physically relevant, de pending on the application area of interest. Usually, both analytical and numerical approaches consider solution profiles that are either spatially localized or (quasi) periodic. The development of numerical techniques for obtaining approxi mate solution of partial differential equations has very much increased in the finite element and finite difference meth ods. Recently, new auxiliary equation method introduced by PANG, BIAN and CHAO is applied to the analytical solu tion of K-dV equation and wavelet methods are applied to the numerical solution of partial differential equations . Pio neer works in this direction are those of Beylkin, Dahmen, Jaffard and Glowinski, among others. In this research we employ the new auxiliary equation method to obtain the effect of dispersion term on travelling wave solution of K-dV and their numerical estimation as well. Our approach views the limit behavior as an invariant measure of the fast motion drifted by the slow component, where the known constants of motion of the fast system are employed as slowly evolv - ing observables; averaging equations for the latter lead to computation of the characteristic features of the motion.