Nonlinear Phenomena in Plasma

**Chapter 6**

**Abstract**

**1. Introduction**

**111**

ordinary differential equations.

Approximate Analytical Solution

of Nonlinear Evolution Equations

*Laxmikanta Mandi, Kaushik Roy and Prasanta Chatterjee*

Analytical solitary wave solution of the dust ion acoustic waves (DIAWs) is studied in the frame-work of Korteweg-de Vries (KdV), damped force Korteweg-de Vries (DFKdV), damped force modified Korteweg-de Vries (DFMKdV) and damped forced Zakharov-Kuznetsov (DFZK) equations in an unmagnetized collisional dusty plasma consisting of negatively charged dust grain, positively charged ions, Maxwellian distributed electrons and neutral particles. Using reductive perturbation technique (RPT), the evolution equations are obtained for DIAWs.

In the field of physics and applied mathematics research getting an exact solution of a nonlinear partial differential equation is very important. The elaboration of many complex phenomena in fluid mechanics, plasma physics, optical fibers, biology, solid-state physics, etc. is possible if analytical solutions can be obtained. Most of the differential equation arises in these field has no explicit solution as popularly known. This problem creates hindrances in the study of nonlinear phenomena and makes it time-consuming in the research of nonlinear models in the plasma and other science. However recent researches in nonlinear differential equations have seen the development of many approximate analytical solutions of partial and

The history behind the discovery of soliton is not only interesting but also significant. In 1834 a Scottish scientist and engineer—John Scott-Russell first noticed the solitary water wave on the Edinburgh Glasgow Canal. In 1844 [1] in "Report on Waves" he accounted his examinations to the British Association. He wrote "I was observing the motion of a boat which was rapidly drawn along a narrow channel by a pair of horses, when the boat suddenly stopped not so the mass of water in the channel which it had put in motion; it accumulated round the prow of the vessel in a state of violent agitation, then suddenly leaving it behind, rolled forward with great velocity, assuming the form of a large solitary elevation, a rounded, smooth and well defined heap of water, which continued its course along the channel apparently without change of form or diminution of speed. I followed it on horseback and overtook it still rolling on at a rate of some eight or nine miles an hour, preserving its original figure some thirty feet long and a foot to a foot and a half in height. Its height gradually diminished and after a chase of one or two miles I lost it in the windings of the channel. Such in the month of August 1834 was my first

**Keywords:** solitary wave, soliton, KdV, DKdV, DFZK
