Exact solutions of cylindrical and spherical dust ion acoustic waves

Abstract

Cylindrical and spherical modified Korteweg–de Vries (KdV) equations are derived for dust ion acoustic waves. It is shown that a suitable coordinate transformation reduces the cylindrical KdV equation into the ordinary KdV equation which can be solved analytically. A completely different analytical solution is obtained using the group analysis method. However, for cylindrical and spherical modified KdV equations group analysis method yields trivial analytical solutions. Numerically, solutions to these modified KdV equations are obtained assuming initial profiles similar to those in one-dimensional soliton solutions.

A note on the cylindrical solitary waves in an electron-acoustic plasma with vortex electron distribution

“…These expressions are formally the same with those of Equation (18). To obtain an additional equation for aðsÞ we shall use (23), but this equation cannot be satisfied point by point.…”

“…In this sub-section, we shall seek a progressive wave solution to the modified cylindrical KdV equation given in (18). As is well-known the planar form of the modified KdV equation for such a plasma has the form 5,9…”

“…[11][12][13][14][15][16][17][18]. In all these works, the nonlinearities are of integer order, and then, the cylindrical KdV equation can be reduced to the conventional KdV equation through the use of some transform techniques.…”

“…where g is the function which represents the right-hand-side of the Equation (18). Starting with the numerical values of the functions at an arbitrary time step n, the numerical values for the next time step n þ 1 can be written as…”

Spherical and Cylindrical Ion Acoustic Solitary Waves in Electron-Positron-Ion Plasmas with Non-Maxwellian Electrons and Positrons

“…To obtain an analytic solution of cylindrical KdV equation, let us first use Hirota's transformation [65] given by…”