Solitary wave and similarity solutions of the combined KdV equation

1988 ◽  
Vol 9 (3) ◽  
pp. 311-316 ◽  
Author(s):  
Pan Xiu-de
Keyword(s):  
Solitary Wave ◽  
Kdv Equation ◽  
Similarity Solutions

Symmetry operators and exact solutions of a type of time-fractional Burgers–KdV equation

2019 ◽  
Vol 16 (02) ◽  
pp. 1950032 ◽  
Author(s):  
Azadeh Naderifard ◽  
S. Reza Hejazi ◽  
Elham Dastranj ◽  
Ahmad Motamednezhad
Keyword(s):  
Exact Solutions ◽  
Fourth Order ◽  
Vector Fields ◽  
Kdv Equation ◽  
Group Analysis ◽  
Invariant Subspaces ◽  
Similarity Solutions ◽  
Optimal System ◽  
Lie Point Symmetries ◽  
Symmetry Operators

In this paper, group analysis of the fourth-order time-fractional Burgers–Korteweg–de Vries (KdV) equation is considered. Geometric vector fields of Lie point symmetries of the equation are investigated and the corresponding optimal system is found. Similarity solutions of the equation are presented by using the obtained optimal system. Finally, a useful method called invariant subspaces is applied in order to find another solutions.

Similarity solutions of the super KdV equation

1995 ◽  
Vol 16 (9) ◽  
pp. 901-904 ◽  
Author(s):  
Yu Huidan ◽  
Zhang Jiefang
Keyword(s):  
Kdv Equation ◽  
Similarity Solutions ◽  
Super Kdv Equation

Dust acoustic solitary waves in non-thermal plasmas consisting of negatively charged dust grains and isothermal electrons

2009 ◽  
Vol 75 (4) ◽  
pp. 455-474 ◽  
Author(s):  
ANIMESH DAS ◽  
ANUP BANDYOPADHYAY
Keyword(s):  
Solitary Wave ◽  
Solitary Waves ◽  
Acoustic Waves ◽  
Kdv Equation ◽  
Thermal Parameter ◽  
Charged Dust ◽  
Average Temperature ◽  
The Kdv Equation ◽  
Dust Acoustic Solitary Waves ◽  
Dust Acoustic

AbstractA Korteweg–de Vries (KdV) equation is derived here, that describes the nonlinear behaviour of long-wavelength weakly nonlinear dust acoustic waves propagating in an arbitrary direction in a plasma consisting of static negatively charged dust grains, non-thermal ions and isothermal electrons. It is found that the rarefactive or compressive nature of the dust acoustic solitary wave solution of the KdV equation does not depend on the dust temperature if σdc < 0 or σdc > σd*, where σdc is a function of β1, α and μ only, and σd*(<1) is the upper limit (upper bound) of σd. This β1 is the non-thermal parameter associated with the non-thermal velocity distribution of ions, α is the ratio of the average temperature of the non-thermal ions to that of the isothermal electrons, μ is the ratio of the unperturbed number density of isothermal electrons to that of the non-thermal ions, Zdσd is the ratio of the average temperature of the dust particles to that of the ions and Zd is the number of electrons residing on the dust grain surface. The KdV equation describes the rarefactive or the compressive dust acoustic solitary waves according to whether σdc < 0 or σdc > σd*. When 0 ≤ σdc ≤ σd*, the KdV equation describes the rarefactive or the compressive dust acoustic solitary waves according to whether σd > σdc or σd < σdc. If σd takes the value σdc with 0 ≤ σdc ≤ σd*, the coefficient of the nonlinear term of the KdV equation vanishes and, for this case, the nonlinear evolution equation of the dust acoustic waves is derived, which is a modified KdV (MKdV) equation. A theoretical investigation of the nature (rarefactive or compressive) of the dust acoustic solitary wave solutions of the evolution equations (KdV and MKdV) is presented with respect to the non-thermal parameter β1. For any given values of α and μ, it is found that the value of σdc completely defines the nature of the dust acoustic solitary waves except for a small portion of the entire range of the non-thermal parameter β1.

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