Copyright © 2016 S. Subhaschandra Singh. This is an open access article distributed under the Creative Commons Attribution License, which
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International Journal of Physical Research, 4 (2) (2016) 37-42
International Journal of Physical Research
Website: www.sciencepubco.com/index.php/IJPR
doi: 10.14419/ijpr.v4i2.6202
Research paper
Solutions of Kudryashov - Sinelshchikov equation and
generalized Radhakrishnan-Kundu-Lakshmanan
equation by the first integral method
S. Subhaschandra Singh *
Department of Physics, Imphal College, Imphal, Manipur, India
*Corresponding author E-mail: subhasic@yahoo.co.in
Abstract
This paper shows the applicability of the First Integral Method in obtaining solutions of Nonlinear Partial Differential Equations
(NLPDEs). The method is applied in constructing solutions of Kudryashov-Sinelshchikov equation (KSE) and Generalized Radhakrish-
nan-Kundu-Lakshmanan Equation (GRKLE). The First Integral Method, which is based on the Ring Theory of Commutative Algebra, is
a direct algebraic method for obtaining exact solutions of NLPDEs. This method is applicable to integrable as well as nonintegrable
NLPDEs. The method is an efficient method for obtaining exact solutions of many Nonlinear Evolution Equations (NLEEs).
Keywords: Division Theorem; First Integral Method; Generalized Radhakrishnan - Kundu- Lakshmanan Equation (GRKLE); Kudryashov – Sinelshchikov
Equation (KSE); NLEEs; Optical Solitons.
1. Introduction
Nonlinear Evolution Equations (NLEEs) are frequently encoun-
tered in the study of many complex phenomena in various branch-
es of physics such as Biophysics, Condensed Matter Physics, Flu-
id Physics, Neurophysics, Nonlinear Optics, Particle Physics,
Plasma Physics, Quantum Field Theory, etc. and in many other
branches of science such as Ecology, Physiology etc. as well as in
Economics and Social Science. In the recent a few decades, quite
a number of methods had been suggested so far for finding exact
solutions of NLEEs in Mathematical Physics and the First Integral
Method is one of them. The first integral method, which is based
on the ring theory of commutative algebra, was first proposed by
Z.S. Feng [1] and was further developed by himself [2 - 4]. This
method has been applied by many authors in solving different
types of NLEEs encountered in science and engineering [5 – 18].
In the present paper, the first integral method is applied in solving
the Kudryashov-Sinelshchikov Equation (KSE) and the General-
ized Radhakrishnan- Kundu-Lakshmanan Equation (GRKLE).
The rest of the article is arranged as in the following. In section 2,
a brief description of the method is presented. In section 3, the
method is applied in obtaining the solutions of KSE [19 – 22] and
GRKLE [23-25]. In section 4, a brief conclusion is presented.
2. The first integral method
Let us consider a general NLPDE in the form
(,
,
,
,
,
,
,...)= 0, (1)
where = (, ) is its solution, and represent the spatial
and the temporal variables.
Let us introduce the transformations,
= (, ) = (), = − , (2)
where v is a constant to be determined latter.
Now, we have,
(. ) =
(. ),
(. ) = −
(. ),
2
2
(. ) =
2
2
(. ) ,
2
(.)= −
2
2
(. ), . (3)
Using equations (3), we transform the NLEE (1) into a nonlinear
ordinary differential equation (NLODE) of the form,
(,
′
,
′′
,
′′′
,...) =0, (4)
where the primes denote derivatives with respect to the same vari-
able (ξ) such that
′
=
,
′′
=
2
2
, .
Let us suppose that the solution of the Non Linear Ordinary Dif-
ferential Equation (NLODE) (4) can be expressed as
(, ) = () = (). (5)
We further introduce the following new independent variables
() = (), () =
′
() =
=
, (6)
leading to a system,
′
() = (),
′
() = ( (), ()). (7)