Nonlinear Dyn
DOI 10.1007/s11071-017-3749-x
ORIGINAL PAPER
An extended Korteweg–de Vries equation: multi-soliton
solutions and conservation laws
Yakup Yıldırım · Emrullah Ya¸ sar
Received: 13 March 2017 / Accepted: 13 August 2017
© Springer Science+Business Media B.V. 2017
Abstract In this paper, we consider an extended
KdV equation, which arises in the analysis of sev-
eral problems in soliton theory. First, we converted
the underlying equation into the Hirota bilinear form.
Then, using the novel test function method, abun-
dant multi-soliton solutions were obtained. Second,
we have performed some distinct methods to extended
KdV equation for getting some exact wave solutions.
In this regard, Kudryashov’s simplest equation meth-
ods were examined. Third, the local conservation laws
are deduced by multiplier/homotopy methods. Finally,
the graphical simulations of the exact solutions are
depicted.
Keywords Extended KdV equation · Exact solutions ·
Conservation laws
1 Introduction
Investigation of analytic or numerical solutions to
nonlinear evolution equations (NLEEs) is important
task in nonlinear science. In last two decades, sev-
Y. Yıldırım · E. Ya¸ sar (B )
Department of Mathematics, Faculty of Arts and Sciences,
Uludag University, 16059 Bursa, Turkey
e-mail: emrullah.yasar@gmail.com
Y. Yıldırım
e-mail: yakupyildirim110@gmail.com
eral researchers made powerful studies on integrability
properties, bilinearization, Backlund transformations
and conservation laws of NLEEs [1–10].
As well known the classical Korteweg–de Vries
(KdV) equation
u
t
+ 6uu
x
+ u
3x
= 0
models weakly nonlinear long waves where first-
order nonlinear and dispersive terms are retained and
are in balance [11, 12]. If second-order terms are
retained, then the extended Korteweg–de Vries (eKdV)
equation
u
t
+ u
x
+ α (λuu
x
+ u
3x
)
+ α
2
c
1
u
2
u
x
+ c
2
u
x
u
2x
+ c
3
uu
3x
+ c
4
u
5x
= 0
(1)
results, where α is a non-dimensional measure of the
(small) wave amplitude. This equation describes the
evolution of steeper waves of shorter wavelength than
does the KdV equation. As stated in [13], the extended
KdV Eq. (1) can reduce to a series of integrable models
or can describe such physical phenomena as the ampli-
tude of the shallow-water waves. As can be seen, Eq.
(2) includes two linear dispersive terms, namely u
3x
and u
5x
. This property cannot emerge in the classical
fifth-order KdV equations. Setting c
1
= 45,c
2
= c
3
=
15,c
4
= 1,λ = 6 leads to a special eKdV equation
given as
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