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Auto-B acklund Transformation and Solitary-wave Solutions to
Non-integrable Generalized Fifth-order Nonlinear Evolution
Equations
Woo-Pyo Hong and Young-Dae Jung a
Department of Physics, Catholic University of
Taegu-Hyosung,Hayang, Kyongsan, Kyungbuk 712-702, South Korea,
e-mail: [email protected] Department of Physics, Hanyang
University, Ansan, Kyunggi-Do 425-791, South Korea,
e-mail: [email protected] requests to Prof. W.-P.
Hong.
Z. Naturforsch. 54 a, 549553 (1999); received June 1, 1999
We show that theapplication of thetruncated Painlev e expansion
and symbolic computation leadsto a new class of analytical
solitary-wave solutions to the general fth-order nonlinear
evolutionequations which include Lax, Sawada-Kotera (SK),
Kaup-Kupershmidt (KK), and Ito equations.Some explicit
solitary-wave solutions are presented.
PACS numbers: 03.40.Kf, 02.30.Jr, 47.20.Ky, 52.35.Mw
Key words: Nonlinear Evolution Equations; General Fifth-order
Nonlinear Evolution Models;
Truncated Painlev e Expansion; B acklund Transformation;
Solitary-wave Solutions;Symbolic Computation.
While it is easy to write down in closed from a soli-tary wave
solution for the simplest standard model,namely the Korteweg-de
Vries (KdV) equation, ithas proved quite difcult to obtain such
solutions forthe problems from which the KdV equation was de-rived
as a First approximation [1]. As such KdV hi-erarchy models, we
investigate the generalized
non-integrablefth-ordernonlinearevolution equations of the form
[2]
u
t
+ u u x x x
+ u x
u
x x
+ u 2 u x
+ u x x x x x
= 0 (1)
where , and are constant model parameters.This model includes
the Lax [3], Kaup-Kupershmidt(KK) [4 - 7], Sawada-Kotela (SK) [9],
and Ito equa-tions [8]. As the constants , , and take
differentvalues, the properties of (1) drastically change.
Forinstance, the Lax equation with = 10 = 20 and
= 30, and the SKequationwhere = = = 5 arecompletely integrable.
These two equations have N -soliton solutions and an innite set of
conservationlaws. The KK equation, with = 10 = 25 and
= 20 is also known to be integrable [6] and to havebilinear
representations [10, 11], but the explicit formof its N -soliton
solution is apparently not known. A
09320784 / 99 / 08000549 $ 06.00 c Verlag der Zeitschrift f ur
Naturforschung, T ubingen www.znaturforsch.com
fourth equation in the model is the Ito equation, with = 3 = 6
and = 2 which is not completelyintegrable but has a limited number
of conservationlaws [8]. More recently, using a simplied version of
Hirotas method, Hereman and Nuseir [2] explicitlyconstructed
multi-soliton solutions of the KK equa-tion for which soliton
solutions were not previouslyknown. However, to our knowledge no
attempt hasbeen made for nding more general solitary-wavesolutions
other than the above mentioned models be-cause of lengthy and
nearly impossible calculationswithout proper symbolic computation
packages. Byutilizing the symbolic package Maple , Hong [12]
re-cently found some analytical solitary-wave solutionsto the
general fth-order water models in [13], withsome constraints on the
model parameters.
In this work, we make use of both the truncatedPainlev e
expansion and the symbolic computationmethod [14 - 17] to obtain an
auto-B acklund trans- formation and certain explicit solitary-wave
solutionsfor the generalized non-integrable fth-order nonlin-ear
evolution equation, which are different from thesolutions of the
models mentioned above [3 - 8].
A non-linear partial differential equation (NPDE)is said to
possess the Painlev e property when the
mailto:[email protected]:[email protected]:[email protected]:[email protected]
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550 W.-P. Hong and Y.-D. Jung Generalized Fifth-order Nonlinear
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solutions of the NPDE are single valued aboutthe movable,
singularity manifold which is non-characteristic. To be more
precise, if the singularitymanifold is determined by
(z
1z
2: : : z
n
) = 0 (2)and u = u (z 1 z 2 : : : z n ) is a solution of the
NPD,thenit is required that
u = 1
X
j =0
u
j
j (3)
where u 0 6 = 0 = (z 1 z 2 : : : z n ) u j = u j (z 1z 2 : : : z
n ) are analytical function of ( z j ) in the neigh-borhood of the
manifold (3), and is a negative,rational number. Substitution of
(3) into the NPDEdetermines the allowed values of and denes the
recursion relation for u j , j = 0 1 2 : : : . When theansatz
(3) is correct, the NPDE is said to possessthe Painlev e property
and is conjectured to be inte-grable [14, 17].
; 7 : ; 720 u 0 x 5 ; 12 u 02 x 3 ; 24 u 02 x 3 ; 2 u 03 x = 0
(6)
; 6 : ; 30 u 1 u 0 x 3 + 18 u 02 x x x + 18 u 0 x 2 u 0 x + 1200
u 0 x 3 x x ; 10 u 0 x 3 u 1 ; 4 u 02 x x x
+ 14 u 0 x 2 u 0 x ; 120 u 1 x 5 ; 5 u 02 u 1 x + u 02 u 0 x +
600 x 4 u 0 x = 0 (7)
; 5 : ; 6 u 0 x u 0 x x ; 4 u 02 x u 2 ; 24 u 2 u 0 x 3 + 6 u 1
x 2 u 0 x ; 2 u 12 x 3 + 4 u 0 x u 1 x x
+ 18 u 1 x 2 u 0 x + 24 u 1 u 0 x x x + 2 u 0 u 1 u 0 x ; 4 u 0
u 12 x + 6 u 0 x 2 u 1 x + 10 u 1 x u 0 x 2
; 2 u 0 x u 0 x x ; 6 u 0 x x u 0 x ; 2 u 0 x u 0 x x ; 4 x u 0
x 2 ; 720 x 2 x x u 0 x + 120 x 4 u 1 x
+ 240 u 1 x 3 x x + u 02 u 1 x ; 240 u 0 x 2 x x x ; 360 u 0 x x
x 2 ; 2 u 02 x x x ; 6 u 12 x 3
; 240 x
3u 0 x x = 0 (8)
; 4 : u 12 u 0 x ; u 13 x + 90 x x 2 u 0 x + 60 x 2 u 0 x x x +
u 0 x u 0 x x ; 6 u 2 u 1 x 3 + 18 u 2 u 0 x x x
+ 2 u 0 u 2 u 0 x ; 3 u 0 u 1 x x x + 2 u 0 u 1 u 1 x + 6 u 1 x
2
u 1 x + 4 u 1 x 2
u 1 x + 18 u 2 x 2
u 0 x
+ 6 u 1
x
u
0x x
; u
1
x
u
0x x
; u
0x
u
1
x x
; 6 u 1
x x
u
0x
+ 6 u 1
2
x
x x
+ u 1
2
x
x x
; 6 u 1 u 2 u 0 x ; 2 u 0 x u 1 x x + 6 u 2 x u 0 x 2 ; 2 u 1 x
u 0 x x ; 3 u 0 u 1 x x x ; 3 u 0 u 1 x x x
+ 180 x
x x
u 0 x x ; 6 u 0 x u 1 x x ; 60 x 3
u 1 x x + 120 x x x x u 0 x + u 02 u 2 x ; 60 u 0 x x x x x
+ u 0 u 0 x x x ; 180 x 2 u 1 x x x ; 60 u 1 x 2 x x x ; 90 u 1
x xx2 + 30 u 0 x x x x = 0 (9)
In order to nd solitonic solutions for (1), we trun-cate the
Painlev e expansion, (3), at the constant-levelterm in the senses
of Tian and Gao [15] and Khateret al. [17],
u ( x t ) = ; J (x t )J
X
l =0
u
l
( x t ) l (x t ): (4)
On balancingthehighest-order contributions from thelinear term
(i.e. u
x x x x x
) with the highest order con-tributions from the nonlinear term
(i.e, u 2 u
x
), we getJ = 2, so that
u ( x t ) =u 0( x t )
2( x t ) +
u 1( x t ) ( x t )
+ u 2( x t ): (5)
We will stay with the general assumption that
x
6= 0 but will not initially impose any constraints on
the model parameters
. When substituting theabove expressions into (1) with the
symbolic programpackage Maple , we let the coefcients of like
powersof vanish, so as to get the set of Painlev e-Backlund(PB)
equations,
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W.-P. Hong and Y.-D. Jung Generalized Fifth-order Nonlinear
Evolution Equations 551
; 3 : u 0 u 1 x x x + u 1 x u 12 ; 2 u 2 x u 0 x x ; 2 u 0 x u 2
x x + 2 u 0 u 1 u 2 x ; 2 u 0 t ; 2 u 0 x x x x x
; 4 u 2 x x u 0 x + 20 u 1 x x x x 2 + 30
x x
2u 1 x ; 20 x x u 0 x x x ; 10 x u 0 x x x x ; 10 x x x x u 0
x
; 20 x x x
u 0 xx + 6 u 2 u 1 x x x + 2 u 1 u 2 u 0 x + 2 u 2 x u 1 x 2 ; 3
u 1 u 1 x x x ; 3 u 1 u 1 x x x
; u 1 x u 1 x x ; u 1 x x u 1 x + 2 u 1 x u 0 u 2 ; 2 u 2 u 0 x
x x ; 2 u 22 u 0 x + 6 u 2 x 2 u 1 x
; 6 u 2 x u 0 x x ; 6 u 2 x x u 0 x ; 2 u 12 u 2 x + 40 x u 1 x
x x x + u 1 u 0 x x x ; 2 u 1 x 2 x
+ u 1 x x u 0 x + u 1 x u 0 x x + 60 x u 1 x x x x + 10 u 1 x x
x x x + 20 u 1 x x x x x ; u 12 x x x
+ 10 x
2u 1 x x x ; 3 x u 0 x x ; 5 x u 0 xxxx ; 4 u 1 x 2 x + 3 x
2u 1 x = 0 (10)
; 2 : ; u 1 t + u 2 u 0 x x x ; u 1 x x x x x ; 2 u 2 x u 1 x x
+ 2 u 1 u 2 u 1 x ; u 22 u 1 x ; u 2 u 1 x x x
+ 2 u 2 x u 0 u 2 ; 3 u 2 u 1 x x x ; 3 u 2 u 1 x x x ; u 2 x x
u 1 x ; u 2 x u 1 x x + u 0 x x x x x + u 0 t
+ u 1 x u 1 x x ; 10 u 1 x x x x x ; 5 u 1 x x x x x + u 22 u 0
x + u 2 x u 12 + u 1 u 1 x x x ; 5 u 1 xxxx x
+u
0u
2 x x x ;
10u
1 x x x
x x
+u
2 x u
0 x x +u
2 xxu
0 x = 0 (11)
; 1 : u 1 u 2 x x x + u 2 u 1 x x x + u 1 x x x x x + 2 u 1 u 2
u 2 x + u 22 u 1 x + u 1 x u 2 x x + u 2 x u 1 x x + u 1 t = 0
(12)
0 : u 2 needs to satisfy the original equation, i.e.,
u 2 x x x x x + u 2 x u 2 x x + u 2 u 2 x x x + u 2 t + u 22 u 2
x = 0 (13)
The set of (5) and (6 - 13) constitutes an auto-B acklund
transformation , if the set is solvable with respectto (x t ) u 0(
x t ) u 1( x t ) and u 2( x t ) [15, 16]. Equation (6) brings out
three possibilities for u 0( x t ):
u
I0 = 0 u
II0 = 3
( ; ; 2 + H ) x
2
u
III0 = 3
( ; ; 2 ; H ) x
2
(14)
where H p
2 + 4 + 4 2 ; 40 . Some complicatedsymbolicmanipulations
arerequiredtondgeneralsolutions for the remaining u 1( x t ) and u
2( x t ). However, due to increasing complexities in the
symboliccomputations, a long CPU time is required. Thus, in the
following, in order to shorten the computation time,we constrain
the model parameters by requiring
H = 0 =) = ; = 2 +p
10 or = ; = 2 ; p
10 : (15)
In the rest of the paper, we consider a solution family for the
case of the non-trivial solution u II0 with the rstconstraint = ; =
2 +
p
10 . Of course, other classes of solitary-wave solutions can be
found from u III andthe second constraint on .
After substituting u II0 and into (7), we obtain
u
1(x t
) = 12 xx(5 + 2
p
10 )
p
10 + 4 :
(16)
Subsequently, we get the following solution for u 2( x t ) from
(9):
u 2( x t ) =; 32
x
xxx + 15 2 xx2 ; 20 2 x xxx ; 4 x xxxp
10 + 24 xx2
x
2(p
10 2 + 8 + 16 3= 2): (17)
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552 W.-P. Hong and Y.-D. Jung Generalized Fifth-order Nonlinear
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(a) (b)
Fig. 1. (a) Typical sech 2-shpaed solitary-wave solution u (x 0)
in (21) with parameters A = 1 B = 1 C 1 = 1 = ; 2 = 10, and = ; = 2
+
p
10 = 11. (b) u (x t ) shows the solitary wave property that the
amplitude becomes nite as j x japproaches innity.
Thus, we are able to nd a class of analytical solu-tions u (x t
) in terms of u II0 u 1 u 2 in (14 - 17) with anarbitrary function
(x t ) constrained by the remain-ing equations (10 - 13). We
notethat,once a B acklundtransformation is discovered and a set of
seed so-lutions is given, one will be able to nd an innitenumber of
solutions by repeated applications of thetransformation, i. e., to
generate a hierarchy of solu-tions with increasing complexity. In
the rest of thepaper we will nd a family of some exact
analyticalsolutions of (1).
Sample Solution
A trial solution
( x t ) = 1+e Q (x t ) where Q (x t ) = A (t )x +B (t ) (18)
is substituted into the remaining constraint equations(10 - 13).
From (10), we nd
; 3& x : A (t )0 = 0 ) A (t ) = A = constant : (19)
Then, we obtain B (t )0 from the terms of ; 3 andintegrate it
over t to get
B ( t ) =; R t ; S
4 T
R 10500 A 5p
10 6 ; 62720 A 5p
10 3 2
+ 13750 A 5 7p
; 97280 A 5 7= 2
; 201600 A 5 3 5= 2 + 14000 A 5 5 3= 2
; 28000 A 5p
10 4 2 ; 6144 A 5p
10 4
+ 625 A 5 8p
10
S ; 224000 C 1 3 5= 2 ; 2500 C 1 7p
; 84000 C 1 5 3= 2 ; 71680 C 1 7 = 2
;
53760C
1
p
10
3
2;
4096C
1
p
10
4
; 7000 C 1p
10 6 ; 56000 C 1p
10 4 2
T 1024 4p
10 + 1750p
10 6 + 14000p
10 4 2
+ 13440p
10 3 2 + 56000 3 5= 2 + 625 7p
+ 21000 5 3= 2 + 17920 7= 2 (20)
where C 1 is the constant of integration. Combiningall terms, we
nd a family of analytical solutions of (1) as
u ( x t ) = 3 ( ; ; 2 + H )
x
2
1(1 + e Q (x t ))2
+ 12
x x
(5 + 2p
10 )
p
10 + 4
11 + e Q (x t )
;
h
32 x
x x x
; 15 2 x x
2 + 20 2 x
x x x
+ 4 x
x x x
p
10 ; 24 x x
2
i
. h
x
2(p
10 2 + 8 + 16 3= 2)i
(21)
with the trial function (x t ) in (18) with Q (x t ) in(19,
20).
In the following we show that our solutions indeedhave
solitary-wave properties by presenting some g-ures from the family:
We choose, as an example, aset of arbitrary constants; A = 1 B = 1
and C 1 = 1
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W.-P. Hong and Y.-D. Jung Generalized Fifth-order Nonlinear
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(a) (b)
Fig. 2. (a) Sech 2-shaped solitary-wave solution u (x 0) with
parameters A = 1 B = 1 C 1 = 1 = 2 = 40, and = 19.(b) u (x t )
shows the breather solitary-wave solution behavior.
with the constraint = ; = 2 +p
10 . Figures 1and 2 correspond to ( = ; 2 = 10 = 11) and( = 2 =
40 = 19), respectively. Firstly, we notethat u (x 0) in Figs. 1(a)
and 2(a) are both sech( x )2-shaped solutions. From Figs. 1(b) and
2(b) we un-derstand that in both cases the solutions have
solitarywave property, i. e., u (x t ) tends to a nite value asj x
j approaches innity. Interestingly, Fig. 2(b) showsa breather
solitary-wave solution.
To sum up, with symbolic computations and thetruncated Painlev e
expansion analysis, we showedthat Backlund ransformations exist for
the general-ized fth-order non-integrable nonlinear evolution
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equation. We found a class of analytical solutionsu ( x t ) to
(1) in terms of u II0 u 1 u 2 in (14 - 17) withan arbitrary
function (x t ) constrained by (10 - 13).A sample solution family
for u (x t ) was found inEq.(21) with the constraint = ; = 2 +
p
10 . Moresolitary-wave families can be found from u III( x t
)and other constraints on in (15).
Acknowledgement
This research was supported by theKoreaResearchFoundation
through the Basic Science Research In-stitute Program
(1998-015-D00128).
