A New Method for Generating Traveling-Wave Solutions of Coupled Nonlinear Equations
We propose a new method of algebraic transformations aimed at finding the traveling-wave solutions of complicated nonlinear wave equations on the basis of simpler equations. The generalized Dullin–Gottwald–Holm (DGH) equation and mKdV equations are chosen to illustrate our method. The solutions of t...
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| Cite this: | A New Method for Generating Traveling-Wave Solutions of Coupled Nonlinear Equations / Jiuli Yin, Shanyu Ding, Lixin Tian, Xinghua Fan // Український математичний журнал. — 2012. — Т. 64, № 10. — С. 1365-1372. — Бібліогр.: 18 назв. — англ. |
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irk-123456789-1652532020-02-13T01:26:05Z A New Method for Generating Traveling-Wave Solutions of Coupled Nonlinear Equations Jiuli Yin Shanyu Ding Lixin Tian Xinghua Fan Статті We propose a new method of algebraic transformations aimed at finding the traveling-wave solutions of complicated nonlinear wave equations on the basis of simpler equations. The generalized Dullin–Gottwald–Holm (DGH) equation and mKdV equations are chosen to illustrate our method. The solutions of the DGH equation can be obtained directly from the solutions of the mKdV equation. The conditions of appearance of different solutions are also presented. Various types of traveling-wave solutions are obtained for the generalized DGH equation, including periodic solutions, smooth solutions with decay, solitary solutions, and kink solutions. Побудовано новий метод алгебраїчних перетворень для знаходження розв’язків типу біжучих хвиль для складних нелінійних хвильових рівнянь на основі більш простих. Для ілюстрації методу використано узагальнене рівняння Далліна – Готвальда – Холма та модифіковане рівняння Кортевега – де Фріза. Розв’язки рівняння Далліна – Готвальда – Холма можна отримати безпосередньо із розв’язків модифікованого рівняння Кортевега – де Фріза. Наведено також умови для отримання різних розв’язків. Отримано чисельні розв’язки типу біжучих хвиль для узагальненого рівняння Далліна – Готвальда – Холма, серед яких періодичні розв’язки, гладкі розв’язки з запізненням, солітонні розв’язки та кінк-розв’язки. 2012 Article A New Method for Generating Traveling-Wave Solutions of Coupled Nonlinear Equations / Jiuli Yin, Shanyu Ding, Lixin Tian, Xinghua Fan // Український математичний журнал. — 2012. — Т. 64, № 10. — С. 1365-1372. — Бібліогр.: 18 назв. — англ. 1027-3190 http://dspace.nbuv.gov.ua/handle/123456789/165253 517.9 en Український математичний журнал Український математичний журнал |
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Статті Статті Jiuli Yin Shanyu Ding Lixin Tian Xinghua Fan A New Method for Generating Traveling-Wave Solutions of Coupled Nonlinear Equations Український математичний журнал |
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We propose a new method of algebraic transformations aimed at finding the traveling-wave solutions of complicated nonlinear wave equations on the basis of simpler equations. The generalized Dullin–Gottwald–Holm (DGH) equation and mKdV equations are chosen to illustrate our method. The solutions of the DGH equation can be obtained directly from the solutions of the mKdV equation. The conditions of appearance of different solutions are also presented. Various types of traveling-wave solutions are obtained for the generalized DGH equation, including periodic solutions, smooth solutions with decay, solitary solutions, and kink solutions. |
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Jiuli Yin Shanyu Ding Lixin Tian Xinghua Fan |
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Jiuli Yin Shanyu Ding Lixin Tian Xinghua Fan |
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Jiuli Yin |
| title |
A New Method for Generating Traveling-Wave Solutions of Coupled Nonlinear Equations |
| title_short |
A New Method for Generating Traveling-Wave Solutions of Coupled Nonlinear Equations |
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A New Method for Generating Traveling-Wave Solutions of Coupled Nonlinear Equations |
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A New Method for Generating Traveling-Wave Solutions of Coupled Nonlinear Equations |
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A New Method for Generating Traveling-Wave Solutions of Coupled Nonlinear Equations |
| title_sort |
new method for generating traveling-wave solutions of coupled nonlinear equations |
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Український математичний журнал |
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2012 |
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A New Method for Generating Traveling-Wave Solutions of Coupled Nonlinear Equations / Jiuli Yin, Shanyu Ding, Lixin Tian, Xinghua Fan // Український математичний журнал. — 2012. — Т. 64, № 10. — С. 1365-1372. — Бібліогр.: 18 назв. — англ. |
| series |
Український математичний журнал |
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2025-07-14T18:14:58Z |
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© JIULI YIN, SHANYU DING, LIXIN TIAN, XINGHUA FAN, 2012
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 10 1365
UDC 517.9
Jiuli Yin, Shanyu Ding, Lixin Tian, Xinghua Fan (Jiangsu Univ., China)
A NEW METHOD OF GENERATING OF TRAVELING WAVE
SOLUTIONS FOR COUPLED NONLINEAR EQUATIONS*
НОВИЙ МЕТОД ГЕНЕРУВАННЯ РОЗВ’ЯЗКІВ ТИПУ
БІЖУЧИХ ХВИЛЬ ДЛЯ ЗЧЕПЛЕНИХ НЕЛІНІЙНИХ РІВНЯНЬ
A new algebraic transformation method is constructed for finding traveling-wave solutions of complicated nonlinear wave
equations on the basis of simpler ones. The generalized Dullin – Gottwald – Holm (DGH) equation and mKdV equations
are chosen to illustrate our method. The solutions of the DGH equation can be obtained directly from solutions of the
mKdV equation. Conditions under which different solutions appear are also given. Abundant traveling-wave solutions of
the generalized DGH equation are obtained, including periodic solutions, smooth solutions with decay, solitary solutions,
and kink solutions.
Побудовано новий метод алгебраїчних перетворень для знаходження розв’язків типу біжучих хвиль для складних
нелінійних хвильових рівнянь на основі більш простих. Для ілюстрації методу використано узагальнене рівняння
Далліна – Готвальда – Холма та модифіковане рівняння Кортевега – де Фріза. Розв’язки рівняння Далліна – Гот-
вальда – Холма можна отримати безпосередньо із розв’язків модифікованого рівняння Кортевега – де Фріза. Наве-
дено також умови для отримання різних розв’язків. Отримано чисельні розв’язки типу біжучих хвиль для узагаль-
неного рівняння Далліна – Готвальда – Холма, серед яких періодичні розв’язки, гладкі розв’язки з запізненням, солі-
тонні розв’язки та кінк-розв’язки.
1. Introduction. Nonlinear wave phenomena appear in a wide variety of scientific applications,
such as fluid mechanics, plasma physics, biology, hydrodynamics, solid state physics and optical
fibers. These nonlinear phenomena are often related to nonlinear wave equations. Investigation of
the traveling wave solutions can make a better understanding of those phenomena and their applica-
tion in real life. Although many methods have been developed to construct traveling wave solutions,
it is still a difficult task to find traveling wave solutions of complicated equations with nonlinear
terms.
The main purpose of this paper is to devise a new method to get traveling wave solutions of the
complicated wave equations from solutions of simpler equations. This method is different from the
classic Miura transformation [1], that is, the classic Miura transformation is between two equations
with a linear dispersive term, and our method has more advantages in that one can obtain abundant
solutions of the aimed equation with a nonlinear dispersive term.
The generalized DGH equation
ut + 2!ux " #2uxxt + aumux + $uxxx ! = !#2 (2uxuxx + uuxxx ) (1.1)
includes two separately integrable soliton equations for water waves. If taking m = 1 and a = 3 ,
Eq. (1.1) becomes the DGH equation [2]
ut + 2!ux " #2uxxt + 3uux + $uxxx ! = !#2 (2uxuxx + uuxxx ) , (1.2)
which arises as a model for the unidirectional shallow water waves over a flat bottom. Here !2
* Research was supported by the National Nature Science Foundation of China (No. 11101191).
1366 JIULI YIN, SHANYU DING, LIXIN TIAN, XINGHUA FAN
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 10
and ! are squares of length scales, and the constant ! is the critical shallow water speed for un-
disturbed water at rest at spatial infinity. Eq. (1.2) has been studied by many researchers. The exis-
tence of several special solutions was proved in [3 – 6]. Those solutions include smooth solitary so-
lutions, peakons, global conservative solutions and low regularity solutions with continuously peri-
odic initial data. In [7 – 11], the authors studied the solution properties of the DGH equation and its
relative equations. In this present letter, we will consider all possible integral constants.
Letting ! = 0 , m = 2 and !2 " 0 , Eq. (1.1) becomes the famous mKdV equation
ut + au2ux + !uxxx ! = !0 , (1.3)
which plays an important role in many nonlinear scientific fields [13 – 16]. Eq. (1.3) has been used
to describe acoustic waves in certain anharmonic lattices and Alfven waves in a collisionless plasma.
It also appears in the models of Schottky barriers transmission lines and traffic congestion. Many
solutions of (1.3) in integral form have been given for some special integral constants. However, for
any integral constants, all the possible solutions have not been determined.
In view of the close relationship between Eq. (1.1) and Eq. (1.3), it is possibble to establish an
explicit connection between solutions of these two equations. If the connection does exist, one can
easily obtain traveling wave solutions of Eq. (1.1) from the already known solutions of Eq. (1.3).
This is one motivation of our work. Another motivation is that, whether the generalized DGH equa-
tion still preserves the integrability, Hamiltonian structure and some important conservative laws,
like the two integral equations (1.2) and (1.3).
The remainder of the paper is organized as follows. In Section 2, a kind of the generalized DGH
equation is firstly proved integrable. Meanwhile, the Hamiltonian structure and some important
conservative laws of Eq. (1.1) are given. In Section 3, under different parameter conditions, the
classification of traveling wave solutions of the mKdV equation is given by a qualitative method in
which all possible integral constants are considered. In Section 4, motivated by the Fan subequation
method [17], we verify directly an explicit connection between the mKdV equation and the general-
ized DGH equation. Furthermore, abundant traveling wave solutions of the generalized DGH equa-
tion are determined from the known solutions of the mKdV equation, and some examples of explicit
solutions are also given. The last section is conclusion.
2. Painleve property and conservative laws. An equation is called Painleve integrable when
it has Painleve property which means its solutions are single valued about an arbitrary singular
manifold. In this section, we will study the Painleve integrability of the generalized DGH equa-
tion.
According to the Kruskal method [18], we expand u in Eq. (1.1) by a local Laurent expansion
in the neighborhood of the singular manifold !(x, t) = 0 as
u ! = ! u j
j=0
!
" # j+$ . (2.1)
Substituting (2.1) into (1.1) leads to conditions on ! and recursion relation for the functions u j .
If ! is a negative integer and the recursion relation is consistent, then we say the system (1.1) is
integrable.
A NEW METHOD OF GENERATING OF TRAVELING WAVE SOLUTIONS … 1367
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 10
Substituting u ~ u j!" into Eq. (1.1), the leading order analysis implies ! = "
2
m " 1
, so we
can deduce m = 2 or m = 3 .
For m = 2 , we obtain the recursion relation of the expansion coefficients u j as
( j + 1)( j ! 6)( j ! 8)u j ! = !Fj ("x ,!"t ,…,!u0,!u1,…,!u j!1) , (2.2)
where Fj is a function of u0,!u1,…,!u j!1 and the derivatives of ! . Therefore the resonances
occur at j = !1,!6,!8 . The resonance at j = !1 represents the arbitrariness of the singular mani-
fold !(x, t) = 0 . For the integability of Eq. (1.1), we only prove the existence of arbitrary functions
at the cases j = 6,!8 with m = 2 . According to the Kruskal’s method, one can take ! =
= x + !(t) , where !(t) is an arbitrary function of t . After a lengthy computation, we obtain
u0 ! = !
24
a
, u1 ! = !u3 ! = !u5 ! = !0 , u2 ! = !!
1
3
" t !
1
3
# ,
u4 ! = !
1
20
! t "
w
10
"
a
180
! t
2 "
a#
90
! t "
a# 2
180
, (2.3)
u7 ! = !
13a
5760
! tt "
a2
8640
! t! tt "
a2#
8640
! tt .
Substituting (2.3) into the recursion relations (2.2), one can find that (2.2) are satisfied identi-
cally. Hence the generalized DGH equation (1.1) with m = 2 is integrable.
Similarly, for m = 3 , we obtain the recursion relation of the expansion coefficients u j as
( j + 1)( j ! 4)( j ! 5)u j ! = !Fj ("x ,!"t ,…,!u0,!u1,…,!u j!1) .
When the resonance occurs at j = 5 , we obtain
F5 ! = !! tt ("32 + 6a#! t + 3a! t
2 + 3a# 2 )! $ !0 .
Therefore the generalized DGH equation (1.1) with m = 3 is not integrable.
The generalized DGH equation, combining the DGH equation and the mKdV equation, still pre-
serve the Hamiltonian structure and some important conservative laws. Indeed, the generalized
DGH equation, analogous to the case of the DGH equation, has the following conservative laws
M (u)! = ! u
R
! dx ,
Q(u)! = !1
2
(u2 + ux2 )
R
! dx ,
H (u)! = !! 1
2
2a
(m + 1)(m + 2)
um+2 + uux2 + 2"u2 ! #ux2
$
%&
'
()R
* dx
and the Hamiltonian structure
1368 JIULI YIN, SHANYU DING, LIXIN TIAN, XINGHUA FAN
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 10
ut ! = !!"x
#H
#u
! = !{u, H} ,
where the Poisson bracket structure is defined as {u(x), u(y)} = !x"(x # y) .
3. Classification of traveling wave solutions of the mKdV equation. In this section, we will
classify traveling wave solutions of the mKdV equation in different regions of parametric space by a
qualitative method.
For a traveling wave u(x, t) = !(") = !(bx # ct) , Eq. (1.3) takes the form
!c"# + ab"2"# + $b3"### ! = !0 , (3.1)
where b and c are constants to be determined. Integrated twice with respect to x , Eq. (3.1)
turns to be the equivalent integrated form
!"
2 ! = !h1(!4 # h2!2 + d1!) + d2 , (3.2)
where h1 = !
"
6b2#
, h2 =
6c
!b
, d1 and d2 are arbitrary constants of integration. For some spe-
cial values of d1 and d2 , it is not difficult to obtain traveling wave solutions of Eq. (3.2). How-
ever, for any integral constants, it is a hard task to determine the type of the solutions. Fortunately,
we can use a qualitative analysis method to deal with this problem, in which we consider Eq. (3.2)
for all possible constants of integration. Let
!x2 ! = !F(!)! = !h1(!4 " h2!2 + d1!) + d2 (3.3)
for determining the solutions of Eq. (3.2). Let the polynomial P(!) = h1!G(!) with a simple root
at ! = 0 , where G(!) = !3 " h2! + d1 . The solutions of Eq. (3.2) correspond to different behav-
iors of this polynomial. Once the integral constant d1 is fixed, a change in d2 will shift the graph
vertically up or down, accordingly change the zero points.
Similar to the method in [12], we can obtain five qualitatively different cases of P(!) when
h1 > 0 (see Figure), and we can also establish the following structure of traveling wave solutions of
Eq. (3.2):
Case 1. When h1 > 0 , any traveling wave solution falls into one of the following categories:
(1) If d1 >
2 3
9
h23/2 , for some negative d2 , there are smooth solutions with decay and peri-
odic solutions.
(2) If 0 < d1 !
2 3
9
h23/2 , there are smooth solutions with decay and periodic solutions for
some positive d2 . For some negative d2 , there are periodic solutions.
(3) If d1 = 0 , there are kink solutions and periodic solutions for some positive d2 .
A NEW METHOD OF GENERATING OF TRAVELING WAVE SOLUTIONS … 1369
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 10
(1) d1 >
2 3
9
h23/2 (2) 0 < d1 !
2 3
9
h23/2 (3) d1 = 0
(4) ! 2 3
9
h23/2 " d1 < 0 (5) d1 < !
2 3
9
h23/2
Five cases of P(!) as h2 > 0 .
(4) If ! 2 3
9
h23/2 " d1 < 0 , there are periodic solutions for some negative d2 . For some
positive d2 , there exists smooth solutions with decay and periodic solutions.
(5) If d1 < !
2 3
9
h23/2 , there are smooth solutions with decay and periodic solutions for some
negative d2 .
For h1 < 0 , we can obtain the following results.
Case 2. When h1 < 0 , any traveling wave solution falls into one of the following categories:
(1) If d1 ! 0 , for some negative d2 , there are periodic solutions. For some positive d2 ,
there are smooth solutions with decay and periodic solutions.
(2) If d1 = 0 , there are solitary solutions for d2 = 0 and periodic solutions for some positive
or negative d2 .
Remark. If h2 ! 0 , there are no bounded solutions of Eq. (3.2).
1370 JIULI YIN, SHANYU DING, LIXIN TIAN, XINGHUA FAN
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 10
4. The new transformation method and traveling wave solutions. For a traveling wave
u(x, t) = u(!) , ! = bx " ct , Eq. (1.1) is reduced to
(2!b " c)u# + ($b + %2c)u### + abumu# ! = !%2b3(2u#u## + uu### ) , (4.1)
where b and c are constants to be determined. The purpose of this section is to find a transfor-
mation between Eq. (4.1) and Eq. (3.2). Furthermore, by this transformation, we can obtain solu-
tions of Eq. (4.1) from those to Eq. (3.2).
The main idea of our method is to expand a solution of Eq. (4.1) in the form
u ! = ! ai!i
i=0
n
" , (4.2)
where ! is a solution of Eq. (3.2). Once the parameters n and ai are determined, one can ob-
tain the solutions of Eq. (4.1) from already known solutions of Eq. (3.2) easily.
Firstly, to determine the parameter n , we give the following results by (3.2):
d!
d"
! = !# h1(!4 $ h2!2 + d1!) + d2 , (4.3)
d
d!
!" !# h1($4 % h2$2 + d1$) + d2
d
d$
, (4.4)
d2
d!2
!" !#2 1
2
h1(4$3 % 2h2$ + d1)
d
d$
+ (h1($4 % h2$2 + d1$) + d2 )
d2
d$2
&
'
(
)
*
+ , (4.5)
……………………………………………………………………………,
where ! = ±1 . Substituting (4.2) together with (4.3) – (4.5) into Eq. (4.1) and balancing the highest
derivative term with the nonlinear convection term, we can obtain n =
2
m ! 1
. Noting that the posi-
tive integer n implies m = 2 or 3 , the parameter n must be taken 2 or 1 correspondingly.
Secondly, to determine the parameters ai , substituting (4.2) into Eq. (4.1) and setting coeffi-
cients of all powers of !i and !i h1(!4 " h2!2 + d1!) + d2 to zero, we can obtain a system of
algebraic equations from which the parameters ai can be found explicitly.
Based on the above method, we obtain the following results with the aid of Maple.
Theorem 1. For m = 2 , corresponding to the traveling wave solution ! of the mKdV equa-
tion (1.3),
u ! = !2c!2 " c
3b
"
c
3!2
" !2#2 (4.6)
is a solution of the generalized DGH equation (1.1), where b , c are arbitrary constants, d1 = 0
and
A NEW METHOD OF GENERATING OF TRAVELING WAVE SOLUTIONS … 1371
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 10
d2 ! = !
9bc!4 " 16ab2 " 8abc!2 " ac2!4 " 18b2c2!8 " 18b2#!4
36b4!8
.
Theorem 2. For m = 3 and d1 ! 0 , corresponding to the traveling wave solution ! of the
mKdV equation (1.3),
u ! = !! 2
"2
!
c
4b
+
24c ! 24abc ! 48b# ! 6ac2"2
ab"2d1
$ (4.7)
is a solution of the generalized DGH equation (1.1) with m = 3 , where d1 and b are arbitrary,
d22 ! = !
1728(8abc2 + 16bc! + 2ac3"2 # 4c2 # 16b2!2 # 16ac!b2 # 4ab!c2"2 # a2b2c3"6 )
5a2b2"6
and c satisfies
!2c2 + (8b!2 " 4b2!6 )c + 16b2 ! = !0 . (4.8)
Theorem 3. For m = 3 and d1 = 0 , corresponding to the traveling wave solution ! of the
mKdV equation (1.3),
u ! = !! 2
"2
!
c
4b
+ a1# , (4.9)
is a solution of the generalized DGH equation (1.1), where b =
4c ! a"2c2
4ac + 8#
and c satisfies (4.8).
Based on the above facts, combining with the results in Section 3, we can obtain that there exist
periodic solutions, kink solutions and smooth solutions with decay to Eq. (1.1) under some parame-
ter conditions.
In the following we will give some examples of explicit solutions.
Example 1. If d1 = d2 = 0 , we can easily get the explicit solution of (3.2). Then substituting
the solution into (4.6), we obtain a smooth solution of Eq. (1.1) with m = 3
u1 ! = !2b3! "2 #
1
3b
#
1
3"2
$
%&
'
() #
6!"2b2
a
sech2 (bx # b3!t) . (4.10)
Furthermore, if b =
!2
3!4 " 1
, (4.10) becomes the solitary wave solution. According to Theo-
rem 1, we find that the parameters of Eq. (1.1) should satisfy
9!b2"4 # 16a # 8a!b2"2 # ab4! 2"4 # 18b6! 2"8 # 18$"4 ! = !0 .
Example 2. If d1 =
!32b3"
a
2"
a
and d2 = 0 , solving (3.2) and using (4.7), we obtain a pe-
riodic solution of Eq. (1.1) with m = 3
1372 JIULI YIN, SHANYU DING, LIXIN TIAN, XINGHUA FAN
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 10
u3 ! = !!
2
"2
! b2# +
8a#b3 + 6$ + 48a"2# 2b5 ! 31#b2
"2b2#
cos2 (bx ! 4b3#t)
3 ! 2 cos2 (bx ! 4b3#t)
,
where the parameters of Eq. (1.1) satisfy
27a(4ab5! 2 + 2!"b2 + 4ab7! 3#2 $ 13b4! 2 $ 2ab3! " –
– !2 " 2a!b5# 2$2 " 2a2$6b9# 3) " 5b6# 3$6 ! = !0 ,
and b is determined by !2" 2b4 # "!6b3 + 2"!2b2 + 1 = 0 .
Example 3. If d1 = 0 and d2 = !
3"b2
2a
, solving (3.2) and using (4.9), we obtain a kink solu-
tion of Eq. (1.1) with m = 3
u2 ! = !!
2
"2
!
c
4b
+ a1 tanh bx + 2b3#t$% &' ,
where b =
4c ! a"2c2
4ac + 8#
and c satisfies (4.8).
5. Conclusions. A new method was devised to construct traveling wave solutions of the com-
plicated nonlinear wave equations from solutions of the simpler equations. Abundant traveling
wave solutions can be obtained by this method easily. As an example, we obtained periodic solu-
tions, smooth solutions with decay, solitary solutions and kink solutions of the generalized DGH
equation. This new method is direct and efficient. It can be widely applied to other nonlinear wave
equations.
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Received 18.10.11,
after revision — 10.05.12
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