Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2013, Article ID 395628, 14 pages
http://dx.doi.org/10.1155/2013/395628
Research Article
Exact Traveling Wave Solutions for
a Nonlinear Evolution Equation of Generalized
Tzitzéica-Dodd-Bullough-Mikhailov Type
Weiguo Rui
College of Mathematics of Honghe University, Mengzi Yunnan 661100, China
Correspondence should be addressed to Weiguo Rui; weiguorhhu@yahoo.com.cn
Received 18 March 2013; Accepted 9 May 2013
Academic Editor: Shiping Lu
Copyright © 2013 Weiguo Rui. This is an open access article distributed under the Creative Commons Attribution License, which
permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
By using the integral bifurcation method, a generalized Tzitzéica-Dodd-Bullough-Mikhailov (TDBM) equation is studied. Under
different parameters, we investigated different kinds of exact traveling wave solutions of this generalized TDBM equation. Many
singular traveling wave solutions with blow-up form and broken form, such as periodic blow-up wave solutions, solitary wave
solutions of blow-up form, broken solitary wave solutions, broken kink wave solutions, and some unboundary wave solutions, are
obtained. In order to visually show dynamical behaviors of these exact solutions, we plot graphs of profiles for some exact solutions
and discuss their dynamical properties.
1. Introduction
In this paper, we consider the following nonlinear evolution
equation:
𝑚𝑢
𝑢𝑥𝑡 = 𝛼𝑒
𝑛𝑢
+ 𝛽𝑒 ,
(1)
where 𝛼, 𝛽 are two non-zero real numbers and 𝑚, 𝑛 are two
integers. We call (1) generalized Tzitzéica-Dodd-BulloughMikhailov equation because it contains Tzitzéica equation,
Dodd-Bullough-Mikhailov equation, and Tzitzéica-DoddBullough equation. When 𝛼 = 1, 𝛽 = −1, 𝑚 = 1, 𝑛 = −2
or 𝛼 = −1, 𝛽 = 1, 𝑚 = −2, 𝑛 = 1, especially (1) becomes
classical Tzitzéica equation [1–3] as follows:
𝑢𝑥𝑡 = 𝑒𝑢 − 𝑒−2𝑢 ,
(2)
which was originally found in the field of geometry in 1907
by G. Tzitzéica and appeared in the fields of mathematics and
physics alike. Equation (2) usually called the “Dodd-Bullough
equation,” which was initiated by Bullough and Dodd [4] and
Žiber and Šabat [5]. Indeed, (2) has another form
𝑢𝑥𝑥 − 𝑢𝑡𝑡 = 𝑒𝑢 − 𝑒−2𝑢 ;
see [6, 7] and the references cited therein.
(3)
When 𝛼 = −1, 𝛽 = −1 and 𝑚 = 1, 𝑛 = −2, (1) becomes
Dodd-Bullough-Mikhailov equation
𝑢𝑥𝑡 + 𝑒𝑢 + 𝑒−2𝑢 = 0.
(4)
When 𝛼 = 1, 𝛽 = 1 and 𝑚 = 1, 𝑛 = −2, (1) becomes TzitzéicaDodd-Bullough equation
𝑢𝑥𝑡 − 𝑒𝑢 − 𝑒−2𝑢 = 0.
(5)
The Dodd-Bullough-Mikhailov equation and TzitzéicaDodd-Bullough equation appeared in many problems
varying from fluid flow to quantum field theory.
Moreover, when 𝑚 = 1, 𝑛 = −1, 𝛼 = 1/2, 𝛽 = −1/2 or
𝑚 = −1, 𝑛 = 1, 𝛼 = −1/2, 𝛽 = 1/2, (1) becomes the sinhGordon equation
𝑢𝑥𝑡 = sinh 𝑢,
(6)
which was shown in [7–10] and the references cited therein.
When 𝑚 = 1, 𝑛 = −1, 𝛼 = 𝛽 = 1/2 or 𝑚 = −1, 𝑛 = 1,
𝛼 = 𝛽 = 1/2, (1) becomes the cosh-Gordon equation
𝑢𝑥𝑡 = cosh 𝑢,
(7)
2
Journal of Applied Mathematics
which was shown in [11, 12] and the references cited therein.
When 𝑚 = 1, 𝑛 = 0, especially (1) becomes the Liouville
equation
𝑢𝑥𝑡 = 𝛼𝑒𝑢 .
2. Two-Dimensional Planar Dynamical System
of (1) and Its First Integral
Applying the transformations 𝑢(𝑥, 𝑡) = ln |V(𝑥, 𝑡)|, we change
(1) into the following form:
(9)
Let V(𝑥, 𝑡) = 𝜙(𝑥 − 𝑐𝑡) = 𝜙(𝜉), substituting 𝜙(𝑥 − 𝑐𝑡) into (9),
respectively, we obtain
2
𝑐(𝜙󸀠 ) − 𝑐𝜙𝜙󸀠󸀠 = 𝛼𝜙𝑚+2 + 𝛽𝜙𝑛+2 ,
𝑑𝜉 = 𝑐𝜙𝑑𝜏,
(8)
All the equations mentioned above play very significant roles
in many scientific applications. And some of them were
studied by many authors in recent years; see the following
brief statements.
In [13], by using the tanh method, Wazwaz considered some solitary wave and periodic wave solutions for
the Dodd-Bullough-Mikhailov and Dodd-Bullough equations. In [14], Andreev studied the Bäcklund transformation
for Bullough-Dodd-Jiber-Shabat equation. In [15], Cherdantzev and Sharipov obtained finite-gap solutions of the
Bullough-Dodd-Jiber-Shabat equation. In [16], Cherdantzev
and Sharipov investigated solitons on the finite-gap background in the Bullough-Dodd-Jiber-Shabat model. In addition, the Darboux transformation, self-dual Einstein spaces
and consistency and general solution of Tzitzéica equation
were studied in [17–19].
In this paper, by using integral bifurcation method [20],
we will study (1). Indeed, the integral bifurcation method
has successfully been combined with the computer method
[21] and some transformations [22, 23] for investigating exact
traveling wave solutions of some nonlinear PDEs. Therefore,
using this method, we will obtain some new results which are
different from those in the above references.
The rest of this paper is organized as follows: in Section 2,
we will derive two-dimensional planar system which is equivalent to (1) and give its first integral equation. In Sections 3
and 4, by using the integral bifurcation method, we will obtain
some new traveling wave solutions of (1) and discuss their
dynamic properties.
VV𝑥𝑡 − V𝑥 V𝑡 = 𝛼V𝑚+2 + 𝛽V𝑛+2 .
and call (11) singular system. In order to obtain an equivalent
system of (10), making a scalar transformation
(12)
equation (11) can be changed into a regular two-dimensional
system as follows:
𝑑𝜙
= 𝑐𝜙𝑦,
𝑑𝜏
𝑑𝑦
= 𝑐𝑦2 − 𝛼𝜙𝑚+2 − 𝛽𝜙𝑛+2 .
𝑑𝜏
(13)
Systems (11) and (13) are two integrable systems, and they
have the same first integral as follows:
𝑦2 = ℎ𝜙2 −
2𝛼 𝑚+2 2𝛽 𝑛+2
−
𝜙
𝜙 ,
𝑐𝑚
𝑐𝑛
(14)
where ℎ is integral constant.
Systems (11) and (13) are planar dynamical systems
defined by the 5-parameter (𝛼, 𝛽, 𝑐, 𝑚, 𝑛). Usually their first
integral (14) can be rewritten as
𝐻 (𝜙, 𝑦) ≡
2𝛼 𝑚 2𝛽 𝑛 𝑦2
𝜙 +
𝜙 + 2 = ℎ.
𝑐𝑚
𝑐𝑛
𝜙
(15)
When 𝑛 − 𝑚 = 2𝑘 − 1, (𝑘 ∈ 𝑁), system (13) has two
equilibrium points 𝑂(0, 0) and 𝐴((−𝛼/𝛽)1/(𝑛−𝑚) , 0) in the 𝜙axes. When 𝑛 − 𝑚 = 2𝑘, (𝑘 ∈ 𝑁) and 𝛼𝛽 < 0, system (13) has
three equilibrium point 𝑂(0, 0) and 𝐵1,2 (±(−𝛼/𝛽)1/(𝑛−𝑚) , 0) in
the 𝜙-axes. When 𝑛−𝑚 = 2𝑘, (𝑘 ∈ 𝑁) and 𝛼𝛽 > 0, system (13)
has only one equilibrium points 𝑂(0, 0). When 𝑛 = 𝑚 = 𝑘,
(𝑘 ∈ 𝑁), system (13) has only one equilibrium point 𝑂(0, 0).
Respectively, substituting every equilibrium point (the
point 𝑂(0, 0) is exception) into (15), we have
𝛼 1/(𝑛−𝑚)
2𝛼 (𝑛 − 𝑚) 𝛼 𝑚/(𝑛−𝑚)
, 0) =
,
(− )
ℎ𝐴 = 𝐻 ((− )
𝛽
𝑐𝑚𝑛
𝛽
𝛼 1/(𝑛−𝑚)
ℎ𝐵1 = 𝐻 (𝐵1 ) = 𝐻 ((− )
, 0)
𝛽
=
2𝛼 (𝑛 − 𝑚) 𝛼 𝑚/(𝑛−𝑚)
,
(− )
𝑐𝑚𝑛
𝛽
𝛼 1/(𝑛−𝑚)
, 0) = (−1)𝑚 ℎ𝐵1 .
ℎ𝐵2 = 𝐻 (𝐵2 ) = 𝐻 (−(− )
𝛽
(10)
(16)
where “󸀠 ” is the derivative with respect to 𝜉 and 𝑐 is wave
speed.
Clearly, (10) is equivalent to the following twodimensional systems:
From the transformation 𝑢(𝑥, 𝑡) = ln |𝜙(𝑥 − 𝑐𝑡)|, we
know that ln |𝜙(𝑥 − 𝑐𝑡)| approaches to −∞ if a solution
𝜙(𝜉) approaches to 0 in (14). In other words, this solution
determines an unboundary wave solution or blow-up wave
solution of (1). It is easy to see that the second equation of
system (11) is not continuous when 𝜙 = 0. In other words,
on such straight line 𝜙 = 0 in the phase plane (𝜙, 𝑦), the
function 𝜙󸀠󸀠 is not defined, and this implies that the smooth
wave solutions of (1) sometimes become nonsmooth wave
solutions.
𝑑𝜙
= 𝑦,
𝑑𝜉
𝑑𝑦 𝑐𝑦2 − 𝛼𝜙𝑚+2 − 𝛽𝜙2+𝑛
=
.
𝑑𝜉
𝑐𝜙
(11)
However, when 𝜙 = 0, (11) is not equivalent to (10) and the
𝑑𝑦/𝑑𝜉 cannot be defined, so we call the 𝜙 = 0 singular line
Journal of Applied Mathematics
3
3. Exact Solutions of (1) and Their Properties
under the Conditions 𝑛 = ± 𝑚, ±2𝑚, 𝑚 ∈ Z+ ,
and 𝑚 = 2, 𝑛 = −1
In this section, we investigate exact solutions of (1) under
different kinds of parametric conditions and their properties.
3.1. Different Kinds of Solitary Wave Solutions and Unboundary Wave Solutions in the Special Cases 𝑛 = 2𝑚 ∈ Z+ or
𝑛 = 𝑚 ∈ Z+ . (i) When ℎ > 0, 𝛽𝑐 < 0, 𝑛 = 2𝑚 and 𝑚 is
even, then (14) can be reduced to
𝑦 = ±𝜙√ ℎ −
𝛽 𝑚 2
2𝛼 𝑚
𝜙 −
(𝜙 ) .
𝑐𝑚
𝑐𝑚
(17)
Taking 𝜉0 = 0 and 𝜙(0) = 𝜙1 as initial values, substituting (17)
into the 𝑑𝜙/𝑑𝜉 = 𝑦 of (11), and then integrating it yield
1/𝑚
V (𝑥, 𝑡) = ±(
𝛾1
)
Ω1 cosh (𝜔1 (𝑥 − 𝑐𝑡)) + 𝛿1
,
(18)
where 𝜙1 = (−𝛼/𝛽 + √(𝛼2 + 𝛽𝑐𝑚ℎ)/𝛽2 )1/𝑚 , Ω1 = 𝜙1𝑚 (𝛼2 +
𝛽𝑐𝑚ℎ), 𝜔1 = −𝑚√ℎ, 𝛾1 = 𝑐𝑚ℎ(𝑐𝑚ℎ − 𝛼𝜙1𝑚 ), 𝛿1 = 𝛼(𝑐𝑚ℎ −
𝛼𝜙1𝑚 ). By using program of Maple, it is easy to validate that
(18) is the solution equation (9) when 𝑚 is given. Substituting
(18) into the transformation 𝑢(𝑥, 𝑡) = ln |V(𝑥, 𝑡)|, we obtain a
solitary wave solutions of (1) as follows:
󵄨
󵄨󵄨
𝛾1
1 󵄨󵄨
󵄨󵄨
󵄨󵄨 .
𝑢 (𝑥, 𝑡) = ln 󵄨󵄨󵄨󵄨
𝑚 󵄨󵄨 Ω1 cosh (𝜔1 (𝑥 − 𝑐𝑡)) + 𝛿1 󵄨󵄨󵄨
(iii) When 𝑛 = 2𝑚, ℎ = 0, 𝛽𝑐 > 0, 𝛼𝑐 < 0 and 𝑚 is even,
(14) can be reduced to
𝑦 = ±𝜙√ −
V (𝑥, 𝑡) = ±(
Taking 𝜉0 = 0 and 𝜙(0) = (𝑚ℎ𝑐/2(𝛼 + 𝛽))1/𝑚 as initial
values, substituting (20) into the 𝑑𝜙/𝑑𝜉 = 𝑦 of (11), and then
integrating it yield
1/𝑚
V (𝑥, 𝑡) = ±(
𝑚ℎ𝑐 sech2 𝜔2 (𝑥 − 𝑐𝑡)
)
2 (𝛼 + 𝛽)
,
(21)
where 𝜔2 = (1/2)𝑚√ℎ. Substituting (21) into the transformation 𝑢(𝑥, 𝑡) = ln |V(𝑥, 𝑡)|, we obtain a solution without
boundary of (1) as follows:
󵄨
󵄨
1 󵄨󵄨 𝑚ℎ𝑐 sech2 𝜔2 (𝑥 − 𝑐𝑡) 󵄨󵄨󵄨
󵄨󵄨 .
𝑢 (𝑥, 𝑡) = ln 󵄨󵄨󵄨󵄨
󵄨󵄨
𝑚 󵄨󵄨
2 (𝛼 + 𝛽)
󵄨
(22)
Equation (21) is smooth solitary wave solution of (9) but (22)
is an exact solution without boundary of (1) because 𝑢 →
−∞ as V → 0. In order to visually show dynamical behaviors
of solutions V and 𝑢 of (21) and (22), we plot graphs of their
profiles which are shown in Figures 1(a) and 1(b), respectively.
2 (−𝛼𝑐)
)
𝛽𝑐 + 𝑚𝛼2 (𝑥 − 𝑐𝑡)2
1/𝑚
.
(24)
Substituting (24) into the transformation 𝑢(𝑥, 𝑡) = ln |V(𝑥, 𝑡)|,
we obtain an exact solutions of rational function type of (1)
as follows:
󵄨󵄨
󵄨
2𝛼𝑐
1 󵄨󵄨
󵄨󵄨
󵄨󵄨 .
𝑢 (𝑥, 𝑡) = ln 󵄨󵄨󵄨󵄨
(25)
2
2
𝑚 󵄨󵄨 𝛽𝑐 + 𝑚𝛼 (𝑥 − 𝑐𝑡) 󵄨󵄨󵄨
(iv) When 𝑛 = 2𝑚, ℎ > 0, 𝛼𝛽 > 0, 𝛽𝑐 > 0 (or 𝛼𝛽 < 0,
𝛽𝑐 > 0), and 𝑚 is odd, (14) can be reduced to
𝑦 = ±√
𝛽
𝑐𝑚ℎ 2𝛼 𝑚
2
−
𝜙 − (𝜙𝑚 ) .
𝜙√
𝑐𝑚
𝛽
𝛽
(26)
Respectively, taking 𝜉0 = 0, 𝜙(0) = 𝜙1 , and 𝜉0 = 0, 𝜙(0) = 𝜙2
as initial values, substituting (26) into the 𝑑𝜙/𝑑𝜉 = 𝑦 of (11),
and then integrating them yield
𝛾1
)
Ω1 cosh (𝜔1 (𝑥 − 𝑐𝑡)) + 𝛿1
1/𝑚
𝛾2
V (𝑥, 𝑡) = (
)
Ω2 cosh (𝜔2 (𝑥 − 𝑐𝑡)) + 𝛿2
1/𝑚
V (𝑥, 𝑡) = (
(19)
(20)
(23)
Taking 𝜉0 = 0 and 𝜙(0) = 0 as initial values, substituting (23)
into the 𝑑𝜙/𝑑𝜉 = 𝑦 of (11), and then integrating it yield
(ii) When 𝑛 = 𝑚, ℎ > 0, 𝑐/(𝛼 + 𝛽) > 0 and 𝑚 is even, (14)
can be reduced to
2 (𝛼 + 𝛽) 𝑚
𝑦 = ±𝜙√ ℎ −
𝜙 .
𝑐𝑚
𝛽 𝑚 2
2𝛼 𝑚
𝜙 −
(𝜙 ) .
𝑐𝑚
𝑐𝑚
𝜙1
where
(−𝛼/𝛽 −
√(𝛼2
is
+
given
above
1/𝑚
𝛽𝑐𝑚ℎ)/𝛽2 ) ,
Ω2
𝛼𝜙2𝑚 ),
and
=
𝜙2
𝜙2𝑚 (𝛼2
,
(27)
,
(28)
=
+ 𝛽𝑐𝑚ℎ),
𝜔2 = 𝑚√ℎ, 𝛾2 = 𝑐𝑚ℎ(𝑐𝑚ℎ −
𝛿2 = 𝛼(𝑐𝑚ℎ − 𝛼𝜙2𝑚 ).
Respectively, substituting (27) and (28) into the
transformation 𝑢(𝑥, 𝑡) = ln |V(𝑥, 𝑡)|, we obtain two smooth
unboundary wave solutions of (1) as follows:
󵄨󵄨
󵄨
𝛾1
1 󵄨󵄨󵄨
󵄨󵄨
󵄨󵄨 ,
ln 󵄨󵄨󵄨
𝑚 󵄨󵄨 Ω1 cosh (𝜔1 (𝑥 − 𝑐𝑡)) + 𝛿1 󵄨󵄨󵄨
󵄨󵄨
󵄨
𝛾2
1 󵄨󵄨
󵄨󵄨
󵄨󵄨 .
𝑢 (𝑥, 𝑡) = ln 󵄨󵄨󵄨󵄨
𝑚 󵄨󵄨 Ω2 cosh (𝜔2 (𝑥 − 𝑐𝑡)) + 𝛿2 󵄨󵄨󵄨
𝑢 (𝑥, 𝑡) =
(29)
(30)
(v) When 𝑛 = 2𝑚, 0 < ℎ < ℎ𝐴 , 𝛼𝛽 < 0, 𝛽𝑐 < 0 (or 𝛼𝛽 > 0,
𝛽𝑐 < 0) and 𝑚 is odd, (14) can be reduced to
𝑦 = ±√ −
𝛽
𝑐𝑚ℎ 2𝛼 𝑚
2
+
𝜙 + (𝜙𝑚 ) .
𝜙√−
𝑐𝑚
𝛽
𝛽
(31)
Similarly, by using (31), we obtain two smooth solitary wave
solutions of (1) which are the same as the solutions (29) and
(30).
4
Journal of Applied Mathematics
2
0.6
1.5
0.4
1
u
0.2
0.5
−2
−1
0
1
2
3
−0.2
0.1
−0.1
0.2
0.3
0.4
0.5
0.6
x
x
(a) Smooth solitary wave
(b) Unboundary wave
Figure 1: The phase portraits of solutions (21) and (22) for 𝑚 = 4, 𝛼 = 3, 𝛽 = −1, 𝑐 = 2, ℎ = 9, 𝑡 = 0.1.
(vi) When 𝑛 = 𝑚, ℎ > 0 and 𝑚 is odd, (14) can be reduced
to (20), so the obtained solution is as the same as the solution
(22).
(vii) When 𝑛 = 2𝑚, ℎ = 0, 𝛽𝑐 > 0 and 𝑚 is odd, (14) can
be reduced to
𝑦 = ±√
𝛽
2𝛼
2
𝜙√− 𝜙𝑚 − (𝜙𝑚 ) .
𝑐𝑚
𝛽
V (𝑥, 𝑡) = (
)
.
(33)
Substituting (33) into the transformation 𝑢(𝑥, 𝑡) = ln |V(𝑥, 𝑡)|,
we obtain a smooth unboundary wave solution of rational
function type of (1) as follows:
󵄨󵄨
󵄨󵄨
󵄨󵄨
2𝛼
1 󵄨󵄨󵄨
󵄨󵄨 .
𝑢 (𝑥, 𝑡) = ln 󵄨󵄨󵄨
(34)
󵄨
2
2
𝑚 󵄨󵄨 𝛽 [1 + (𝑚𝛼 /𝛽𝑐) (𝑥 − 𝑐𝑡) ] 󵄨󵄨󵄨
󵄨
󵄨
3.2. Periodic Blow-Up Wave Solutions, Broken Kink Wave, and
Antikink Wave Solutions in the Special Case 𝑛 = 2𝑚 ∈ Z+ . (1)
When 𝑛 = 2𝑚, 𝛼𝛽 < 0, 𝛽𝑐 > 0, ℎ𝐵1 < ℎ < 0 (or ℎ𝐵2 < ℎ < 0)
and 𝑚 is even, (14) can be reduced to
𝑦 = ±√
𝛽
𝑐𝑚ℎ 2𝛼 𝑚
2
𝜙√
−
𝜙 − (𝜙𝑚 ) .
𝑐𝑚
𝛽
𝛽
Taking 𝜉0 = 0, 𝜙(0) = (−𝛼/𝛽 + √𝛼2 /𝛽2 + 𝑚ℎ𝑐/𝛽)
𝜉0 = 0, 𝜙(0) = (−𝛼/𝛽 − √𝛼2 /𝛽2 + 𝑚ℎ𝑐/𝛽)
𝑐𝑚ℎ
)
𝛼 − 𝛽Ω3 sin (arcsin 𝐷1 − 𝜔3 (𝑥 − 𝑐𝑡))
1/𝑚
𝑐𝑚ℎ
V (𝑥, 𝑡) = −(
)
𝛼 + 𝛽Ω3 sin (arcsin 𝐷2 + 𝜔3 (𝑥 − 𝑐𝑡))
= √(𝛼2 + 𝑐𝑚ℎ𝛽)/𝛽2 , 𝜔3
where Ω3
1/𝑚
𝛽 [1 + (𝑚𝛼2 /𝛽𝑐) (𝑥 − 𝑐𝑡)2 ]
1/𝑚
V (𝑥, 𝑡) = (
(32)
Taking 𝜉0 = 0 and 𝜙(0) = (−2𝛼/𝛽)1/𝑚 as initial values,
substituting (32) into the 𝑑𝜙/𝑑𝜉 = 𝑦 of (11), and then
integrating it yield
−2𝛼
values, substituting (35) into the 𝑑𝜙/𝑑𝜉 = 𝑦 of (11), and then
integrating them yield
2
as the initial
1/𝑚
,
(37)
= 𝑚√−ℎ, 𝐷1
=
2
𝑢 (𝑥, 𝑡) =
󵄨󵄨
󵄨
𝑐𝑚ℎ
1 󵄨󵄨󵄨
󵄨󵄨
󵄨󵄨 ,
ln 󵄨󵄨󵄨
𝑚 󵄨󵄨 𝛼 − 𝛽Ω3 sin (arcsin 𝐷1 − 𝜔3 (𝑥 − 𝑐𝑡)) 󵄨󵄨󵄨
(38)
𝑢 (𝑥, 𝑡) =
󵄨󵄨
󵄨
1 󵄨󵄨󵄨
𝑐𝑚ℎ
󵄨󵄨
󵄨󵄨 .
ln 󵄨󵄨󵄨
𝑚 󵄨󵄨 𝛼 + 𝛽Ω3 sin (arcsin 𝐷2 + 𝜔3 (𝑥 − 𝑐𝑡)) 󵄨󵄨󵄨
(39)
(2) When 𝑛 = 2𝑚, 𝛼𝛽 < 0, 𝛽𝑐 > 0, ℎ𝐵2 < ℎ < 0 and 𝑚 is
odd, (14) can be reduced to
𝑦 = ±√
and
(36)
(𝛼 + 𝛼𝛽Ω3 + 𝛽𝑐𝑚ℎ)/𝛽Ω3 (𝛼 + 𝛽Ω3 ), 𝐷2 = (𝛼 − 𝛼𝛽Ω3 +
𝛽𝑐𝑚ℎ)/𝛽Ω3 (−𝛼 + 𝛽Ω3 ). Respectively, substituting (36) and
(37) into the transformation 𝑢(𝑥, 𝑡) = ln |V(𝑥, 𝑡)|, we obtain
two periodic blow-up wave solutions of (1) as follows:
(35)
1/𝑚
,
𝛽
𝑐𝑚ℎ 2𝛼 𝑚
2
𝜙√
−
𝜙 − (𝜙𝑚 ) ,
𝑐𝑚
𝛽
𝛽
(40)
which is the same as (35), so the obtained solution is the same
as solution (39). Similarly, when 𝑛 = 2𝑚, 𝛼𝛽 < 0, 𝛽𝑐 > 0,
Journal of Applied Mathematics
5
ℎ𝐵1 < ℎ < 0 and 𝑚 is odd, the obtained solution is also the
same as solution (38).
(3) when 𝑛 = 2𝑚, 𝛼𝛽 < 0, 𝛽𝑐 < 0, ℎ = ℎ𝐵1 = ℎ𝐵2 =
2
−𝛼 /𝛽𝑐𝑚 and 𝑚 is even, (14) can be reduced to
𝑦 = ±√−
1
𝜙 (𝛽𝜙𝑚 + 𝛼) .
𝑐𝑚𝛽
(41)
Substituting (41) into the 𝑑𝜙/𝑑𝜉 = 𝑦 of (11) and then
integrating it and setting the integral constants as zero yield
V (𝑥, 𝑡) = (−
1/𝑚
Ω
𝛼
(1 ± tanh 4 (𝑥 − 𝑐𝑡))) ,
2𝛽
2
(42)
where Ω4 = 𝛼√−𝑚/𝑐𝛽. Substituting (42) into the transformation 𝑢(𝑥, 𝑡) = ln |V(𝑥, 𝑡)|, we obtain an unboundary wave
solution of (1) as follows:
𝑢 (𝑥, 𝑡) =
󵄨󵄨
Ω
1 󵄨󵄨󵄨󵄨 𝛼
󵄨
ln 󵄨󵄨 [1 ± tanh 4 (𝑥 − 𝑐𝑡)]󵄨󵄨󵄨 .
󵄨󵄨
𝑚 󵄨󵄨 2𝛽
2
(43)
Equation (42) is smooth kink wave solution of (9), but (43) is
a broken kink wave solution without boundary of (1) because
𝑢 → −∞ as V → 0. In order to visually show dynamical
behaviors of solutions V and 𝑢 of (42) and (43), we plot graphs
of their profiles which are shown in Figures 2(a) and 2(b),
respectively.
(4) When 𝑛 = 2𝑚, 𝛼𝛽 > 0, 𝛽𝑐 < 0, ℎ = ℎ𝐴 or 𝛼𝛽 < 0,
𝛽𝑐 < 0, ℎ = ℎ𝐴 and 𝑚 is odd, (14) can be reduced to (41). So
the obtained solution is also the same as the solution (43).
3.3. Periodic Blow-Up Wave Solutions and Solitary Wave
Solutions of Blow-Up Form in the Special Case 𝑚 = 2, 𝑛 = −1.
When 𝑛 = −1, (14) can be reduced to
𝑦 = ±√ ℎ𝜙2 −
2𝛼 𝑚+2 2𝛽
+
𝜙
𝜙.
𝑐𝑚
𝑐
(44)
(a) Under the conditions 𝛼𝛽 > 0, 𝛼𝑐 > 0, ℎ ∈ (−∞, +∞)
and 𝑚 = 2, (44) becomes
𝛼
𝑦 = ±√ √(𝜗1 − 𝜙) (𝜙 − 0) (𝜙 − 𝛾1 ) (𝜙 − 𝛿1 ),
𝑐
𝜙 ∈ (0, 𝜗1 ] ,
(45)
where 𝜗1 , 𝛾1 , 𝛿1 are three roots of equation 2𝛽/𝛼 + (𝑐ℎ/𝛼)𝜙 −
𝜙3 and 𝜗1 > 0 > 𝛾1 > 𝛿1 . These three roots 𝜗1 , 𝛾1 , 𝛿1 can
be obtained by Cardano formula as long as the parameters
𝛼, 𝛽, 𝑐, ℎ are fixed (or given) concretely. For example, taking
𝛼 = 4.0, 𝛽 = 2.0, 𝑐 = 4.0, ℎ = 9.0, the 𝜗1 ≐ 3.054084215,
𝛾1 ≐ −0.1112641576, 𝛿1 ≐ −2.942820058. Taking (0, 𝜗1 ) as
the initial values of the variables (𝜉, 𝜙), substituting (45) into
the 𝑑𝜙/𝑑𝜉 = 𝑦 of (11), and then integrating it yield
𝜗 (−𝛿1 ) cn2 (Ω1 (𝜉) , 𝑘1 )
,
V (𝑥, 𝑡) = 𝜙 (𝜉) = 1
(−𝛿1 ) + 𝜗1 sn2 (Ω1 𝜉, 𝑘1 )
=
(1/2)√−𝛼𝛿1 (𝜗1 − 𝛿1 )/𝑐, 𝑘1
=
where Ω1
√𝜗1 (𝛾1 − 𝛿1 )/ − 𝛿1 (𝜗1 − 𝛾1 ), 𝑇1 = (1/Ω1 )sn−1 (1, 𝑘1 ). By
using the transformation 𝑢(𝑥, 𝑡) = ln |V(𝑥, 𝑡)| and (46), we
obtain a periodic blow-up wave solution of (1) as follows:
󵄨󵄨 𝜗 (−𝛿 ) cn2 (Ω (𝑥−𝑐𝑡) , 𝑘 ) 󵄨󵄨
󵄨
1
1
1 󵄨󵄨
󵄨󵄨 , 𝑥 − 𝑐𝑡=𝑇
̸ 1.
𝑢 (𝑥, 𝑡) = ln 󵄨󵄨󵄨󵄨 1
2
󵄨󵄨 (−𝛿1 )+𝜗1 sn (Ω1 (𝑥−𝑐𝑡) , 𝑘1 ) 󵄨󵄨󵄨
(47)
(b) Under the conditions 𝛼𝛽 > 0, 𝛼𝑐 > 0, ℎ ∈ (ℎ𝐴 , +∞)
and 𝑚 = 2, (44) becomes
𝛼
𝑦 = ±√ √(𝜗1 − 𝜙) (0 − 𝜙) (𝛾1 − 𝜙) (𝜙 − 𝛿1 ),
𝑐
𝜙 ∈ [𝛿1 , 𝛾1 ] .
(48)
Similarly, taking (0, 𝛿1 ) as the initial values of the variables
(𝜉, 𝜙), substituting (48) into the 𝑑𝜙/𝑑𝜉 = 𝑦 of (11), and then
integrating it yield
V (𝑥, 𝑡) = 𝜙 (𝜉) =
𝛾1 𝛿1
.
𝛿1 + (𝛾1 − 𝛿1 ) sn2 (Ω1 𝜉, 𝑘1 )
(49)
By using the transformation 𝑢(𝑥, 𝑡) = ln |V(𝑥, 𝑡)| and (49), we
obtain a periodic blow-up wave solution of (1) as follows:
󵄨󵄨
󵄨󵄨
𝛾1 𝛿1
󵄨󵄨
󵄨
󵄨󵄨 .
𝑢 (𝑥, 𝑡) = ln 󵄨󵄨󵄨󵄨
2
󵄨󵄨 𝛿1 + (𝛾1 − 𝛿1 ) sn (Ω1 (𝑥 − 𝑐𝑡) , 𝑘1 ) 󵄨󵄨󵄨
(50)
(c) Under the conditions 𝛼𝛽 < 0, 𝛼𝑐 > 0, ℎ ∈ (−∞, +∞)
and 𝑚 = 2, (44) becomes
𝛼
𝑦 = ±√ √(𝜗2 − 𝜙) (𝜂2 − 𝜙) (0 − 𝜙) (𝜙 − 𝛿2 ),
𝑐
𝜙 ∈ [𝛿2 , 0) ,
(51)
where 𝜗2 , 𝜂2 , 0, 𝛿2 are roots of equation 𝜙(2𝛽/𝛼 + (𝑐ℎ/𝛼)𝜙 −
𝜙3 ) = 0 and 𝜗2 > 𝜂2 > 0 > 𝛿2 . As in the above cases, these
three roots can be obtained by Cardano formula as long as
the parameters 𝛼, 𝛽, 𝑐, ℎ are given concretely, so the similar
cases will be not commentated anymore in the following
discussions. Similarly, taking (0, 𝛿2 ) as the initial values of the
variables (𝜉, 𝜙), substituting (51) into the 𝑑𝜙/𝑑𝜉 = 𝑦 of (11),
and then integrating it yield
V (𝑥, 𝑡) = 𝜙 (𝜉) =
𝜂2 𝛿2 sn2 (Ω2 (𝑥 − 𝑐𝑡) , 𝑘2 )
,
(𝜂2 − 𝛿2 ) + 𝛿2 sn2 (Ω2 𝜉, 𝑘2 )
𝜉 ≠ 𝑇2 ,
(52)
=
(1/2)√𝛼𝜗2 (𝜂2 − 𝛿2 )/𝑐, 𝑘2
=
where Ω2
√−𝛿2 (𝜗2 − 𝜂2 )/𝜗2 (𝜂2 − 𝛿2 ), 𝑇2 = (1/Ω2 )sn−1 (0, 𝑘2 ). By
using the transformation 𝑢(𝑥, 𝑡) = ln |V(𝑥, 𝑡)| and (52), we
obtain a periodic blow-up wave solution of (1) as follows:
󵄨󵄨󵄨 𝜂 𝛿 sn2 (Ω2 (𝑥 − 𝑐𝑡) , 𝑘2 ) 󵄨󵄨󵄨
󵄨󵄨 , 𝑥 − 𝑐𝑡 ≠ 𝑇 .
𝑢 (𝑥, 𝑡) = ln 󵄨󵄨󵄨󵄨 2 2
󵄨
2
󵄨󵄨 (𝜂2 − 𝛿2 ) + 𝛿2 sn2 (Ω2 𝜉, 𝑘2 ) 󵄨󵄨󵄨
(53)
(d) Under the conditions 𝛼𝛽 < 0, 𝛼𝑐 > 0, ℎ ∈ (ℎ𝐴 , +∞)
and 𝑚 = 2, (44) becomes
𝜉 ≠ 𝑇1 ,
(46)
𝛼
𝑦 = ±√ √(𝜗2 − 𝜙) (𝜙 − 𝜂2 ) (𝜙 − 0) (𝜙 − 𝛿2 ).
𝑐
(54)
6
Journal of Applied Mathematics
−30
x
−10
−20
x
0
10
−4
−2
0
2
4
6
8
−0.1
−0.2
−0.2
−0.3
u −0.4
−0.4
−0.6
−0.5
−0.8
−0.6
(a) Smooth kink wave
(b) Unboundary wave
Figure 2: The phase portraits of solutions (42) and (43) for 𝑚 = 4, 𝛼 = 0.3, 𝛽 = −0.3, 𝑐 = 2, 𝑡 = 0.1.
Taking the (0, 𝜗2 ) as the initial values of the variables (𝜉, 𝜙),
substituting (54) into the 𝑑𝜙/𝑑𝜉 = 𝑦 of (11), and then
integrating it yield
V (𝑥, 𝑡) = 𝜙 (𝜉) =
2
𝜗2 (𝜂2 − 𝛿2 ) dn (Ω2 (𝑥 − 𝑐𝑡) , 𝑘2 )
. (55)
(𝜂2 − 𝛿2 ) + (𝜗2 − 𝜂2 ) sn2 (Ω2 𝜉, 𝑘2 )
By using the transformations 𝑢(𝑥, 𝑡) = ln |V(𝑥, 𝑡)| and (55), we
obtain a periodic blow-up wave solution of (1) as follows:
󵄨󵄨
󵄨󵄨
𝜗2 (𝜂2 − 𝛿2 ) dn2 (Ω2 (𝑥 − 𝑐𝑡) , 𝑘2 )
󵄨󵄨
󵄨
󵄨󵄨 .
𝑢 (𝑥, 𝑡) = ln 󵄨󵄨󵄨󵄨
2
󵄨󵄨 (𝜂2 − 𝛿2 ) + (𝜗2 − 𝜂2 ) sn (Ω2 (𝑥 − 𝑐𝑡) , 𝑘2 ) 󵄨󵄨󵄨
(56)
(e) Under the conditions 𝛼𝛽 < 0, 𝛼𝑐 < 0, ℎ ∈ (−∞, ℎ𝐴 )
and 𝑚 = 2, (44) becomes
𝛼
𝑦 = ±√− √(𝜗3 − 𝜙) (𝜂3 − 𝜙) (𝜙 − 0) (𝜙 − 𝛿3 ),
𝑐
𝜙 ∈ (0, 𝜂3 ] ,
(57)
where 𝜗3 , 𝜂3 , 0, 𝛿3 are roots of equation 𝜙(−2𝛽/𝛼 − (𝑐ℎ/𝛼)𝜙 +
𝜙3 ) = 0 and 𝜗3 > 𝜂3 > 0 > 𝛿3 . Taking the (0, 𝜂3 ) as the
initial values of the variables (𝜉, 𝜙), substituting (57) into the
𝑑𝜙/𝑑𝜉 = 𝑦 of (11), and then integrating it yield
V (𝑥, 𝑡) = 𝜙 (𝜉) =
2
𝜗3 𝜂3 cn (Ω3 𝜉, 𝑘3 )
,
𝛿3 − 𝜂3 sn2 (Ω3 𝜉, 𝑘3 )
𝜉 ≠ 𝑇3 ,
(58)
=
(1/2)√−𝛼𝜗3 (𝜂3 − 𝛿3 )/𝑐, 𝑘3
=
where Ω3
√𝜂3 (𝜗3 − 𝛿3 )/𝜗3 (𝜂3 − 𝛿3 ), 𝑇3 = (1/Ω3 )sn−1 (1, 𝑘3 ). By
using the transformation 𝑢(𝑥, 𝑡) = ln |V(𝑥, 𝑡)| and (58), we
obtain a periodic blow-up wave solution of (1) as follows:
󵄨󵄨 𝜗 𝜂 cn2 (Ω (𝑥 − 𝑐𝑡) , 𝑘 ) 󵄨󵄨
󵄨
3
3 󵄨󵄨
󵄨󵄨 , 𝑥 − 𝑐𝑡 ≠ 𝑇3 .
𝑢 (𝑥, 𝑡) = ln 󵄨󵄨󵄨󵄨 3 3
2
󵄨󵄨 𝛿3 − 𝜂3 sn (Ω3 (𝑥 − 𝑐𝑡) , 𝑘3 ) 󵄨󵄨󵄨
(59)
(f) Under the conditions 𝛼𝛽 < 0, 𝛼𝑐 < 0, ℎ ∈ (−∞, ℎ𝐴 )
and 𝑚 = 2, (44) becomes
𝛼
𝑦 = ±√− √(𝜗4 − 𝜙) (0 − 𝜙) (𝜙 − 𝛾4 ) (𝜙 − 𝛿4 ),
𝑐
𝜙 ∈ [𝛾4 , 0) ,
(60)
where 𝜗4 , 0, 𝛾4 , 𝛿4 are roots of equation 𝜙(−2𝛽/𝛼 − (𝑐ℎ/𝛼)𝜙 +
𝜙3 ) = 0 and 𝜗4 > 0 > 𝛾4 > 𝛿4 . Taking the (0, 𝛾4 ) as the
initial values of the variables (𝜉, 𝜙), substituting (60) into the
𝑑𝜙/𝑑𝜉 = 𝑦 of (11), and then integrating it yield
V (𝑥, 𝑡) = 𝜙 (𝜉) =
𝛿4 𝛾4 cn2 (Ω4 𝜉, 𝑘4 )
,
𝛿4 − 𝛾4 cn2 (Ω4 𝜉, 𝑘4 )
𝜉 ≠ 𝑇4 ,
(61)
=
(1/2)√𝛼𝛿4 (𝜗4 − 𝛾4 )/𝑐, 𝑘4
=
where Ω4
√𝛾4 (𝜗4 − 𝛿4 )/𝜗4 (𝜗4 − 𝛾4 ), 𝑇4
= (1/Ω4 )sn−1 (1, 𝑘4 ). By
using the transformation 𝑢(𝑥, 𝑡) = ln |V(𝑥, 𝑡)| and (61), we
obtain a periodic below-up wave solutions of (1) as follows:
󵄨󵄨 𝛿 𝛾 cn2 (Ω (𝑥 − 𝑐𝑡) , 𝑘 ) 󵄨󵄨
󵄨󵄨
󵄨
4
4
󵄨󵄨 , 𝑥 − 𝑐𝑡 ≠ 𝑇4 .
𝑢 (𝑥, 𝑡) = ln 󵄨󵄨󵄨󵄨 4 4
2
󵄨󵄨 𝛿4 − 𝛾4 cn (Ω4 (𝑥 − 𝑐𝑡) , 𝑘4 ) 󵄨󵄨󵄨
(62)
(g) Under the conditions 𝑚 = 2, 𝛼𝑐 < 0, ℎ = ℎ𝐴 =
−𝛼2 /2𝑐𝛽, (44) becomes
𝛼󵄨
󵄨
𝑦 = ±√− 󵄨󵄨󵄨𝜙1 − 𝜙󵄨󵄨󵄨 √𝜙 (𝜙 + 2𝜙1 ),
𝑐
𝜙 ≠ 0,
(63)
where 𝜙1 = (−𝛽/𝛼)1/3 . Taking (0, (1/2)𝜙1 ) as the initial values
of the variables (𝜉, 𝜙), substituting (63) into the 𝑑𝜙/𝑑𝜉 = 𝑦 of
(11), and then integrating it yield
V (𝑥, 𝑡) = 𝜙 (𝜉) = 𝜙1 [1 −
3
],
cosh (Ω5 𝜉) + 2
𝜉 ≠ 𝑇5 ,
(64)
Journal of Applied Mathematics
7
where Ω5 = 𝜙1 √−3𝛼/𝑐, 𝑇5 = (1/Ω5 )cosh−1 (1). Clearly, we
have 𝜙(𝑇5 ) = 0, 𝜙(±∞) = 𝜙1 . By using the transformation
𝑢(𝑥, 𝑡) = ln |V(𝑥, 𝑡)| and (64), we obtain a solitary wave
solution of blow-up form of (1) as follows:
󵄨󵄨
󵄨󵄨
3
󵄨
󵄨
𝑢 (𝑥, 𝑡) = ln 󵄨󵄨󵄨󵄨𝜙1 [1 −
]󵄨󵄨󵄨󵄨 , 𝑥 − 𝑐𝑡 ≠ 𝑇5 .
cosh (Ω5 (𝑥 − 𝑐𝑡)) + 2 󵄨󵄨
󵄨󵄨
(65)
Equation (64) is smooth solitary wave solution of (9), but (65)
is a solitary wave solution of blow-up form for (1) because
𝑢 → −∞ as V → 0. In order to visually show dynamical
behaviors of solutions V and 𝑢 of (64) and (65), we plot graphs
of their profiles which are shown in Figures 3(a) and 3(b),
respectively.
3.4. Different Kinds of Periodic Wave, Broken Solitary Wave
Solutions in the Special Case 𝑚 = 1, 𝑛 = −1. (1) Under the
conditions ℎ ∈ (ℎ𝐴 , +∞), 𝛼𝛽 > 0, 𝛽𝑐 > 0 (or 𝛼𝛽 < 0, 𝛽𝑐 > 0),
𝛼𝑐 > 0 and 𝑚 = 1, (44) becomes
2𝛼
𝑦 = ±√ √(𝜙𝑀 − 𝜙) (𝜙 − 𝜙𝑚 ) (𝜙 − 0),
𝑐
where Ω7 = √−𝛼𝜙𝑀/2𝑐, 𝑘7 = √𝜙𝑀/𝜙𝑚 , 𝑇7 =
(1/Ω7 )sn−1 (0, 𝑘7 ). By using the transformation 𝑢(𝑥, 𝑡) =
ln |V(𝑥, 𝑡)| and (70), we obtain a periodic blow-up wave
solution of (1) as follows:
󵄨
󵄨
𝑢 (𝑥, 𝑡) = ln 󵄨󵄨󵄨󵄨𝜙𝑀sn2 (Ω7 (𝑥 − 𝑐𝑡) , 𝑘7 )󵄨󵄨󵄨󵄨 , 𝑥 − 𝑐𝑡 ≠ 𝑇7 .
(71)
(3) Under the conditions ℎ ∈ (ℎ𝐴 , +∞), 𝛼𝛽 < 0, 𝛽𝑐 > 0
(or 𝛼𝛽 > 0, 𝛽𝑐 > 0) and 𝑚 = 1, (44) becomes
𝑦 = ±√ −
2𝛼
√(0 − 𝜙) (𝜙𝑀 − 𝜙) (𝜙 − 𝜙𝑚 ),
𝑐
where 𝜙𝑀,𝑚 = (ℎ𝑐 ± √ℎ2 𝑐2 + 16𝛼𝛽)/4𝛼 and 0 are three
roots of equation ℎ𝜙2 − (2𝛼/𝑐)𝜙3 + (2𝛽/𝑐)𝜙 = 0 and 𝜙𝑀 >
𝜙𝑚 > 0. Taking (0, 𝜙𝑚 ) as the initial values of the variables
(𝜉, 𝜙), substituting (66) into the 𝑑𝜙/𝑑𝜉 = 𝑦 of (11), and then
integrating it yield
V (𝑥, 𝑡) = 𝜙 (𝜉) = 𝜙𝑚 − (𝜙𝑀 − 𝜙𝑚 ) sn2 (Ω6 𝜉, 𝑘6 ) ,
V (𝑥, 𝑡) = 𝜙 (𝜉) =
Equation (67) is smooth periodic wave solution of (9), but
(68) is a double periodic wave solution of blow-up form of
(1) because 𝑢 → −∞ as V → 0. In order to visually show
dynamical behaviors of solutions V and 𝑢 of (67) and (68), we
plot graphs of their profiles which are shown in Figures 4(a)
and 4(b), respectively.
(2) Under the condition ℎ ∈ (−∞, ℎ𝐴 ), 𝛼𝛽 < 0, 𝛽𝑐 > 0 (or
𝛼𝛽 < 0, 𝛽𝑐 < 0),𝛼𝑐 > 0 and 𝑚 = 1, (44) becomes
𝑦 = ±√
2𝛼
√(0 − 𝜙) (𝜙 − 𝜙𝑀) (𝜙 − 𝜙𝑚 ),
𝑐
𝜙 ∈ [𝜙𝑀, 0) ,
(69)
where 𝜙𝑀, 𝜙𝑚 , 0 are roots of equation ℎ𝜙2 − (2𝛼/𝑐)𝜙3 +
(2𝛽/𝑐)𝜙 = 0 and 𝜙𝑚 < 𝜙𝑀 < 0. Taking (0, 𝜙𝑀) as the
initial values of the variables (𝜉, 𝜙), substituting (69) into the
𝑑𝜙/𝑑𝜉 = 𝑦 of (11), and then integrating it yield
V (𝑥, 𝑡) = 𝜙 (𝜉) = 𝜙𝑀sn2 (Ω7 𝜉, 𝑘7 ) ,
𝜉 ≠ 𝑇7 ,
(70)
dn2
𝜙𝑀
,
(Ω8 𝜉, 𝑘8 )
(73)
where Ω8 = √𝛼𝜙𝑚 /2𝑐, 𝑘8 = √(𝜙𝑀 − 𝜙𝑚 )/ − 𝜙𝑚 . By using
the transformation 𝑢(𝑥, 𝑡) = ln |V(𝑥, 𝑡)| and (73), we obtain a
periodic blow-up wave solutions of (1) as follows:
󵄨󵄨
󵄨󵄨
𝜙𝑀
󵄨󵄨
󵄨
󵄨󵄨 .
𝑢 (𝑥, 𝑡) = ln 󵄨󵄨󵄨󵄨 2
󵄨󵄨 dn (Ω8 (𝑥 − 𝑐𝑡) , 𝑘8 ) 󵄨󵄨󵄨
(74)
(4) Under the conditions ℎ ∈ (−∞, ℎ𝐴 ), 𝛼𝛽 < 0, 𝛽𝑐 > 0
(or 𝛼𝛽 < 0, 𝛽𝑐 < 0) and 𝑚 = 1, (44) becomes
(67)
where Ω6 = √𝛼𝜙𝑀/2𝑐, 𝑘6 = √(𝜙𝑀 − 𝜙𝑚 )/𝜙𝑀. By using the
transformation 𝑢(𝑥, 𝑡) = ln |V(𝑥, 𝑡)| and (67), we obtain a
double periodic wave solution of (1) as follows:
󵄨
󵄨
𝑢 (𝑥, 𝑡) = ln 󵄨󵄨󵄨󵄨𝜙𝑚 − (𝜙𝑀 − 𝜙𝑚 ) sn2 (Ω6 (𝑥 − 𝑐𝑡) , 𝑘6 )󵄨󵄨󵄨󵄨 .
(68)
(72)
where 𝜙𝑀, 𝜙𝑚 , 0 are roots of equation ℎ𝜙2 − (2𝛼/𝑐)𝜙3 +
(2𝛽/𝑐)𝜙 = 0 and 0 > 𝜙𝑀 > 𝜙𝑚 . Taking (0, 𝜙𝑀) as the
initial values of the variables (𝜉, 𝜙), substituting (72) into the
𝑑𝜙/𝑑𝜉 = 𝑦 of (11), and then integrating it yield
𝜙 ∈ [𝜙𝑚 , 𝜙𝑀] ,
(66)
𝜙 ∈ [𝜙𝑚 , 𝜙𝑀] ,
𝑦 = ±√ −
2𝛼
√(𝜙𝑀 − 𝜙) (𝜙𝑚 − 𝜙) (𝜙 − 0),
𝑐
𝜙 ∈ (0, 𝜙𝑚 ] ,
(75)
where 𝜙𝑀, 𝜙𝑚 , 0 are roots of equation ℎ𝜙2 − (2𝛼/𝑐)𝜙3 +
(2𝛽/𝑐)𝜙 = 0 and 𝜙𝑀 > 𝜙𝑚 > 0. Taking (0, 𝜙𝑚 ) as the
initial values of the variables (𝜉, 𝜙), substituting (75) into the
𝑑𝜙/𝑑𝜉 = 𝑦 of (11), and then integrating it yield
V (𝑥, 𝑡) = 𝜙 (𝜉) =
𝜙𝑚 cn2 (Ω9 𝜉, 𝑘9 )
,
dn2 (Ω9 𝜉, 𝑘9 )
𝜉 ≠ 𝑇9 ,
(76)
where Ω9 = √−𝛼𝜙𝑀/2𝑐, 𝑘9 = √𝜙𝑚 /𝜙𝑀, 𝑇9 =
(1/Ω9 )sn−1 (0, 𝑘9 ). By using the transformation 𝑢(𝑥, 𝑡) =
ln |V(𝑥, 𝑡)| and (76), we obtain periodic blow-up wave solutions of (1) as follows:
󵄨󵄨 𝜙 cn2 (Ω (𝑥 − 𝑐𝑡) , 𝑘 ) 󵄨󵄨
󵄨
9
9 󵄨󵄨
󵄨󵄨 ,
𝑢 (𝑥, 𝑡) = ln 󵄨󵄨󵄨󵄨 𝑚 2
󵄨󵄨 dn (Ω9 (𝑥 − 𝑐𝑡) , 𝑘9 ) 󵄨󵄨󵄨
𝑥 − 𝑐𝑡 ≠ 𝑇9 .
(77)
(5) Under the conditions ℎ = ℎ𝐴 = −4𝛼𝜙1 /𝑐, 𝛼𝛽 < 0,
𝛽𝑐 > 0 and 𝑚 = 1, (44) becomes
𝑦 = ±√
2𝛼 󵄨󵄨
󵄨
󵄨𝜙 + 𝜙󵄨󵄨󵄨 √−𝜙,
𝑐 󵄨 1
𝜙 ∈ [−𝜙1 , 0) ,
(78)
8
Journal of Applied Mathematics
x
1
−15
−10
−5
5
0
0.5 10
−0.5
−1
−20
−15
−10
5
−5
10
u
15
x
−1.5
−0.5
−2
−2.5
−1
(a) Smooth solitary wave
(b) Solitary wave of blow-up form
Figure 3: The phase portraits of solutions (64) and (65) for 𝑚 = 2, 𝛼 = 0.3, 𝛽 = −0.3, 𝑐 = −2, 𝑡 = 0.1.
1
x
−8
−6
−4
−2
0
2
4
0.5
−6
−4
x
2
−2
4
6
8
−0.5
−1
u
−0.5
−1
−1.5
−1.5
−2
−2
(a) Smooth periodic wave
(b) Double periodic wave of blow-up type
Figure 4: The phase portraits of solutions (67) and (68) for ℎ = 8, 𝛼 = 2.5, 𝛽 = −2.5, 𝑐 = 2, 𝑡 = 0.1.
where 𝜙1 = (−𝛽/𝛼)1/2 . Taking (0, −𝜙1 ) as the initial values of
the variables (𝜉, 𝜙), substituting (78) into the 𝑑𝜙/𝑑𝜉 = 𝑦 of
(11), and then integrating it yield
V (𝑥, 𝑡) = 𝜙 (𝜉) = −𝜙1 tanh2 (0.5Ω10 𝜉) ,
𝜉 ≠ 0,
(79)
where Ω10 = √2𝛼𝜙1 /𝑐. By using the transformation 𝑢(𝑥, 𝑡) =
ln |V(𝑥, 𝑡)| and (79), we obtain a solitary wave solution of
blow-up form of (1) as follows:
󵄨
󵄨
𝑢 (𝑥, 𝑡) = ln 󵄨󵄨󵄨󵄨𝜙1 tanh2 (0.5Ω10 (𝑥 − 𝑐𝑡))󵄨󵄨󵄨󵄨 ,
𝑥 − 𝑐𝑡 ≠ 0.
(80)
(6) Under the conditions ℎ = ℎ𝐴 = 4𝛼𝜙1 /𝑐, 𝛼𝛽 < 0, 𝛽𝑐 >
0 and 𝑚 = 1, (44) becomes
𝑦 = ±√ −
2𝛼 󵄨󵄨
󵄨
󵄨𝜙 − 𝜙1 󵄨󵄨󵄨 √𝜙,
𝑐 󵄨
𝜙 ∈ (0, 𝜙1 ] ,
(81)
where 𝜙1 = (−𝛽/𝛼)1/2 . Taking (0, 𝜙1 ) as the initial values of
the variables (𝜉, 𝜙), substituting (81) into the 𝑑𝜙/𝑑𝜉 = 𝑦 of
(11), and then integrating it yield
V (𝑥, 𝑡) = 𝜙 (𝜉) = 𝜙1 tanh2 (0.5Ω11 𝜉) ,
𝜉 ≠ 0,
(82)
Journal of Applied Mathematics
9
where Ω11 = √−2𝛼𝜙1 /𝑐. By using the transformation
𝑢(𝑥, 𝑡) = ln |V(𝑥, 𝑡)| and (82), we obtain a broken solitary wave
solution of (1) as follows:
󵄨
󵄨
𝑢 (𝑥, 𝑡) = ln 󵄨󵄨󵄨󵄨𝜙1 tanh2 (0.5Ω11 (𝑥 − 𝑐𝑡))󵄨󵄨󵄨󵄨 , 𝑥 − 𝑐𝑡 ≠ 0. (83)
(7) Under the conditions ℎ ∈ (−∞, 0], 𝛼𝛽 > 0, 𝛽𝑐 < 0 and
𝑚 = 1, (44) becomes
𝑦 = ±√ −
2𝛼
√(𝜙𝑀 − 𝜙) (0 − 𝜙) (𝜙 − 𝜙𝑚 ),
𝑐
𝜙 ∈ [𝜙𝑚 , 0) ,
(84)
where 𝜙𝑀, 𝜙𝑚 , 0 are roots of equation 𝜙2 ℎ − (2𝛼/𝑐)𝜙3 +
(2𝛽/𝑐)𝜙 = 0 and 𝜙𝑀 > 0 > 𝜙𝑚 . Taking (0, 𝜙𝑀) as the
initial values of the variables (𝜉, 𝜙), substituting (84) into the
𝑑𝜙/𝑑𝜉 = 𝑦 of (11), and then integrating it yield
V (𝑥, 𝑡) = 𝜙 (𝜉) =
2
sn2 (Ω12 𝜉, 𝑘12 )
−𝜙𝑀𝑘12
,
dn2 (Ω12 𝜉, 𝑘12 )
𝜉 ≠ 𝑇12 ,
(85)
where Ω12 = √−𝛼(𝜙𝑀 − 𝜙𝑚 )/2𝑐, 𝑘12 = √−𝜙𝑚 /(𝜙𝑀 − 𝜙𝑚 ),
𝑇12 = (1/Ω12 )sn−1 (0, 𝑘12 ). By using the transformation
𝑢(𝑥, 𝑡) = ln |V(𝑥, 𝑡)| and (85), we obtain a periodic blow-up
wave solution of (1) as follows:
󵄨󵄨 −𝜙 𝑘2 sn2 (Ω (𝑥 − 𝑐𝑡) , 𝑘 ) 󵄨󵄨
󵄨
12
12 󵄨󵄨
󵄨󵄨 ,
𝑢 (𝑥, 𝑡) = ln 󵄨󵄨󵄨󵄨 𝑀 12
󵄨󵄨
dn2 (Ω12 (𝑥 − 𝑐𝑡) , 𝑘12 )
󵄨󵄨
󵄨
𝑥 − 𝑐𝑡 ≠ 𝑇12 .
(86)
(8) Under the conditions ℎ ∈ (−∞, 0], 𝛼𝛽 > 0, 𝛽𝑐 > 0 and
𝑚 = 1, (44) becomes
𝑦 = ±√
2𝛼
√(𝜙𝑀 − 𝜙) (𝜙 − 0) (𝜙 − 𝜙𝑚 ),
𝑐
𝜙 ∈ (0, 𝜙𝑀] ,
(87)
2
3
where 𝜙𝑀, 𝜙𝑚 , 0 are roots of equation ℎ𝜙 − (2𝛼/𝑐)𝜙 +
(2𝛽/𝑐)𝜙 = 0 and 𝜙𝑀 > 0 > 𝜙𝑚 . Taking (0, 𝜙𝑀) as the
initial values of the variables (𝜉, 𝜙), substituting (87) into the
𝑑𝜙/𝑑𝜉 = 𝑦 of (11), and then integrating it yield
V (𝑥, 𝑡) = 𝜙𝑀cn2 (Ω13 𝜉, 𝑘13 ) ,
𝜉 ≠ 𝑇13 ,
(88)
where Ω13 = √𝛼(𝜙𝑀 − 𝜙𝑚 )/2𝑐, 𝑘13 = √𝜙𝑀/(𝜙𝑀 − 𝜙𝑚 ),
𝑇13 = (1/Ω13 )sn−1 (0, 𝑘13 ). By using the transformation
𝑢(𝑥, 𝑡) = ln |V(𝑥, 𝑡)| and (88), we obtain a periodic blow-up
wave solution of (1) as follows:
󵄨
󵄨
𝑢 (𝑥, 𝑡) = ln 󵄨󵄨󵄨󵄨𝜙𝑀cn2 (Ω13 (𝑥 − 𝑐𝑡) , 𝑘13 )󵄨󵄨󵄨󵄨 , 𝑥 − 𝑐𝑡 ≠ 𝑇13 .
(89)
3.5. Periodic Blow-Up Wave, Broken Solitary Wave, Broken
Kink, and Antikink Wave Solutions in the Special Cases 𝑚 = 2
or 𝑚 = 1 and 𝑛 = −2. When 𝑚 = 2 is even and 𝑛 = −2, (14)
can be reduced to
𝑦 = ±√ ℎ𝜙2 −
2𝛼 𝑚+2 𝛽
+ .
𝜙
𝑐𝑚
𝑐
(90)
Because the singular straight line 𝜙 = 0, this implies that
(1) has broken kink and antikink wave solutions: see the
following discussions.
(i) Under the conditions 𝑚 = 2, 𝑛 = −2, ℎ = √−4𝛼𝛽/𝑐2 ,
𝛼𝛽 < 0, 𝛼𝑐 < 0, (90) becomes
𝑦 = ±√−
𝛼 󵄨󵄨 2
󵄨
󵄨󵄨𝜙 − 𝜙12 󵄨󵄨󵄨 ,
󵄨
𝑐󵄨
(𝜙 ≠ 0) ,
(91)
where 𝜙1 = √−𝛽/𝛼. Taking (0, 𝜙1 ) as the initial values of the
variables (𝜉, 𝜙), substituting (91) into the 𝑑𝜙/𝑑𝜉 = 𝑦 of (11),
and then integrating it yield
V (𝑥, 𝑡) = 𝜙 (𝜉) = ±𝜙1 tanh [Ω14 𝜉] ,
(92)
where Ω14 = 𝜙1 √−𝛼/𝑐. By using the transformations
𝑢(𝑥, 𝑡) = ln |V(𝑥, 𝑡)| and (92), we obtain a broken kink wave
solutions of (1) as follows:
󵄨
󵄨
𝑢 (𝑥, 𝑡) = ln 󵄨󵄨󵄨𝜙1 tanh [Ω14 (𝑥 − 𝑐𝑡)]󵄨󵄨󵄨 ,
(93)
where (𝑥 − 𝑐𝑡) ∈ (−∞, 0) and (𝑥 − 𝑐𝑡) ∈ (0, +∞).
(ii) Under the conditions ℎ ∈ (ℎ𝐵1 , +∞), 𝛼𝛽 > 0, 𝛼𝑐 > 0
and 𝑚 = 2, 𝑛 = −2, (90) becomes
𝛼
𝑦 = ±√ √(𝑎2 − 𝜙2 ) (𝜙2 − 𝑏2 ),
𝑐
(94)
where 𝑎2 = (1/2𝛼)(𝑐ℎ + √𝑐2 ℎ2 + 4𝛼𝛽), 𝑏2 = (1/2𝛼)(𝑐ℎ −
√𝑐2 ℎ2 + 4𝛼𝛽). Taking (0, 𝑎) as the initial values of the variables (𝜉, 𝜙), substituting (94) into the 𝑑𝜙/𝑑𝜉 = 𝑦 of (11), and
then integrating it yield
V (𝑥, 𝑡) = 𝜙 (𝜉) = ±√𝑎2 − (𝑎2 − 𝑏2 ) sn2 (Ω15 𝜉, 𝑘15 ),
(95)
where 𝑎2 = (1/2𝛼)(𝑐ℎ + √𝑐2 ℎ2 + 4𝛼𝛽), 𝑏2 = (1/2𝛼)(𝑐ℎ −
√𝑐2 ℎ2 + 4𝛼𝛽), Ω15 = 𝑎√𝛼/𝑐, 𝑘15 = √(𝑎2 − 𝑏2 )/𝑎2 . By using
the transformations 𝑢(𝑥, 𝑡) = ln |V(𝑥, 𝑡)| and (95), we obtain
a periodic blow-up wave solution of (1) as follows:
󵄨
󵄨
𝑢 (𝑥, 𝑡) = 0.5 ln 󵄨󵄨󵄨󵄨𝑎2 − (𝑎2 − 𝑏2 ) sn2 (Ω15 𝜉, 𝑘15 )󵄨󵄨󵄨󵄨 .
(96)
(iii) Under the conditions 𝑚 = 1, 𝑛 = −2, ℎ = ℎ𝐵1 =
3𝛼𝜙2 /𝑐, 𝛼𝛽 > 0, 𝛼𝑐 > 0, (90) becomes
𝑦 = ±√
𝛼 󵄨󵄨
󵄨
󵄨−𝜙 + 𝜙󵄨󵄨󵄨 √−𝜙2 − 2𝜙,
𝑐󵄨 2
(𝜙 ≠ 0) ,
(97)
where 𝜙2 = (−𝛽/𝛼)1/3 . Taking (0, −0.5𝜙2 ) as the initial values
of the variables (𝜉, 𝜙), substituting (97) into the 𝑑𝜙/𝑑𝜉 = 𝑦 of
(11), and then integrating it yield
V (𝑥, 𝑡) = 𝜙 (𝜉) = −𝜙2 [
3
− 1] ,
cosh (Ω16 𝜉) + 1
𝜉 ≠ 𝑇16 ,
(98)
where Ω16 = √−3𝛼𝜙2 /𝑐, 𝑇16 = (1/Ω16 )cosh−1 (2). By using
the transformations 𝑢(𝑥, 𝑡) = ln |V(𝑥, 𝑡)| and (98), when
10
Journal of Applied Mathematics
(𝑥 − 𝑐𝑡) ∈ (−∞, 𝑇16 ) ⋃ (𝑇16 , +∞), we obtain a broken
solitary wave solution of (1) as follows:
󵄨󵄨
󵄨󵄨
3𝜙2
󵄨
󵄨
𝑢 (𝑥, 𝑡) = ln 󵄨󵄨󵄨󵄨
− 𝜙2 󵄨󵄨󵄨󵄨 .
(99)
󵄨󵄨
󵄨󵄨 cosh (Ω16 (𝑥 − 𝑐𝑡)) + 1
(iv) Under the conditions ℎ = ℎ𝐵1 = 3𝛼𝜙2 /𝑐, 𝛼𝛽 < 0,
𝛼𝑐 < 0 and 𝑚 = 1, 𝑛 = −2, (90) becomes
𝛼󵄨
󵄨
𝑦 = ±√− 󵄨󵄨󵄨𝜙2 − 𝜙󵄨󵄨󵄨 √𝜙2 + 2𝜙,
𝑐
(𝜙 ≠ 0) ,
𝜉 ≠ 𝑇17 ,
(101)
where Ω17 = √3𝛼𝜙1 /𝑐, 𝑇17 = (1/Ω17 )cosh−1 (2). By using the
transformations 𝑢(𝑥, 𝑡) = ln |V(𝑥, 𝑡)| and (101), when (𝑥−𝑐𝑡) ∈
(−∞, 𝑇17 ) ⋃ (𝑇17 , +∞), we obtain a broken solitary wave
solution of (1) as follows:
󵄨󵄨
󵄨󵄨
3𝜙2
󵄨
󵄨󵄨
󵄨󵄨 .
𝑢 (𝑥, 𝑡) = ln 󵄨󵄨󵄨󵄨𝜙2 −
(102)
cosh (Ω17 (𝑥 − 𝑐𝑡)) + 1 󵄨󵄨󵄨
󵄨󵄨
(v) Under the conditions ℎ ∈ (ℎ𝐵1 , +∞), 𝛼𝛽 < 0, 𝛽𝑐 < 0
and 𝑚 = 1, 𝑛 = −2, (90) becomes
2𝛼
𝑦 = ±√ √(𝜗5 − 𝜙) (𝜙 − 𝜂5 ) (𝜙 − 𝛾5 ),
𝑐
𝜙 ∈ [𝜂5 , 𝜗5 ] ,
V (𝑥, 𝑡) = 𝜙 (𝜉) = 𝜗5 − (𝜗5 − 𝜂5 ) sn2 (Ω18 𝜉, 𝑘18 ) ,
(104)
where Ω18 = √𝛼(𝜗5 − 𝛾5 )/2𝑐, 𝑘18 = √(𝜗5 − 𝜂5 )/(𝜗5 − 𝛾5 ). By
using the transformations 𝑢(𝑥, 𝑡) = ln |V(𝑥, 𝑡)| and (104), we
obtain a periodic blow-up wave solution of (1) as follows:
󵄨
󵄨
𝑢 (𝑥, 𝑡) = ln 󵄨󵄨󵄨󵄨𝜗5 − (𝜗5 − 𝜂5 ) sn2 (Ω18 (𝑥 − 𝑐𝑡) , 𝑘18 )󵄨󵄨󵄨󵄨 . (105)
(vi) Under the conditions ℎ ∈ (ℎ𝐵1 , +∞), 𝛼𝛽 > 0, 𝛽𝑐 < 0
and 𝑚 = 1, 𝑛 = −2, (90) becomes
2𝛼
√(𝜗5 − 𝜙) (𝜂5 − 𝜙) (𝜙 − 𝛾5 ),
𝑐
√(2𝛽/𝑐𝑚) 𝜙𝑚 + ℎ(𝜙𝑚 )2 − (2𝛼/𝑐𝑚) (𝜙𝑚 )3
𝜙𝑚−1
=±
√2𝛼/𝑐𝑚√(𝜙𝑔𝑚
−
𝜙𝑚 ) (𝜙𝑚
(109)
−
𝜙𝑙𝑚 ) (𝜙𝑚
𝜙𝑚−1
− 0)
,
where 𝜙𝑔,𝑙 = [(1/4𝛼)(𝑐𝑚ℎ ± √𝑐2 𝑚2 ℎ2 + 16𝛼𝛽)]1/𝑚 . Taking
(0, 𝜙𝑔 ) as the initial values of the variables (𝜉, 𝜙), substituting
(109) into the 𝑑𝜙/𝑑𝜉 = 𝑦 of (11), and then integrating it yields
V (𝑥, 𝑡) = 𝜙 (𝜉) = ±[𝜙𝑔 − (𝜙𝑔 − 𝜙𝑙 ) sn2 (Ω19 𝜉, 𝑘19 )]
1/𝑚
,
(110)
where Ω19 = 𝑚√𝛼𝜙𝑔 /2𝑐𝑚, 𝑘19 = √(𝜙𝑔 − 𝜙𝑙 )/𝜙𝑔 . By using
the transformations 𝑢(𝑥, 𝑡) = ln |V(𝑥, 𝑡)| and (110), we obtain
a periodic wave solution of (1) as follows:
𝑢 (𝑥, 𝑡) =
1 󵄨󵄨
󵄨
ln 󵄨󵄨󵄨[𝜙𝑔 − (𝜙𝑔 − 𝜙𝑙 ) sn2 (Ω19 (𝑥 − 𝑐𝑡) , 𝑘19 )]󵄨󵄨󵄨󵄨 .
𝑚
(111)
(103)
where 𝜗5 , 𝜂5 , 𝛾5 are roots of equation 𝛽/2𝛼 + (𝑐ℎ/2𝛼)𝜙2 − 𝜙3
and 𝛾5 < 0 < 𝜂5 < 𝜗5 . Taking (0, 𝜗5 ) as the initial values of the
variables (𝜉, 𝜙), substituting (103) into the 𝑑𝜙/𝑑𝜉 = 𝑦 of (11),
and then integrating it yield
𝑦 = ±√
𝑦=±
(100)
where 𝜙2 = (−𝛽/𝛼)1/3 . Taking (0, −0.5𝜙2 ) as the initial values
of the variables (𝜉, 𝜙), substituting (100) into the 𝑑𝜙/𝑑𝜉 = 𝑦
of (11), and then integrating it yield
3
V (𝑥, 𝑡) = 𝜙 (𝜉) = 𝜙2 [1 −
],
cosh (Ω17 𝜉) + 1
3.6. Periodic Wave, Broken Solitary Wave, Broken Kink, and
Antikink Wave Solutions in the Special Cases 𝑛 = −𝑚 or
𝑛 = −2𝑚 and 𝑚 ∈ Z+ . (a) When 𝑛 = −𝑚, 𝑚 ∈ Z+ and 𝛼𝛽 < 0,
𝛽𝑐 < 0, 𝛼𝑐 > 0, ℎ > 4√−𝛼𝛽/𝑐𝑚, (14) can be reduced to
𝜙 ∈ [𝛾5 , 𝜂5 ] ,
(106)
where 𝛾5 < 𝜂5 < 0 < 𝜗5 . Taking (0, 𝜗5 ) as the initial values of
the variables (𝜉, 𝜙), substituting (106) into the 𝑑𝜙/𝑑𝜉 = 𝑦 of
(11), and then integrating it yield
V (𝑥, 𝑡) = 𝜙 (𝜉) = 𝛾5 + (𝜂5 − 𝛾5 ) sn2 (Ω19 𝜉, 𝑘19 ) ,
(107)
where 𝛾5 < 𝜂5 < 0 < 𝜗5 , Ω19 = √−𝛼(𝜗5 − 𝛾5 )/2𝑐,
𝑘19 = √(𝜗5 − 𝜂5 )/(𝜗5 − 𝛾5 ). By using the transformations
𝑢(𝑥, 𝑡) = ln |V(𝑥, 𝑡)| and (113), we obtain a periodic blow-up
wave solution of (1) as follows:
󵄨
󵄨
𝑢 (𝑥, 𝑡) = ln 󵄨󵄨󵄨󵄨𝛾5 + (𝜂5 − 𝛾5 ) sn2 (Ω19 (𝑥 − 𝑐𝑡) , 𝑘19 )󵄨󵄨󵄨󵄨 . (108)
(b) When 𝑛 = −2𝑚, 𝑚 ∈ Z+ and 𝛼𝛽 < 0, 𝛽𝑐 < 0, 𝛼𝑐 > 0,
ℎ > ℎ𝐵1 , (14) can be reduced to
𝑦=±
=±
√𝛽/𝑐𝑚 + ℎ(𝜙𝑚 )2 − (2𝛼/𝑐𝑚) (𝜙𝑚 )3
𝜙𝑚−1
√2𝛼/𝑐𝑚√(𝜗𝑚 − 𝜙𝑚 ) (𝜙𝑚 − 𝜂𝑚 ) (𝜙𝑚 − 𝛾𝑚 )
𝜙𝑚−1
(112)
,
where 𝜗𝑚 , 𝜂𝑚 , 𝛾𝑚 are roots of equation 𝛽/𝑐𝑚 + ℎ(𝜙𝑚 )2 −
(2𝛼/𝑐𝑚)(𝜙𝑚 )3 = 0 and 𝜗𝑚 > 𝜂𝑚 > 0 > 𝛾𝑚 . Taking (0, 𝜗)
as the initial values of the variables (𝜉, 𝜙), substituting (112)
into the 𝑑𝜙/𝑑𝜉 = 𝑦 of (11), and then integrating it yield
V (𝑥, 𝑡) = 𝜙 (𝜉) = [𝜗𝑚 − (𝜗𝑚 − 𝜂𝑚 ) sn2 (Ω20 𝜉, 𝑘20 )]
1/𝑚
,
(113)
where Ω20
=
𝑚√𝛼(𝜗𝑚 − 𝛾𝑚 )/2𝑐𝑚, 𝑘20
=
𝑚
𝑚
𝑚
𝑚
√(𝜗 − 𝜂 )/(𝜗 − 𝛾 ). By using the transformations
𝑢(𝑥, 𝑡) = ln |V(𝑥, 𝑡)| and (113), we obtain periodic wave
solutions of (1) as follows:
𝑢 (𝑥, 𝑡) =
1 󵄨󵄨 𝑚
󵄨
ln 󵄨󵄨󵄨𝜗 − (𝜗𝑚 − 𝜂𝑚 ) sn2 (Ω20 (𝑥 − 𝑐𝑡) , 𝑘20 )󵄨󵄨󵄨󵄨 .
𝑚
(114)
Journal of Applied Mathematics
11
(c) When 𝑛 = −2𝑚, 𝑚 ∈ Z+ and 𝛼𝛽 > 0, 𝛽𝑐 < 0, 𝛼𝑐 > 0,
ℎ > ℎ𝐵2 , (14) can be reduced to
𝑦=±
=±
√𝛽/𝑐𝑚 + ℎ(𝜙𝑚 )2 − (2𝛼/𝑐𝑚) (𝜙𝑚 )3
𝜙𝑚−1
(115)
√−2𝛼/𝑐𝑚√(𝜗𝑚 − 𝜙𝑚 ) (𝜂𝑚 − 𝜙𝑚 ) (𝜙𝑚 − 𝛾𝑚 )
𝜙𝑚−1
we only discuss the front case here. When 𝑝 = 2, 𝑞 = 1, under
the transformation 𝑑𝜉 = 𝑐𝜙𝑑𝜏, system (118) can be reduced to
the following regular system:
V (𝑥, 𝑡) = [
𝜂 −
1/𝑚
2
𝜗𝑚 𝑘21
sn2 (Ω21 𝜉, 𝑘21 )
] ,
dn2 (Ω21 𝜉, 𝑘21 )
󵄨
󵄨 𝑚
𝑚 2
2
1 󵄨󵄨󵄨 𝜂 − 𝜗 𝑘21 sn (Ω21 𝜉, 𝑘21 ) 󵄨󵄨󵄨
󵄨󵄨 .
ln 󵄨󵄨󵄨
󵄨󵄨
𝑚 󵄨󵄨
dn2 (Ω21 𝜉, 𝑘21 )
󵄨
(116)
(117)
4. Exact Traveling Wave Solutions of (1) and
Their Properties under the Conditions
𝑚 ∈ Z− , 𝑛 ∈ Z−
In this sections, we consider the case 𝑚 ∈ Z− , 𝑛 ∈ Z− of (1);
that is, we will investigate different kinds of exact traveling
wave solutions of (1) under the conditions 𝑚 ∈ Z− , 𝑛 ∈ Z− .
Letting 𝑚 = −𝑞, 𝑛 = −𝑝, 𝑝 ∈ Z+ , 𝑞 ∈ Z+ , (11) becomes
𝑑𝑦 𝑐𝑦2 − 𝛼𝜙2−𝑞 − 𝛽𝜙2−𝑝
=
.
𝑑𝜉
𝑐𝜙
𝑑𝜙
= 𝑦,
𝑑𝜉
𝐻 (𝜙, 𝑦) =
(119)
which is an integrable system and it has the following first
integral:
(120)
𝑦2 2 (𝛼 + 𝛽) ̃
−
= ℎ.
𝜙2
𝑐𝜙
(126)
𝑦2 2𝛼 −𝑞 2𝛽 −𝑝 ̃
−
𝜙 −
𝜙 = ℎ.
𝜙2 𝑐𝑞
𝑐𝑝
(127)
Obviously, when 𝑝 = 𝑞, system (125) has not any equilibrium
point in the 𝜙-axis. When 𝑝 − 𝑞 is even number and 𝛼𝛽 < 0,
system (125) has two equilibrium points 𝐵1,2 (±𝜙+ , 0) in the 𝜙axis, where 𝜙+ = (−𝛽/𝛼)1/(𝑝−𝑞) . When 𝑝 − 𝑞 is odd number,
system (125) has only one equilibrium point at 𝐵1 (𝜙+ , 0) in
the 𝜙-axis.
Respectively substituting the above equilibrium points
into (121), (124), and (127), we have
2
where ̃ℎ is integral constant. When 𝜙 ≠ 0, we define
𝐻 (𝜙, 𝑦) =
(125)
2𝛼 2−𝑞 2𝛽 2−𝑝
𝜙 +
𝜙 .
𝑐𝑞
𝑐𝑝
𝑦2 = 𝜙2 ̃ℎ +
𝐻 (𝜙, 𝑦) =
2 (𝛼 + 𝛽)
𝜙,
𝑐
(124)
which is an integrable system and it has the following first
integral:
First, we consider a special case 𝑝 = 𝑞 = 1. When 𝑝 = 𝑞 =
1, under the transformation 𝑑𝜉 = 𝑐𝜙𝑑𝜏, system (118) can be
reduced to the following regular system:
𝑦2 = 𝜙2 ̃ℎ +
𝛽
𝑦2 2𝛼
−
= ̃ℎ.
−
𝜙2 𝑐𝜙 𝑐𝜙2
𝑑𝑦
= 𝑐𝜙𝑝−2 𝑦2 − 𝛼𝜙𝑝−𝑞 − 𝛽
𝑑𝜏
𝑑𝜙
= 𝑐𝜙𝑝−1 𝑦,
𝑑𝜏
When 𝜙 ≠ 0, we define
𝑑𝑦
= 𝑐𝑦2 − (𝛼 + 𝛽) 𝜙,
𝑑𝜏
(123)
̃ 0 , 0)
Indeed, system (122) has only one equilibrium point 𝐴(𝜙
in the 𝜙-axis, where 𝜙0 = −𝛽/𝛼. When 𝑐𝛽 > 0, system (122)
has two equilibrium points 𝑆± (0, 𝑌± ) in the singular line 𝜙 =
0, where 𝑌± = ±√𝛽/𝑐.
Finally, we consider the general case 𝑝 > 2. When 𝑝 > 2,
under the transformation 𝑑𝜉 = 𝑐𝜙𝑝−1 𝑑𝜏, system (118) can be
reduced to the following regular system:
(118)
𝑑𝜙
= 𝑐𝜙𝑦,
𝑑𝜏
𝛽
2𝛼
𝜙+ ,
𝑐
𝑐
𝑦2 = ̃ℎ𝜙2 +
also we define
=
𝑚√−𝛼(𝜗𝑚 − 𝛾𝑚 )/2𝑐𝑚, 𝑘21
=
where Ω21
𝑚
𝑚
𝑚
𝑚
√(𝜂 − 𝜂 )/(𝜗 − 𝛾 ). By using the transformations
𝑢(𝑥, 𝑡) = ln |V(𝑥, 𝑡)| and (116), we obtain a periodic wave
solution of (1) as follows:
𝑢 (𝑥, 𝑡) =
(122)
which is an integrable system and it has the following first
integral:
,
where 𝜗𝑚 > 0 > 𝜂𝑚 > 𝛾𝑚 . Taking (0, 𝜂) as the initial values of
the variables (𝜉, 𝜙), substituting (115) into the 𝑑𝜙/𝑑𝜉 = 𝑦 of
(11), and then integrating it yield
𝑚
𝑑𝑦
= 𝑐𝑦2 − 𝛼𝜙 − 𝛽,
𝑑𝜏
𝑑𝜙
= 𝑐𝜙𝑦,
𝑑𝜏
(121)
Obviously, when 𝛼 ≠ − 𝛽, system (119) has only one equilibrium point 𝑂(0, 0) in the 𝜙-axis.
Second, we consider the special case 𝑝 = 2, 𝑞 = 1, and
another case 𝑝 = 1, 𝑞 = 2 is very similar to the front case, so
̃ℎ ̃ = 𝐻 (𝜙 , 0) = − 𝛼 ,
0
𝐴
𝑐𝛽
(128)
̃ℎ = 𝐻 (𝜙 , 0) = − 2𝛼 𝜙−𝑞 − 2𝛽 𝜙−𝑝 ,
1
+
𝑐𝑞 +
𝑐𝑝 +
(129)
̃ℎ = 𝐻 (−𝜙 , 0) = (−1)𝑞+1 2𝛼 𝜙−𝑞 + (−1)𝑝+1 2𝛽 𝜙−𝑝 .
2
+
𝑐𝑞 +
𝑐𝑝 +
(130)
12
Journal of Applied Mathematics
4.1. Nonsmooth Peakon Solutions and Periodic Blow-Up Wave
Solutions under Two Special Cases 𝑝 = 𝑞 = 1 and 𝑝 = 2, 𝑞 = 1.
(1) When 𝑝 = 𝑞 = 1 and 𝛼𝑐 > 0, ̃ℎ < 0, (120) can be reduced
to
2 (𝛼 + 𝛽)
𝑦 = ±√̃ℎ𝜙2 −
𝜙.
𝑐
(131)
Substituting (131) into the first equation of (118), taking
(0, 𝑐̃ℎ/(2(𝛼 + 𝛽))) as initial value conditions, and integrating
it, we obtain
V (𝑥, 𝑡) = 𝜙 (𝜉) =
𝛼+𝛽
(cos √−̃ℎ𝜉 − 1) .
𝑐ℎ
(132)
where 𝑇1 = (1/√−̃ℎ)arccos((𝜙𝑀 + 𝜙𝑚 )/(𝜙𝑀 − 𝜙𝑚 )). From the
transformation 𝑢(𝑥, 𝑡) = ln |V(𝑥, 𝑡)| and (138), we obtain a
periodic blow-up wave solution of (1) as follows:
𝑢 (𝑥, 𝑡)
󵄨󵄨 1
󵄨󵄨󵄨
󵄨
= ln 󵄨󵄨󵄨 [(𝜙𝑀 + 𝜙𝑚 ) − (𝜙𝑀 − 𝜙𝑚 ) cos (√−̃ℎ (𝑥 − 𝑐𝑡))]󵄨󵄨󵄨 ,
󵄨󵄨 2
󵄨󵄨
(139)
where 𝑥 − 𝑐𝑡 ≠ 𝑇1 + 𝐾𝜋, 𝐾 ∈ Z.
(4) When 𝑝 = 2, 𝑞 = 1 and 𝛼𝛽 > 0, 𝛽𝑐 < 0 (or 𝛼𝛽 < 0,
𝛽𝑐 < 0), ̃ℎ > −𝛼2 /𝑐𝛽, (123) can be reduced to
From the transformation 𝑢(𝑥, 𝑡) = ln |V(𝑥, 𝑡)| and (132), we
obtain a periodic blow-up wave solution of (1) as follows:
󵄨󵄨
󵄨󵄨 𝛼 + 𝛽
󵄨
󵄨
𝑢 (𝑥, 𝑡) = ln 󵄨󵄨󵄨
[cos (√−̃ℎ (𝑥 − 𝑐𝑡)) − 1]󵄨󵄨󵄨 ,
󵄨󵄨
󵄨󵄨 𝑐ℎ
(133)
where (𝑥 − 𝑐𝑡) ≠ 2𝐾𝜋/√−̃ℎ, 𝐾 ∈ Z.
(2) When 𝑝 = 2, 𝑞 = 1 and 𝛼𝛽 > 0, 𝛽𝑐 > 0 (or 𝛼𝛽 < 0,
𝛽𝑐 > 0), ̃ℎ = ̃ℎ𝐴̃ = 𝛼2 /𝛽𝑐, (123) can be reduced to
𝑦=±
𝛽 󵄨󵄨󵄨
𝛼 󵄨󵄨󵄨󵄨
󵄨󵄨𝜙 + 󵄨󵄨󵄨 .
𝛼 󵄨󵄨
√𝛽𝑐 󵄨󵄨
(134)
Similarly, substituting (134) into the first equation of (118) and
then integrating it, and setting the integral constant as zero,
we have
󵄨󵄨 󵄨󵄨
𝛽
󵄨𝛼𝜉󵄨
V (𝑥, 𝑡) = 𝜙 (𝜉) = − [1 − exp (− 󵄨 󵄨 )] .
(135)
𝛼
√𝛽𝑐
From the transformation 𝑢(𝑥, 𝑡) = ln |V(𝑥, 𝑡)| and (135), we
obtain a broken solitary cusp wave solution of (1) as follows:
󵄨
󵄨󵄨󵄨 𝛽
|𝛼 (𝑥 − 𝑐𝑡)| 󵄨󵄨󵄨
𝑢 (𝑥, 𝑡) = ln 󵄨󵄨󵄨󵄨− [1 − exp (−
)]󵄨󵄨󵄨 , (𝑥 − 𝑐𝑡) ≠ 0.
󵄨󵄨 𝛼
󵄨󵄨
√𝛽𝑐
(136)
(3) When 𝑝 = 2, 𝑞 = 1 and 𝛼𝛽 > 0, 𝛽𝑐 > 0 (or 𝛼𝛽 < 0,
𝛽𝑐 > 0), ̃ℎ < 0, (123) can be reduced to
(140)
Similarly, we have
V (𝑥, 𝑡) = 𝜙 (𝜉) = −
2
𝛼 √𝛼 − 𝛽𝑐̃ℎ
−
cos (√−̃ℎ𝜉) .
𝑐̃ℎ
𝑐̃ℎ
(141)
From the transformation 𝑢(𝑥, 𝑡) = ln |V(𝑥, 𝑡)| and (141), we
obtain a periodic blow-up wave solution of (1) as follows:
󵄨󵄨
󵄨󵄨
󵄨󵄨
󵄨󵄨
2
󵄨
󵄨󵄨 𝛼 √𝛼 − 𝛽𝑐̃ℎ
cos √−̃ℎ (𝑥 − 𝑐𝑡)󵄨󵄨󵄨󵄨 .
𝑢 (𝑥, 𝑡) = ln 󵄨󵄨󵄨 +
̃
̃
󵄨󵄨
󵄨󵄨 𝑐ℎ
𝑐ℎ
󵄨󵄨
󵄨󵄨
(142)
4.2. Periodic Blow-Up Wave Solutions in the Special Case
𝑝 = 2𝑞 > 1. When 𝑝 = 2𝑞 > 2 and 𝛼𝛽 < 0, 𝛼𝑐 > 0 (or 𝛼𝛽 > 0,
𝛼𝑐 < 0), ̃ℎ ∈ (̃ℎ1 , 0), (118) becomes
2𝛼 −𝑞 𝛽 −2𝑞
𝑦 = ±√ 𝜙2 (̃ℎ +
𝜙 + 𝜙 )
𝑐𝑞
𝑐𝑞
=±
√−̃ℎ√−𝛽/𝑐𝑞̃ℎ − (2𝛼/𝑐𝑞̃ℎ) 𝜙𝑞 − (𝜙𝑞 )2
𝜙𝑞−1
(143)
.
Substituting (143) into the first equation of (118) and then
integrating it, we have
𝛽 2𝛼
𝑦 = ±√−̃ℎ√ − −
𝜙 − 𝜙2
𝑐̃ℎ 𝑐̃ℎ
(137)
= ±√−̃ℎ√(𝜙𝑀 − 𝜙) (𝜙 − 𝜙𝑚 ),
where 𝜙𝑀 = (−𝛼+ √𝛼2 − 𝑐̃ℎ𝛽)/𝑐̃ℎ, 𝜙𝑚 = (−𝛼− √𝛼2 − 𝑐̃ℎ𝛽)/𝑐̃ℎ
are roots of equation −𝛽/𝑐̃ℎ − (2𝛼/𝑐̃ℎ)𝜙 − 𝜙2 = 0. Substituting
(137) into the first equation of (118), taking (0, 𝜙𝑀) as initial
value conditions, and then integrating it, we have
V (𝑥, 𝑡) = 𝜙 (𝜉) =
2𝛼
𝛼
𝑦 = ±√̃ℎ𝜙2 +
𝜙+ .
𝑐
𝑐
1
[ (𝜙𝑀 + 𝜙𝑚 ) − (𝜙𝑀 − 𝜙𝑚 ) cos (√−ℎ𝜉)] ,
2
𝜉 ∈ (−𝑇1 , 𝑇1 ) ,
(138)
1/𝑞
𝛼2 − 𝑐𝑞̃ℎ𝛽
𝛼
cos (𝑞√−̃ℎ𝜉)] .
+√
V (𝑥, 𝑡) = 𝜙 (𝜉) = [−
𝑐𝑞̃ℎ
𝑐2 𝑞2 ̃ℎ2
]
[
(144)
From the transformation 𝑢(𝑥, 𝑡) = ln |V(𝑥, 𝑡)| and (144), we
obtain periodic blow-up wave solutions of (1) as follows:
󵄨󵄨
󵄨󵄨
󵄨󵄨
󵄨
𝛼2 − 𝑐𝑞̃ℎ𝛽
1 󵄨󵄨󵄨 𝛼
󵄨
cos (𝑞√−̃ℎ (𝑥 − 𝑐𝑡))󵄨󵄨󵄨󵄨 .
+√
𝑢 (𝑥, 𝑡) = ln 󵄨󵄨󵄨−
2
2
2
̃
𝑞 󵄨󵄨 𝑐𝑞̃ℎ
󵄨󵄨
𝑐𝑞ℎ
󵄨󵄨
󵄨󵄨
(145)
Journal of Applied Mathematics
5. Conclusion
In this work, by using the integral bifurcation method,
a generalized Tzitzéica-Dodd-Bullough-Mikhailov (TDBM)
equation is studied. In some special cases, we obtained
many exact traveling wave solutions of TDBM equation.
These exact solutions include smooth solitary wave solutions,
singular periodic wave solutions of blow-up form, singular
solitary wave solutions of blow-up form, broken solitary wave
solutions, broken kink wave solutions, and some unboundary
wave solutions. Comparing with the results in many references (such as [24–27]), these types of solutions in this
paper are very different than those types of solutions which
are obtained by Exp-function method. Though the types of
solutions derived from two methods are different, they are
all useful in nonlinear science. The smooth solitary wave
solutions and singular solitary wave solutions obtained in
this paper especially can be explained phenomenon of soliton
surface that the total curvature at each point is proportional
to the fourth power of the distance from a fixed point to the
tangent plane.
Acknowledgments
The authors would like to thank the reviewers very much
for their useful comments and helpful suggestions. This
work is supported by the Natural Science Foundation of
Yunnan Province (no. 2011FZ193) and the National Natural
Science Foundation of China (no. 11161020). The e-mail
address “weiguorhhu@aliyun.com” in the author’s former
publications (published articles) has been replaced by new
e-mail address “weiguorhhu@aliyun.com” because Yahoo
Company in China will be closed on August 19, 2013. Please
use the new E-mail address “weiguorhhu@aliyun.com” if the
readers contact him.
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