Hindawi Publishing Corporation
Mathematical Problems in Engineering
Volume 2012, Article ID 320163, 10 pages
doi:10.1155/2012/320163
Research Article
Periodic Loop Solutions and Their Limit Forms for
the Kudryashov-Sinelshchikov Equation
Bin He,1 Qing Meng,2 Jinhua Zhang,1 and Yao Long1
1
2
College of Mathematics, Honghe University, Mengzi, Yunnan 661100, China
Department of Physics, Honghe University, Mengzi, Yunnan 661100, China
Correspondence should be addressed to Bin He, hebinmtc@163.com
Received 5 December 2011; Revised 24 January 2012; Accepted 30 January 2012
Academic Editor: Stefano Lenci
Copyright q 2012 Bin He et al. 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.
The Kudryashov-Sinelshchikov equation is studied by using the bifurcation method of dynamical
systems and the method of phase portraits analysis. We show that the limit forms of periodic loop
solutions contain loop soliton solutions, smooth periodic wave solutions, and periodic cusp wave
solutions. Also, some new exact travelling wave solutions are presented through some special
phase orbits.
1. Introduction
A mixture of liquid and gas bubbles of the same size may be considered as an example of
a classic nonlinear medium. In practice, analysis of propagation of the pressure waves in a
liquid with gas bubbles is important problem. We know that there are solitary and periodic
waves in a mixture of a liquid and gas bubbles and these waves can be described by nonlinear
partial differential equations. As for examples of nonlinear differential equations to describe
the pressure waves in bubbly liquids, we can point out the Burgers equation, the Korteweg-de
Vries equation, the Burgers-Korteweg-de Vries equation, and so on 1.
In 2010, Kudryashov and Sinelshchikov 1 obtained a more common nonlinear partial
differential equation for describing the pressure waves in a mixture liquid and gas bubbles
taking into consideration the viscosity of liquid and the heat transfer, and the equation reads
as follows:
ut αuux uxxx − uuxx x − βux uxx 0,
1.1
where u is a density and which model heat transfer and viscosity, α, β are real parameters.
Equation 1.1 is called Kudryashov-Sinelshchikov equation, it is generalization of the
2
Mathematical Problems in Engineering
KdV and the BKdV equation and similar but not identical to the Camassa-Holm equation.
Undistorted waves are governed by a corresponding ordinary differential equation which, for
special values of some integration constant, is solved analytically in 1. Ryabov 2 obtained
some exact solutions for β −3 and β −4 using a modification of the truncated expansion
method. Solutions are derived in a more straightforward manner and cast into a simpler form,
and some new types of solutions which contain solitary wave and periodic wave solutions
are presented in 3.
In this paper, we focus on the case β 2 of 1.1 using the bifurcation theory and the
method of phase portraits analysis 4–6, we will investigate periodic loop solutions and their
limit forms and give some new exact travelling wave solutions.
2. Preliminary
In this paper, we always consider the case β 2, so from now on we assume β 2 in 1.1
without mentioning it further.
Substituting ux, t 1 − φx αt 1 − φξ into 1.1 and integrating the resulting
equation once with respect to ξ, we obtain
g − 2αφ 2
α 2
φ − φφ − φ 0,
2
2.1
where g is the integral constant.
Letting y dφ/dξ, we get the following planar system:
dφ
y,
dξ
dy g − 2αφ α/2φ2 − y2
.
dξ
φ
2.2
Using the transformation dξ φdτ, it carries 2.2 into the Hamiltonian system:
dφ
φy,
dτ
dy
α
g − 2αφ φ2 − y2 .
dτ
2
2.3
Since both system 2.2 and 2.3 have the same first integral:
1
4
φ2 y2 − g αφ − αφ2 h,
3
4
2.4
then the two systems above have the same topological phase portraits except the line φ 0.
Therefore, we can obtain the bifurcation phase portraits of system 2.2 from that of system
2.3.
Write Δ 2α2α − g. Clearly, when
√ Δ > 0, system 2.3 has two equilibrium points
at φ1,2 , 0 in φ-axis, where φ1,2 2α ± Δ/α. When Δ 0, system 2.3 has only one
equilibrium point at 2, 0 in φ-axis. When Δ < 0, system 2.3 has no any equilibrium point
in φ-axis. When g > 0, there exist two equilibrium points of system 2.3 in line φ 0 at
√
0, ± g.
Let Mφe , ye be the coefficient matrix of the linearized system of 2.3 at equilibrium
point φe , ye , J detMφe , ye , and T traceMφe , ye . By the theory of planar
Mathematical Problems in Engineering
3
dynamical systems, we know that for an equilibrium point φe , ye of a planar integrable
system, φe , ye is a saddle point if J < 0, a center point if J > 0 and T 0, a cusp if J 0
and the Poincaré index of φe , ye is zero. By using the properties of equilibrium points and
bifurcation method of dynamical systems, we can show that the bifurcation phase portraits
of systems 2.2 and 2.3 is as drawn in Figure 1.
From Figures 1b, 1c, 1d, 1e, and 1l, we have the following results.
3. Main Results
Proposition 3.1. i When α < 0, g 2α, for h −4/3α defined by 2.4, 1.1 has a loop-soliton
solution.
ii When α < 0, g 2α, for h ∈ 0, −4/3α, there exists a family of uncountably infinite
many periodic loop solutions of 1.1. Moreover, the periodic loop solutions converge to the loop-soliton
solution as h approaches −4/3α.
iii When α < 0, g 2α, for h ∈ −4/3α, ∞, there exists a family of uncountably
infinite many periodic loop solutions of 1.1. Moreover, the periodic loop solutions converge to the
loop-soliton solution as h approaches −4/3α.
Proposition 3.2. Denote that h1 Hφ1 , 0 and h2 Hφ2 , 0.
i When α < 0, 2α < g < 0, for h h1 defined by 2.4, 1.1 has a loop-soliton solution and
has a solitary wave solution.
ii When α < 0, 2α < g < 0, for h ∈ h2 , h1 , there exist a family of uncountably infinite
many periodic loop solutions and a family of uncountably infinite many smooth periodic wave solutions
of 1.1. Moreover, the periodic loop solutions converge to the loop-soliton solution and the smooth
periodic wave solutions converge to the solitary wave solution as h approaches h1 .
iii When α < 0, 2α < g < 0, for h ∈ h1 , ∞, there exists a family of uncountably infinite
many periodic loop solutions of 1.1. Moreover, the periodic loop solutions converge to the loop-soliton
solution as h approaches h1 .
iv When α < 0, 2α < g < 0, for h ∈ 0, h2 , there exists a family of uncountably infinite
many periodic loop solutions of 1.1.
Proposition 3.3. i When α < 0, g 0, for h 0 defined by 2.4, 1.1 has a smooth periodic
wave solution.
ii When α < 0, g 0, for h ∈ 0, ∞, there exists a family of uncountably infinite many
periodic loop solutions of 1.1. Moreover, the periodic loop solutions converge to the smooth periodic
wave solution as h approaches 0.
Proposition 3.4. i When α < 0, g > 0, for h 0 defined by 2.4, 1.1 has two cusp periodic
wave solutions.
ii When α < 0, g > 0, for h ∈ 0, ∞, there exists a family of uncountably infinite many
periodic loop solutions of 1.1. Moreover, the periodic loop solutions converge to the cusp periodic
wave solutions as h approaches 0.
Proposition 3.5. Denote that h2 Hφ2 , 0.
i When α > 0, g < 0, for h h2 defined by 2.4, 1.1 has a loop-soliton solution.
ii When α > 0, g < 0, for h ∈ 0, h2 , there exists a family of uncountably infinite many
periodic loop solutions of 1.1. Moreover, the periodic loop solutions converge to the loop-soliton
solution as h approaches h2 .
4
Mathematical Problems in Engineering
y
y
y
φ
φ
(a) α < 0, g < 2α
2
φ
(b) α < 0, g = 2α
(c ) α < 0, 2α < g < 0
y
y
y
5
φ
5
φ
(d) α < 0, g = 0
(e) α < 0, g > 0
y
y
5
5
φ
(h) α > 0, (16/9) α < g < 2α
(g) α > 0, g = 2α
y
y
φ
5
(j ) α > 0, 0 < g < (16/9) α
φ
(f ) α > 0, g > 2α
y
φ
2
( i ) α > 0, g = (16/9) α
y
φ
(k) α > 0, g = 0
5
φ
φ
(l ) α > 0, g < 0
Figure 1: The bifurcation phase portraits of systems 2.2 and 2.3.
Mathematical Problems in Engineering
5
4. Exact Traveling Wave Solutions of
the Kudryashov-Sinelshchikov Equation
Corresponding to Figure 1b, the graph defined by Hφ, y −4/3α consist of two
hyperbolic sectors of the cusp 2, 0 and an open-end curve Γ0 passing through the point
−2/3, 0. It follows from 2.4 that
y±
2−φ
−α 2 − φ φ 2/3
2φ
−
,
2
≤ φ < 2, φ /
0.
3
4.1
Substituting 4.1 into the dφ/dξ y and integrating along the curve Γ0 and noting that
ux, t 1 − φx αt 1 − φξ, we obtain the following representation of loop-soliton
solution:
2
u χ 1 − 2sin2 χ cos2 χ ,
3
4
3
tan χ − χ ,
ξ χ √
−α 4
4.2
where χ is a new parametric variable.
Corresponding to Figure 1c, the graph defined by Hφ, y Hφ1 , 0 consists of an
with
open-end curve Γ1 passing through the point φm , 0 and a homoclinic orbit
connecting
√
√
saddle point φ1 , 0 and passing point φM , 0, where φm 3 Δ − 2α − 2 2α2α − 3 Δ/ −
√
√
3α, φM 3 Δ − 2α 2 2α2α − 3 Δ/ − 3α. It follows from 2.4 that
y±
φ1 − φ
y±
−α φM − φ φ − φm
2φ
,
φ − φ1 −α φM − φ φ − φm
2φ
φm ≤ φ < φ1 , φ /
0,
4.3
φ1 ≤ φ ≤ φM .
4.4
,
Substituting 4.3 into the dφ/dξ y and integrating along the curve Γ1 , we can obtain the
following representation of loop-soliton solution:
2
2
φm φM
φ
sinh
cosh
ωχ
−
φ
ωχ
φ
1
M
m
,
u χ 1−
φM cosh2 ωχ − φm sinh2 ωχ − φ1
⎛
⎞⎞
⎛
φ
−
φ
2 ⎝
1
m
χ − 2 arctan⎝
tanh ωχ ⎠⎠,
ξ χ √
φM − φ1
−α
where ω φ1 − φm φM − φ1 /2φ1 .
4.5
6
Mathematical Problems in Engineering
Substituting 4.4 into the dφ/dξ y and integrating along the homoclinic orbit, we
can obtain the following representation of solitary wave solution:
φ1 φM cosh2 ωχ − φm sinh2 ωχ − φm φM
u χ 1−
,
φM sinh2 ωχ − φm cosh2 ωχ φ1
⎞⎞
⎛
⎛
φ
−
φ
2 ⎝
M
1
χ − 2 arctan⎝
tanh ωχ ⎠⎠,
ξ χ √
φ1 − φm
−α
4.6
where ω φ1 − φm φM − φ1 /2φ1 .
Moreover, the graph defined by Hφ, y h, h ∈ Hφ2 , 0, Hφ1 , 0, consists of two
open-end curves Γ2 , Γ3 , and a periodic orbit, say Ψ, enclosing the center point φ2 , 0. The
curve Ψ passes through the points γ1 , 0 and γ2 , 0, while Γ2 , Γ3 pass through the points
γ3 , 0 and γ4 , 0, respectively, where γ1 , γ2 , γ3 , γ4 γ4 < 0 < γ3 < γ2 < γ1 are four real roots of
ψ 4 − 16/3ψ 3 4g/αψ 2 4h/α 0. It follows from 2.4 that
y±
−α γ1 − φ γ2 − φ γ3 − φ φ − γ4
y±
2φ
,
−α γ1 − φ φ − γ2 φ − γ3 φ − γ4
2φ
γ4 ≤ φ ≤ γ3 , φ /
0,
4.7
γ2 ≤ φ ≤ γ1 .
4.8
,
Let us denote by F·, k and Π·, ·, k the Legendre’s incomplete elliptic integrals of the
first and third kinds, respectively, with the modulus k see 7.
Substituting 4.7 into the dφ/dξ y and integrating along the curve Γ2 , we can obtain
the implicit representation of periodic loop solution for u ∈ 1 − γ3 , 1 − γ4 :
4γ3
α21 −α γ1 − γ3 γ2 − γ4
⎛
⎛
⎞
⎡
⎞
γ2 −γ4 γ3 u−1
2
2
2
× ⎣ α1 −α2 Π⎝arcsin⎝ ⎠, α1 , k⎠
γ3 −γ4 γ2 u −1
⎛
⎛
⎞ ⎞⎤
γ2 −γ4 γ3 u−1
2 ⎝
α2 F arcsin⎝ ⎠, k⎠⎦ ±ξ,
γ3 −γ4 γ2 u−1
where α21
γ3 − γ4 /γ2 − γ4 ,
γ1 − γ2 γ3 − γ4 /γ1 − γ3 γ2 − γ4 .
α22
γ2 γ3 − γ4 /γ3 γ2 − γ4 ,
4.9
k
Mathematical Problems in Engineering
7
Substituting 4.8 into the dφ/dξ y and integrating along the periodic orbit, we can
obtain the implicit representation of smooth periodic wave solution for u ∈ 1 − γ1 , 1 − γ2 :
4γ2
α21 −α γ1 − γ3 γ2 − γ4
⎛
⎛
⎞
⎡
⎞
γ1 − γ3 1 − u − γ2
× ⎣ α21 − α22 Π⎝arcsin⎝ ⎠, α21 , k⎠
γ1 − γ2 1 − u − γ3
⎛
⎛
⎞ ⎞⎤
γ1 − γ3 1 − u − γ2
α22 F⎝arcsin⎝ ⎠, k⎠⎦ ±ξ,
γ1 − γ2 1 − u − γ3
4.10
where α21
γ1 − γ2 /γ1 − γ3 , α22
γ3 γ1 − γ2 /γ2 γ1 − γ3 , k
γ1 − γ2 γ3 − γ4 /γ1 − γ3 γ2 − γ4 .
Corresponding to Figure 1d, the
graph defined by Hφ, y 0 is a periodic
orbit enclosing the center point 2α − αα − 1/α, 0 and passing through the points
0, 0, 16/3, 0. It follows from 2.4 that
1
y±
2
−αφ
16
−φ
3
,
0≤φ≤
16
.
3
4.11
Substituting 4.11 into the dφ/dξ y and integrating along the periodic orbit, we can obtain
the following representation of smooth periodic wave solution:
ux, t 1 −
16
cos2 ωx αt,
3
4.12
√
where ω 1/4 −α.
Corresponding to Figure 1e, the graph defined by Hφ, y 0 consists of four
√
heteroclinic orbits: two of them connecting the saddle points 0, ± g with
φm , 0, and the
√
g
with
φ
,
0,
where
φ
24α
α16α − 9g/3α,
others connecting
saddle
points
0,
±
M
m
φM 24α − α16α − 9g/3α. It follows from 2.4 that
1 −α φ − φm φM − φ ,
2
1 −α φ − φm φM − φ ,
y±
2
y±
φm ≤ φ ≤ 0,
4.13
0 ≤ φ ≤ φM .
4.14
Substituting 4.13 into the dφ/dξ y and integrating along the heteroclinic orbit, we can
obtain the following representation of cusp periodic wave solution:
ux, t 1 − φM sin2 Ω − ω|x αt − 2nT | − φm cos2 Ω − ω|x αt − 2nT |,
4.15
√
where ω 1/4 −α, Ω arctan −φm /φM , T 2|Ω|, n 0, ±1, ±2, . . . , 2n − 1T ≤ x αt ≤
2n 1T .
8
Mathematical Problems in Engineering
Substituting 4.14 into the dφ/dξ y and integrating along the heteroclinic orbit, we
can obtain the following representation of cusp periodic wave solution:
ux, t 1 − φM cos2 Ω − ω|x αt − 2nT | − φm sin2 Ω − ω|x αt − 2nT |,
4.16
√
where ω 1/4 −α, Ω arctan −φM /φm , T 2|Ω|, n 0, ±1, ±2, . . ., 2n − 1T ≤ x αt ≤
2n 1T .
Moreover, the graph defined by Hφ, y h, h ∈ 0, ∞ consists of two open-end
curves Γ4 , Γ5 passing through the points φm , 0 and φM , 0, respectively, where φM − φφ −
φm φ − b1 2 a21 −φ4 16/3φ3 − 4g/αφ2 − 4h/α. It follows from 2.4 that
y±
2
−α φM − φ φ − φm φ − b1 a21
2φ
,
φm ≤ φ ≤ φM , φ /
0.
4.17
Substituting 4.17 into the dφ/dξ y and integrating along the curve Γ4 , we can obtain the
implicit representation of periodic loop solution for u ∈ 1 − φM , 1 − φm :
α21
2 φm A φM B
α1 − α2
α2 F ϕ, k Π ϕ, 2
, k − α1 f1 − η0 ±ξ,
√
1 − α21
α1 − 1
A − B AB
4.18
φM − b1 2 a21 , B φm − b1 2 a21 , α1 A − B/A B, α2 φm A − φM B/
√
φm A φM B, k φM − φm 2 − A − B2 /4AB, k1 1 − k2 , η0 α22 Fϕ, k α1 − α2 /
1 − α21 Πϕ, α21 /α21 − 1, k − α1 f1 |u1−φM , ϕ
arccosφM u −
u
−
1A/φ
u
−
1B
−
φ
u
− 1A, f1
1B
φ
m
M
m
where A 1 − α21 /k2 k12 α21 arctan sin2 ϕk2 k12 α21 /1 − k2 sin2 ϕ1 − α21 .
Corresponding to Figure 1l, the graph defined by Hφ, y Hφ2 , 0 consists of two
through
hyperbolic sectors of the saddle point φ2 , 0 and two open-end curves
Γ6 , Γ7 passing
√
√
the points φm , 0, φM , 0, respectively, where φm 2α 3 Δ − 2 2α2α 3 Δ/3α, φM √
√
2α 3 Δ 2 2α2α 3 Δ/3α. It follows from 2.4 that
y±
φ − φ2
α φm − φ φM − φ
2φ
,
φ2 < φ ≤ φm , φ /
0.
4.19
Substituting 4.19 into the dφ/dξ y and integrating along the curve Γ6 , we can obtain the
following representation of loop-soliton solution:
φ2 φM sinh2 ωχ − φm cosh2 ωχ φm φM
,
u χ 1−
φM cosh2 ωχ − φm sinh2 ωχ − φ2
⎞⎞
⎛
⎛
φ
−
φ
2
m
2
tanh ωχ ⎠⎠,
ξ χ √ ⎝χ − 2 tan h−1 ⎝
φM − φ2
α
4.20
Mathematical Problems in Engineering
9
where ω φm − φ2 φM − φ2 /2φ2 , tan h−1 · is the inverse function of the hyperbolic
function tanh·, see 7.
Moreover, the graph defined by Hφ, y h, h ∈ 0, Hφ2 , 0 consist of four
open-end curves Γ8 , Γ9 , Γ10 and Γ11 passing through the points γ4 , 0, γ3 , 0, γ2 , 0, γ1 , 0
respectively, where γ1 , γ2 , γ3 , γ4 γ4 < γ3 < 0 < γ2 < γ1 are four real roots of ψ 4 − 16/3ψ 3 4g/αψ 2 4h/α 0. It follows from 2.4 that
y±
α γ1 − φ γ2 − φ φ − γ3 φ − γ4
2φ
,
γ3 ≤ φ ≤ γ2 , φ /
0.
4.21
Substituting 4.21 into dφ/dξ y and integrating along the curve Γ10 , we can obtain the
implicit representation of periodic loop solution for u ∈ 1 − γ2 , 1 − γ3 :
4γ2
α21 α γ1 − γ3 γ2 − γ4
⎛
⎛
⎡
⎞
⎞
γ1 − γ3 γ2 u − 1
2
2
2
× ⎣ α1 − α2 Π⎝arcsin⎝ ⎠, α1 , k⎠
γ2 − γ3 γ1 u − 1
⎛
⎛
⎞ ⎞⎤
γ1 − γ3 γ2 u − 1
2 ⎝
α2 F arcsin⎝ ⎠, k⎠⎦ ±ξ,
γ2 − γ3 γ1 u − 1
where
α21
γ2 − γ3 /γ1 − γ3 ,
γ2 − γ3 γ1 − γ4 /γ1 − γ3 γ2 − γ4 .
α22
γ1 γ2 − γ3 /γ2 γ1 − γ3 ,
4.22
k
Remark 4.1. Denote that i α < 0, g 2α, h ∈ 0, −4/3α, ii α < 0, g 2α, h ∈
−4/3α, ∞, iii α < 0, 2α < g < 0, h ∈ 0, Hφ2 , 0, iv α < 0, 2α < g < 0, h ∈
Hφ1 , 0, ∞, v α < 0, g 0, h ∈ 0, ∞, we can obtain the implicit representation of
periodic loop solution similar to 4.18 when β, α, g, and h satisfy one and only one of above
conditions, we omit it for brevity.
Example 4.2. Taking α −1, g 1 and h 1, we get the approximations of
.
.
A, B, φm , φM , a1 , b1 , α1 , α2 , k, k1 in the formula 4.18, where A 5.846662930, B .
.
.
.
1.525667184, φm −1.117067993, φM 6.016534182, a1 0.7403378831, b1 0.2169335722,
.
.
.
.
α1 0.586109911, α2 −5.932667554, k 0.9502338139, k1 0.3115376364.
5. Conclusion
In this paper, using the bifurcation theory and the method of phase portraits analysis, we
investigated periodic loop solutions and their limit forms of the Kudryashov-Sinelshchikov
equation and show that the limit forms contain loop soliton solutions, smooth periodic
wave solutions, and periodic cusp wave solutions. We also obtain the exact parametric
representations above travelling wave solutions. The results of this paper have enriched
results of 1–3. We would like to study the Kudryashov-Sinelshchikov equation further.
10
Mathematical Problems in Engineering
Acknowledgments
The authors thank the referees for some perceptive comments and for some valuable
suggestions. This work is supported by the National Natural Science Foundation of China
no. 11161020.
References
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2 P. N. Ryabov, “Exact solutions of the Kudryashov-Sinelshchikov equation,” Applied Mathematics and
Computation, vol. 217, no. 7, pp. 3585–3590, 2010.
3 M. Randrüüt, “On the Kudryashov-Sinelshchikov equation for waves in bubbly liquids,” Physics
Letters, Section A, vol. 375, no. 42, pp. 3687–3692, 2011.
4 L. Zhang and J. Li, “Dynamical behavior of loop solutions for the 2, 2 equation,” Physics Letters A,
vol. 375, no. 33, pp. 2965–2968, 2011.
5 J. Li, Y. Zhang, and G. Chen, “Exact solutions and their dynamics of traveling waves in three
typical nonlinear wave equations,” International Journal of Bifurcation and Chaos in Applied Sciences and
Engineering, vol. 19, no. 7, pp. 2249–2266, 2009.
6 B. He, “Bifurcations and exact bounded travelling wave solutions for a partial differential equation,”
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7 P. F. Byrd and M. D. Friedman, Handbook of Elliptic Integrals for Engineers and Physicists, Springer, Berlin,
Germany, 1954.
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