- Email: [email protected]
Bright and dark soliton solutions of the strain wave equation in microstructured solids
Accepted Manuscript Title: Bright and dark soliton solutions of the strain wave equation in microstructured solids Author: Heng Wang Shuhua Zheng PII: DOI: Reference:
S0030-4026(17)31040-9 http://dx.doi.org/doi:10.1016/j.ijleo.2017.08.132 IJLEO 59595
To appear in: Received date: Accepted date:
10-5-2017 28-8-2017
Please cite this article as: Heng Wang, Shuhua Zheng, Bright and dark soliton solutions of the strain wave equation in microstructured solids, (2017), http://dx.doi.org/10.1016/j.ijleo.2017.08.132 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Bright and dark soliton solutions of the strain wave equation in microstructured solids Heng Wang1
ip t
State Key Laboratory of Earth Surface Processes and Resource Ecology, Faculty of Geographical Science, Beijing Normal University, Beijing 100875, China Institute of Land Surface System and Sustainable Development, Faculty of Geographical Science, Beijing Normal University, Beijing 100875, China
Shuhua Zheng
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Investment and Development Department, Market and Investment Center, Yunnan Water Investment Co., Limited, Kunming, Yunnan 650106, China
Introduction
d
1
M
an
us
Abstract: By using the approach of dynamical system, the exact travelling wave solutions of the strain wave equation in microstructured solids are solved. As results, all kinds of phase portraits in the parametric space are shown. The exact solutions are obtained including bright and dark soliton solutions. With the aid of Maple, the graphical representations and physical explanations of the bright and dark soliton solutions are given. Keywords: Strain wave equation; Microstructured solids; Dynamical system method; Phase portrait; Bright and dark soliton solution.
te
In this paper, we consider the travelling wave solutions of the strain wave equation in microstructured solids
Ac
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utt −uxx −εα1 (u2 )xx −γα2 uxxt +δα3 uxxxx −(δα4 −γ 2 α7 )uxxtt +γδ(α5 uxxxxt +α6 uxxttt ) = 0, (1.1) where ε accounts for elastic strains, δ characterizes the ratio between the microstructure size and the wave length, γ characterizes the influence of dissipation and α1 , α2 , α3 , α4 , α5 , α6 are constants [1]. The balance between nonlinearity and dispersion takes place when δ = O(ε). If we set γ = 0, we have the non-dissipative case, and governed by the double dispersive equation as follows [2]: utt − uxx − ε(α1 (u2 )xx − α3 uxxxx + α4 uxxtt ) = 0.
(1.2)
The strain wave equation in microstructured solids is a class of important nonlinear wave solutions and plays a crucial role in nonlinear physics fields. Eq.(1.2) has been studied by many researchers. In [1], M.G. Hafeza and M.A. Akbar used an exponential expansion method to obtain some exact solutions of Eq.(1.2). In [3], the general traveling wave solutions were obtained by Md. Nur Alam et al. by using the new generalized (G0 /G)-expansion method. However, we notice that the previous authors 1
Corresponding author. E-mail: [email protected].
1
Page 1 of 10
did not study the dynamical behavior of Eq.(1.2). Therefore, it is essential to study the dynamical behavior of Eq.(1.2) and find some new exact solutions by qualitative analysis of the dynamic properties of Eq.(1.2). The dynamical system method is an effective method which is based on the bifurcation theory of planar dynamical systems and the analytical solutions of the nonlinear wave equations are solved by qualitative analysis [4-10]. Here, we use the approach of dynamical system to solve Eq.(1.2) and try to look for some new travelling wave solutions. To look for travelling wave solutions of (1.2), we assume that u(x, t) = P (ξ), ξ = x − V t,
ip t
(1.3)
where V is a travelling wave parameter. Substituting (1.3) into (1.2), we have
(1.4)
cr
(V 2 − 1)P 00 − εα1 (P 2 )00 + ε(α3 − V 2 α4 )P 0000 = 0.
Obviously, Eq.(1.4) is integrable. Therefore, integrating twice for (1.4), we have
us
(V 2 − 1)P − εα1 P 2 + ε(α3 − V 2 α4 )P 00 + g = 0.
(1.5)
where g is an integration constant. Suppose ε(α3 − V 2 α4 ) 6= 0 and write that −εα1 V2−1 g , b = ,c = . 2 2 ε(α3 − V α4 ) ε(α3 − V α4 ) ε(α3 − V 2 α4 )
an
a=
Finally, we have the following equation
M
P 00 + aP 2 + bP + c = 0,
(1.6)
(1.7)
which corrsponds to the two-dimensional Hamiltonian system
with the Hamiltonian
te
d
dP dy = y, = −aP 2 − bP − c, dξ dξ
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1 1 1 H(P, y) = y 2 + αP 3 + βP 2 + γP = h, 2 3 2
(1.8)
(1.9)
Ac
where h is the Hamiltonian constant. According to the Hamiltonian, we can get all kinds of phase portraits in the parametric space of (1.8). By analyzing the bifurcations of phase portraits of system (1.8), we can obtain the travelling wave solutions of Eq.(1.2). This rest paper is organized as follows. In section 2, the theoretical description of dynamical system method is given. In section 3, we give all phase portraits of system (1.8) and discuss the bifurcations of phase portraits of system (1.8). According to the dynamics of the phase orbits of system (1.8), we solve Eq.(1.2) for two cases of c 6= 0 and c = 0. Finally, a short conclusion is given in section 4.
2
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2
Theory: the dynamical system method
In this section, we describe the dynamical system method for solving traveling wave solutions of nonlinear wave equations [11]. A (n+1)-dimensional nonlinear partial differential equation is given as follows. P (t, xi , ut , uxi , uxi xi , uxi xj , utt , · · · ) = 0
i, j = 1, 2, ..., n.
(2.1)
ip t
The main steps of the dynamical system method are as follows. Step1. Reduction of Eq.(2.1). ∑ Making a transformation u(t, x1 , x2 , ..., xn ) = φ(ξ), ξ = ni=1 ki xi − ct, Eq.(2.1) can be reduced to a non-linear ordinary differential equation D(ξ, φ, φξ , φξξ , φξξξ , · · · ) = 0,
cr
(2.2)
us
where ki are non-zero constant and c is the wave speed. Integrating several times for Eq.(2.2), if it can be reduced to the following second-order nonlinear ordinary differential equation E(φ, φξ , φξξ ) = 0, (2.3)
an
then let φξ = dφ/dξ = y, Eq.(2.3) can be reduced to a two-dimensional dynamical system dy dφ = y, = f (φ, y), (2.4) dξ dξ
M
where f (φ, y) is an integral expression or a fraction. If f (φ, y) is a fraction such as f (φ, y) = F (φ, y)/g(φ) and g(φs ) = 0, dy dξ does not exist when φ = φs . Then we shall make a transformation dξ = g(φ)dζ, thus system (2.4) can be rewritten as dφ = g(φ)y, dζ
(2.5)
te
d
dy = F (φ, y), dζ
ce p
where ζ is a parameter. If Eq.(2.1) can be reduced to the above system (2.4) or (2.5), then we can go on to the next step. Step2. Discussion of bifurcations of phase portraits of system (2.4). If system (2.4) is an integral system, systems (2.4) and (2.5) can be reduced the differential equation
Ac
f (φ, y) dy = , dφ y
dy F (φ, y) f (φ, y) = = , dφ g(φ)y y
(2.6)
then systems (2.4) and (2.5) have the same first integral (that is Hamiltonian) as follows H(φ, y) = h,
(2.7)
where h is an integral constant. According to the first integral, we can get all kinds of phase portraits in the parametric space. Because the phase orbits defined the vector fields of system (2.4) (or system (2.5)) determine the travelling wave solutions of Eq.(2.1), we can investigate the bifurcations of phase portraits of system (2.4) (or system (2.5)) to seek the travelling wave solutions of Eq.(2.1). Usually, a periodic orbit always corresponds to a periodic wave solution; A homoclinic orbit always corresponds 3
Page 3 of 10
to a solitary wave solution; A heteroclinic orbit (or so called connecting orbit) always corresponds to kink (or anti-kink) wave solution. When we find all phase orbits, we can get the value of h or its range. Step3. Calculation of the first equation of system (2.4). After h is determined, we can get the following relationship from Eq.(2.7) y = y(φ, h),
(2.8)
ip t
that is, dφ/dξ = y(φ, h). If the expression (2.8) is an integral expression, then substituting it into the first term of Eq.(2.4) and integrating it, we obtain ∫ ξ ∫ φ dϕ dτ. (2.9) = 0 φ0 y(ϕ, h)
3
an
us
cr
where φ(0) and 0 are initial constants. Usually, the initial constants can be taken by a root of Eq. (2.8) or inflection points of the travelling waves. Taking proper initial constants and integrating Eq.(2.9), through the Jacobian elliptic functions, we can obtain the exact travelling wave solutions of Eq.(2.1). This is the whole process of the dynamical system method. The different nonlinear wave equations correspond to different dynamical systems. The different dynamical systems correspond to different travelling wave solutions.
Application: bright and dark soliton solutions of Eq.(1.2)
d
M
In this section, based on the theoretical description in section 2, we solve Eq.(1.2). According to the dynamical system and Hamiltonian in section 1, we let right hand terms of system (1.8) be zeros, i.e. y = 0 and −aP 2 − bP − c = 0. Obviously, the abscissas of equilibrium points of system (1.8) are the real roots of f (P ) = aP√2 +bP +c. ∆ , 0) and Then, we obtain that the system (1.8) has two equilibrium points at S1 ( −b+ 2a √
te
∆ , 0) if ∆ > 0, where ∆ = b2 − 4ac. If ∆ = 0, the system (1.8) has a unique S2 ( −b− 2a b equilibrium at O(− 2a , 0). If ∆ < 0, the system (1.8) has no equilibrium. For the √
Hamiltonian H(P, y) = 12 y 2 + 13 aP 3 + 21 bP 2 + cP = h, we write h1 = H( −b+ 2a √
√
√
∆
, 0) =
∆ , h2 = H( −b− . Then, we consider the bifurcations , 0) = 2a of phase portraits of (1.8) in the case c = 0, we can obtain that the system (1.8) has two equilibrium points S(− ab , 0) and O(0,0). For the Hamiltonian H(P, y) = 12 y 2 + 1 b b3 1 3 2 3 aP + 2 bP = h, we write h3 = H(0, 0) = 0, h4 = H(− a , 0) = 6a2 . Because only bounded travelling waves are meaningful to a physical model, we just pay attention to the bounded orbit of the system (1.8). With the change of the parameter group of a, b and c, the phase portraits for (1.8) are shown in Figs.1-6. From Figs.1-6, we summarize the following conclusions: (1) When ∆ > 0, the system (1.8) has bounded orbits. When ∆ ≤ 0, the system (1.8) has no bounded orbits. (2) When ∆ > 0, the system (1.8) has a unique homoclinic orbit Γ which is asymptotic to the saddle and enclosing the center. There is a family of periodic orbits which are enclosing the center and filling up the interior of the homoclinic orbit Γ. (3) The system (1.11) has a unique homoclinic orbit Γ which is asymptotic to the saddle and enclosing the center. There is a family of periodic orbits which are enclosing the center and filling up the interior of the homoclinic orbit Γ. b3 −6abc+ 12a2
ce p
∆3
∆3
Ac
b3 −6abc− 12a2
4
Page 4 of 10
ip t
d
M
an
us
cr
Fig. 1: The bifurcations of phase portraits Fig. 2: The bifurcations of phase portraits of (1.8) when a < 0 and 4 > 0. of (1.8) when a > 0 and 4 > 0.
Ac
ce p
te
Fig. 3: The bifurcations of phase portraits Fig. 4: The bifurcations of phase portraits of (1.8) when a < 0,b > 0 and c = 0. of (1.8) when a < 0, b < 0 and c = 0.
Fig. 5: The bifurcations of phase portraits Fig. 6: The bifurcations of phase portraits of (1.8) when a > 0, b > 0 and c = 0. of (1.8) when a > 0, b < 0 and c = 0.
5
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According to the bounded orbit of the system (1.8), we obtain the travelling wave solutions of Eqs.(1.1). In fact, various possible results have been discussed by Wang et al [12]. Here we will cite the main results in Ref. [12]. So we have the following travelling wave solutions of Eqs.(1.1). (1)When a < 0 and h = h2 , there is a dark soliton solution which corresponds to a smooth homoclinic orbit Γ of (1.8) defined by H(P, y) = h2 , we have P (ξ) =
b+
√
√ 4 √ ∆ − 3 ∆sech2 ( 2∆ ξ) , 2|a|
(3.1)
us
cr
ip t
where ∆ = b2 − 4ac. (2)When a < 0 and h ∈ (h1 , h2 ), there is a family of periodic solutions which correspond to the family of periodic orbits Γh of (1.8) defined by H(P, y) = h, h ∈ (h1 , h2 ), we have √ √ 6|a|(z1 − z3 ) z2 − z3 2 ξ, ), (3.2) P (ξ) = z3 + (z2 − z3 )sn ( 6 z1 − z3
√
√ 4 √ ∆ + 3 ∆sech2 ( 2∆ ξ) , 2a
M
P (ξ) =
−b −
an
where z1 > z2 > z3 and the parameters z1 , z2 , z3 are defined by y 2 = 2h − cP − 12 bP 2 − 1 1 3 3 aP = − 3 α(z1 − P )(z2 − P )(P − z3 ). (3) When a > 0 and h = h2 , there exists a bright soliton solution which corresponds to a smooth homoclinic orbit Γ of (1.8) defined by H(P, y) = h2 , we have (3.3)
ce p
te
d
where ∆ = b2 − 4ac. (4) When a > 0 and h ∈ (h1 , h2 ), there is a family of periodic solutions which correspond to the family of periodic orbits Γh of (1.8) defined by H(P, y) = h, h ∈ (h1 , h2 ), we have √ √ 6a(z1 − z3 ) z1 − z2 2 P (ξ) = z1 − (z1 − z2 )sn ( ξ, ), (3.4) 6 z1 − z3
Ac
where z1 > z2 > z3 and the parameters z1 , z2 , z3 are defined by y 2 = 2h − cP − 12 bP 2 − 1 1 3 3 aP = 3 α(z1 − P )(P − z2 )(P − z3 ). (5) When b < 0, there is a smooth solitary wave solution which corresponds to a smooth homoclinic orbit Γ of (1.8) defined by H(P, y) = h3 = 0, we have 3b − 3btanh2 ( P (ξ) = −2a
√
−b 2 ξ)
.
(3.5)
(6) When b > 0, there is a smooth solitary wave solution which corresponds to a smooth homoclinic orbit Γ of (1.8) defined by H(P, y) = h4 , we have √ b 3 b 2 (3.6) P (ξ) = − (1 − sech ( ξ)). a 2 2 In addition, when a < 0, b < 0(b > 0) and h ∈ (0, h4 )(h ∈ (h4 , 0)), there exists a family of periodic solutions which correspond to the family of periodic orbits Γh of (1.8) 6
Page 6 of 10
defined by H(P, y) = h, h ∈ (0, h4 )(h ∈ (h4 , 0)). Here, we have the same parametric representation as the case (2). When a > 0, b < 0(b > 0) and h ∈ (h4 , 0)(h ∈ (0, h4 )), there exists a family of periodic solutions which correspond to the family of periodic orbits Γh of (1.8) defined by H(P, y) = h, h ∈ (h4 , 0)(h ∈ (0, h4 )). Here, we have the same parametric representation as the case (4). By using above results, we can obtain exact travelling wave solutions of Eq.(1.2). (1) When a < 0 and h = h2 √
√ 4 √ ∆ − 3 ∆sech2 ( 2∆ (x − V t)) . 2|a|
(3.7)
ip t
u(x, t) =
b+
(2) When a < 0 and h ∈ (h1 , h2 ) u(x, t) = z3 + (z2 − z3 )sn (
6|a|(z1 − z3 ) (x − V t), 6
√
√ 4 √ ∆ + 3 ∆sech2 ( 2∆ (x − V t)) . 2a
an
u(x, t) =
−b −
z2 − z3 ). z1 − z3
us
(3) When a > 0 and h = h2
√
(4) When a > 0 and h ∈ (h1 , h2 ) √ u(x, t) = z1 − (z1 − z2 )sn (
6|b|(z1 − z3 ) (x − V t), 6
M
2
d
(5) When b < 0 and h = h3 = 0
√
z1 − z2 ). z1 − z3
(3.9)
(3.10)
√
3b − 3btanh2 ( 2−b (x − V t)) . u(x, t) = − 2a
te
(3.8)
cr
√ 2
(3.11)
(6) When b > 0 and h = h4
ce p
√ b b 3 2 u(x, t) = − (1 − sech ( (x − V t))). a 2 2
(3.12)
Ac
In physics, the bright and dark soliton solutions are important traveling wave solutions which have important physical properties. Based on the above results, by using the numerical simulation method, we will simulate all the bright and dark soliton solutions of Eq.(1.2). With the aid of Maple software, the three-dimensional graphics and their orbits of bright and dark soliton solutions of Eq.(1.2) are given in Figs.7-18. Usually, a periodic wave solution of a travelling wave equation corresponds to a periodic orbit of a dynamical system. A solitary wave solution of a travelling wave equation corresponds to a homoclinic orbit of a dynamical system. From Figs.7-18, it’s easy to see that (3.7), (3.11)(a < 0, b < 0) and (3.12)(a < 0, b > 0) are dark soliton solutions. The solution (3.9), (3.11)(a > 0, b < 0) and (3.12)(a > 0, b > 0) are bright soliton solutions.
7
Page 7 of 10
an
us
cr
ip t
Fig. 7: The three-dimensional graphics of Fig. 8: The homoclinic orbit defined by solution (3.7) (α1 = 1, α3 = 5, α4 = 1, ε = H(P, y) = h2 corresponding to (3.7) (α1 = 3, V = 2, g = 3). 1, α3 = 5, α4 = 1, ε = 3, V = 2, g = 3).
Ac
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te
d
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Fig. 9: The three-dimensional graphics of Fig. 10: The homoclinic orbit defined by solution (3.9) (α1 = −1, α3 = 5, α4 = H(P, y) = h2 corresponding to (3.9) (α1 = 1, ε = 1, V = 2, g = 1). −1, α3 = 5, α4 = 1, ε = 1, V = 2, g = 1).
Fig. 11: The three-dimensional graphics Fig. 12: The homoclinic orbit defined of solution (3.11) (α1 = 21 , α3 = 5, α4 = by H(P, y) = h3 corresponding to (3.11) (α1 = 12 , α3 = 5, α4 = 1, ε = −1, V = 1, ε = −1, V = 2, g = 0). 2, g = 0).
8
Page 8 of 10
ip t
Fig. 16: The homoclinic orbit defined by H(P, y) = h4 corresponding to (3.12) (α1 = 2, α3 = 5, α4 = 1, ε = 12 , V = √ 3, g = 0).
Ac
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te
d
Fig. 15: The three-dimensional of solution (3.12) (α1 = 2, α3 = 5, α4 = 1, ε = 21 , V = √ 3, g = 0).
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an
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Fig. 13: The three-dimensional graphics Fig. 14: The homoclinic orbit defined graphics of solution (3.11) (α1 = − 21 , α3 = by H(P, y) = h3 corresponding to (3.11) (α1 = − 12 , α3 = 5, α4 = 1, ε = −1, V = 5, α4 = 1, ε = −1, V = 2, g = 0). 2, g = 0).
Fig. 17: The three-dimensional graphics of Fig. 18: The homoclinic orbit defined solution (3.12) √ (α1 = −2, α3 = 5, α4 = by H(P, y) = h4 corresponding to1 (3.12) 1 1, ε = 2 , V = 3, g = 0). (α1 = −2, α3 = 5, α4 = 1, ε = 2 , V = √ 3, g = 0).
9
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4
Conclusion
In summary, by using the dynamical system method, the nonlinear dynamics of the strain wave equation in microstructured solids are studied and the periodic travelling wave solutions which are expressed by Jacobian elliptic functions and solitary wave solutions which are expressed by the hyperbolic functions are obtained. Among them, (3.8) and (3.10) are periodic travelling wave solutions. (3.7), (3.11)(a < 0, b < 0) and (3.12)(a < 0, b > 0) are dark soliton solutions. (3.9), (3.11)(a > 0, b < 0) and (3.12)(a > 0, b > 0) are bright soliton solutions.
ip t
Acknowledgements
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The authors gratefully acknowledge the support of the Intergovernmental Key International S&T Innovation Cooperation Program (No. 2016YFE0102400) and the Fundamental Research Funds for the Central Universities-Beijing Normal University Research Fund (No. 12500-310421103).
References
an
[1] Hafez, M. G., and M. A. Akbar, Ain Shams Engineering Journal 6(2014)683-690. [2] Porubov AV, Pastrone F, Int J Non-Lin Mech 39(2004) 1289-1299.
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[3] Alam MN, Akbar MA, Mohyud-Din ST, Alexandria Eng J 53(2014)233-41. [4] S.L. Xie, L. Wang, Y.Z. Zhang, Commun Nonlinear Sci Numer Simulat 17(2012)1130-1141.
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[5] H.Z. Liu, J.B. Li, Journal of Computational and Applied Mathematics 257(2014)144-156.
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[6] H. Wang, S.H. Zheng, Chaos Solitons and Fractals 82(2016)83-86.
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[7] H. Wang, S.H. Zheng, Analysis and Mathematical Physics (DOI 10.1007/s13324017-0178-4). [8] H. Wang, S.H. Zheng, Optik-International Journal for Light and Electron Optics. 140(2017)730-734.
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[9] J.B. Li, L. Zhang, Chaos Solitons Fractals 14(2002)581-593. [10] B. He, M. Qing, Commun.Theor.Phys. 65(2016)1-10. [11] H. Wang, L.W. Chen, H.J. Liu, et al, Mathematical Problems in Engineering 2016(2016)1-10. [12] H. Wang, S.H. Zheng, L.W. Chen, et al, Pramana. 87(2016)77.
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S0030-4026(17)31040-9 http://dx.doi.org/doi:10.1016/j.ijleo.2017.08.132 IJLEO 59595
To appear in: Received date: Accepted date:
10-5-2017 28-8-2017
Please cite this article as: Heng Wang, Shuhua Zheng, Bright and dark soliton solutions of the strain wave equation in microstructured solids, (2017), http://dx.doi.org/10.1016/j.ijleo.2017.08.132 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Bright and dark soliton solutions of the strain wave equation in microstructured solids Heng Wang1
ip t
State Key Laboratory of Earth Surface Processes and Resource Ecology, Faculty of Geographical Science, Beijing Normal University, Beijing 100875, China Institute of Land Surface System and Sustainable Development, Faculty of Geographical Science, Beijing Normal University, Beijing 100875, China
Shuhua Zheng
cr
Investment and Development Department, Market and Investment Center, Yunnan Water Investment Co., Limited, Kunming, Yunnan 650106, China
Introduction
d
1
M
an
us
Abstract: By using the approach of dynamical system, the exact travelling wave solutions of the strain wave equation in microstructured solids are solved. As results, all kinds of phase portraits in the parametric space are shown. The exact solutions are obtained including bright and dark soliton solutions. With the aid of Maple, the graphical representations and physical explanations of the bright and dark soliton solutions are given. Keywords: Strain wave equation; Microstructured solids; Dynamical system method; Phase portrait; Bright and dark soliton solution.
te
In this paper, we consider the travelling wave solutions of the strain wave equation in microstructured solids
Ac
ce p
utt −uxx −εα1 (u2 )xx −γα2 uxxt +δα3 uxxxx −(δα4 −γ 2 α7 )uxxtt +γδ(α5 uxxxxt +α6 uxxttt ) = 0, (1.1) where ε accounts for elastic strains, δ characterizes the ratio between the microstructure size and the wave length, γ characterizes the influence of dissipation and α1 , α2 , α3 , α4 , α5 , α6 are constants [1]. The balance between nonlinearity and dispersion takes place when δ = O(ε). If we set γ = 0, we have the non-dissipative case, and governed by the double dispersive equation as follows [2]: utt − uxx − ε(α1 (u2 )xx − α3 uxxxx + α4 uxxtt ) = 0.
(1.2)
The strain wave equation in microstructured solids is a class of important nonlinear wave solutions and plays a crucial role in nonlinear physics fields. Eq.(1.2) has been studied by many researchers. In [1], M.G. Hafeza and M.A. Akbar used an exponential expansion method to obtain some exact solutions of Eq.(1.2). In [3], the general traveling wave solutions were obtained by Md. Nur Alam et al. by using the new generalized (G0 /G)-expansion method. However, we notice that the previous authors 1
Corresponding author. E-mail: [email protected].
1
Page 1 of 10
did not study the dynamical behavior of Eq.(1.2). Therefore, it is essential to study the dynamical behavior of Eq.(1.2) and find some new exact solutions by qualitative analysis of the dynamic properties of Eq.(1.2). The dynamical system method is an effective method which is based on the bifurcation theory of planar dynamical systems and the analytical solutions of the nonlinear wave equations are solved by qualitative analysis [4-10]. Here, we use the approach of dynamical system to solve Eq.(1.2) and try to look for some new travelling wave solutions. To look for travelling wave solutions of (1.2), we assume that u(x, t) = P (ξ), ξ = x − V t,
ip t
(1.3)
where V is a travelling wave parameter. Substituting (1.3) into (1.2), we have
(1.4)
cr
(V 2 − 1)P 00 − εα1 (P 2 )00 + ε(α3 − V 2 α4 )P 0000 = 0.
Obviously, Eq.(1.4) is integrable. Therefore, integrating twice for (1.4), we have
us
(V 2 − 1)P − εα1 P 2 + ε(α3 − V 2 α4 )P 00 + g = 0.
(1.5)
where g is an integration constant. Suppose ε(α3 − V 2 α4 ) 6= 0 and write that −εα1 V2−1 g , b = ,c = . 2 2 ε(α3 − V α4 ) ε(α3 − V α4 ) ε(α3 − V 2 α4 )
an
a=
Finally, we have the following equation
M
P 00 + aP 2 + bP + c = 0,
(1.6)
(1.7)
which corrsponds to the two-dimensional Hamiltonian system
with the Hamiltonian
te
d
dP dy = y, = −aP 2 − bP − c, dξ dξ
ce p
1 1 1 H(P, y) = y 2 + αP 3 + βP 2 + γP = h, 2 3 2
(1.8)
(1.9)
Ac
where h is the Hamiltonian constant. According to the Hamiltonian, we can get all kinds of phase portraits in the parametric space of (1.8). By analyzing the bifurcations of phase portraits of system (1.8), we can obtain the travelling wave solutions of Eq.(1.2). This rest paper is organized as follows. In section 2, the theoretical description of dynamical system method is given. In section 3, we give all phase portraits of system (1.8) and discuss the bifurcations of phase portraits of system (1.8). According to the dynamics of the phase orbits of system (1.8), we solve Eq.(1.2) for two cases of c 6= 0 and c = 0. Finally, a short conclusion is given in section 4.
2
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2
Theory: the dynamical system method
In this section, we describe the dynamical system method for solving traveling wave solutions of nonlinear wave equations [11]. A (n+1)-dimensional nonlinear partial differential equation is given as follows. P (t, xi , ut , uxi , uxi xi , uxi xj , utt , · · · ) = 0
i, j = 1, 2, ..., n.
(2.1)
ip t
The main steps of the dynamical system method are as follows. Step1. Reduction of Eq.(2.1). ∑ Making a transformation u(t, x1 , x2 , ..., xn ) = φ(ξ), ξ = ni=1 ki xi − ct, Eq.(2.1) can be reduced to a non-linear ordinary differential equation D(ξ, φ, φξ , φξξ , φξξξ , · · · ) = 0,
cr
(2.2)
us
where ki are non-zero constant and c is the wave speed. Integrating several times for Eq.(2.2), if it can be reduced to the following second-order nonlinear ordinary differential equation E(φ, φξ , φξξ ) = 0, (2.3)
an
then let φξ = dφ/dξ = y, Eq.(2.3) can be reduced to a two-dimensional dynamical system dy dφ = y, = f (φ, y), (2.4) dξ dξ
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where f (φ, y) is an integral expression or a fraction. If f (φ, y) is a fraction such as f (φ, y) = F (φ, y)/g(φ) and g(φs ) = 0, dy dξ does not exist when φ = φs . Then we shall make a transformation dξ = g(φ)dζ, thus system (2.4) can be rewritten as dφ = g(φ)y, dζ
(2.5)
te
d
dy = F (φ, y), dζ
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where ζ is a parameter. If Eq.(2.1) can be reduced to the above system (2.4) or (2.5), then we can go on to the next step. Step2. Discussion of bifurcations of phase portraits of system (2.4). If system (2.4) is an integral system, systems (2.4) and (2.5) can be reduced the differential equation
Ac
f (φ, y) dy = , dφ y
dy F (φ, y) f (φ, y) = = , dφ g(φ)y y
(2.6)
then systems (2.4) and (2.5) have the same first integral (that is Hamiltonian) as follows H(φ, y) = h,
(2.7)
where h is an integral constant. According to the first integral, we can get all kinds of phase portraits in the parametric space. Because the phase orbits defined the vector fields of system (2.4) (or system (2.5)) determine the travelling wave solutions of Eq.(2.1), we can investigate the bifurcations of phase portraits of system (2.4) (or system (2.5)) to seek the travelling wave solutions of Eq.(2.1). Usually, a periodic orbit always corresponds to a periodic wave solution; A homoclinic orbit always corresponds 3
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to a solitary wave solution; A heteroclinic orbit (or so called connecting orbit) always corresponds to kink (or anti-kink) wave solution. When we find all phase orbits, we can get the value of h or its range. Step3. Calculation of the first equation of system (2.4). After h is determined, we can get the following relationship from Eq.(2.7) y = y(φ, h),
(2.8)
ip t
that is, dφ/dξ = y(φ, h). If the expression (2.8) is an integral expression, then substituting it into the first term of Eq.(2.4) and integrating it, we obtain ∫ ξ ∫ φ dϕ dτ. (2.9) = 0 φ0 y(ϕ, h)
3
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us
cr
where φ(0) and 0 are initial constants. Usually, the initial constants can be taken by a root of Eq. (2.8) or inflection points of the travelling waves. Taking proper initial constants and integrating Eq.(2.9), through the Jacobian elliptic functions, we can obtain the exact travelling wave solutions of Eq.(2.1). This is the whole process of the dynamical system method. The different nonlinear wave equations correspond to different dynamical systems. The different dynamical systems correspond to different travelling wave solutions.
Application: bright and dark soliton solutions of Eq.(1.2)
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In this section, based on the theoretical description in section 2, we solve Eq.(1.2). According to the dynamical system and Hamiltonian in section 1, we let right hand terms of system (1.8) be zeros, i.e. y = 0 and −aP 2 − bP − c = 0. Obviously, the abscissas of equilibrium points of system (1.8) are the real roots of f (P ) = aP√2 +bP +c. ∆ , 0) and Then, we obtain that the system (1.8) has two equilibrium points at S1 ( −b+ 2a √
te
∆ , 0) if ∆ > 0, where ∆ = b2 − 4ac. If ∆ = 0, the system (1.8) has a unique S2 ( −b− 2a b equilibrium at O(− 2a , 0). If ∆ < 0, the system (1.8) has no equilibrium. For the √
Hamiltonian H(P, y) = 12 y 2 + 13 aP 3 + 21 bP 2 + cP = h, we write h1 = H( −b+ 2a √
√
√
∆
, 0) =
∆ , h2 = H( −b− . Then, we consider the bifurcations , 0) = 2a of phase portraits of (1.8) in the case c = 0, we can obtain that the system (1.8) has two equilibrium points S(− ab , 0) and O(0,0). For the Hamiltonian H(P, y) = 12 y 2 + 1 b b3 1 3 2 3 aP + 2 bP = h, we write h3 = H(0, 0) = 0, h4 = H(− a , 0) = 6a2 . Because only bounded travelling waves are meaningful to a physical model, we just pay attention to the bounded orbit of the system (1.8). With the change of the parameter group of a, b and c, the phase portraits for (1.8) are shown in Figs.1-6. From Figs.1-6, we summarize the following conclusions: (1) When ∆ > 0, the system (1.8) has bounded orbits. When ∆ ≤ 0, the system (1.8) has no bounded orbits. (2) When ∆ > 0, the system (1.8) has a unique homoclinic orbit Γ which is asymptotic to the saddle and enclosing the center. There is a family of periodic orbits which are enclosing the center and filling up the interior of the homoclinic orbit Γ. (3) The system (1.11) has a unique homoclinic orbit Γ which is asymptotic to the saddle and enclosing the center. There is a family of periodic orbits which are enclosing the center and filling up the interior of the homoclinic orbit Γ. b3 −6abc+ 12a2
ce p
∆3
∆3
Ac
b3 −6abc− 12a2
4
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ip t
d
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cr
Fig. 1: The bifurcations of phase portraits Fig. 2: The bifurcations of phase portraits of (1.8) when a < 0 and 4 > 0. of (1.8) when a > 0 and 4 > 0.
Ac
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Fig. 3: The bifurcations of phase portraits Fig. 4: The bifurcations of phase portraits of (1.8) when a < 0,b > 0 and c = 0. of (1.8) when a < 0, b < 0 and c = 0.
Fig. 5: The bifurcations of phase portraits Fig. 6: The bifurcations of phase portraits of (1.8) when a > 0, b > 0 and c = 0. of (1.8) when a > 0, b < 0 and c = 0.
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According to the bounded orbit of the system (1.8), we obtain the travelling wave solutions of Eqs.(1.1). In fact, various possible results have been discussed by Wang et al [12]. Here we will cite the main results in Ref. [12]. So we have the following travelling wave solutions of Eqs.(1.1). (1)When a < 0 and h = h2 , there is a dark soliton solution which corresponds to a smooth homoclinic orbit Γ of (1.8) defined by H(P, y) = h2 , we have P (ξ) =
b+
√
√ 4 √ ∆ − 3 ∆sech2 ( 2∆ ξ) , 2|a|
(3.1)
us
cr
ip t
where ∆ = b2 − 4ac. (2)When a < 0 and h ∈ (h1 , h2 ), there is a family of periodic solutions which correspond to the family of periodic orbits Γh of (1.8) defined by H(P, y) = h, h ∈ (h1 , h2 ), we have √ √ 6|a|(z1 − z3 ) z2 − z3 2 ξ, ), (3.2) P (ξ) = z3 + (z2 − z3 )sn ( 6 z1 − z3
√
√ 4 √ ∆ + 3 ∆sech2 ( 2∆ ξ) , 2a
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P (ξ) =
−b −
an
where z1 > z2 > z3 and the parameters z1 , z2 , z3 are defined by y 2 = 2h − cP − 12 bP 2 − 1 1 3 3 aP = − 3 α(z1 − P )(z2 − P )(P − z3 ). (3) When a > 0 and h = h2 , there exists a bright soliton solution which corresponds to a smooth homoclinic orbit Γ of (1.8) defined by H(P, y) = h2 , we have (3.3)
ce p
te
d
where ∆ = b2 − 4ac. (4) When a > 0 and h ∈ (h1 , h2 ), there is a family of periodic solutions which correspond to the family of periodic orbits Γh of (1.8) defined by H(P, y) = h, h ∈ (h1 , h2 ), we have √ √ 6a(z1 − z3 ) z1 − z2 2 P (ξ) = z1 − (z1 − z2 )sn ( ξ, ), (3.4) 6 z1 − z3
Ac
where z1 > z2 > z3 and the parameters z1 , z2 , z3 are defined by y 2 = 2h − cP − 12 bP 2 − 1 1 3 3 aP = 3 α(z1 − P )(P − z2 )(P − z3 ). (5) When b < 0, there is a smooth solitary wave solution which corresponds to a smooth homoclinic orbit Γ of (1.8) defined by H(P, y) = h3 = 0, we have 3b − 3btanh2 ( P (ξ) = −2a
√
−b 2 ξ)
.
(3.5)
(6) When b > 0, there is a smooth solitary wave solution which corresponds to a smooth homoclinic orbit Γ of (1.8) defined by H(P, y) = h4 , we have √ b 3 b 2 (3.6) P (ξ) = − (1 − sech ( ξ)). a 2 2 In addition, when a < 0, b < 0(b > 0) and h ∈ (0, h4 )(h ∈ (h4 , 0)), there exists a family of periodic solutions which correspond to the family of periodic orbits Γh of (1.8) 6
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defined by H(P, y) = h, h ∈ (0, h4 )(h ∈ (h4 , 0)). Here, we have the same parametric representation as the case (2). When a > 0, b < 0(b > 0) and h ∈ (h4 , 0)(h ∈ (0, h4 )), there exists a family of periodic solutions which correspond to the family of periodic orbits Γh of (1.8) defined by H(P, y) = h, h ∈ (h4 , 0)(h ∈ (0, h4 )). Here, we have the same parametric representation as the case (4). By using above results, we can obtain exact travelling wave solutions of Eq.(1.2). (1) When a < 0 and h = h2 √
√ 4 √ ∆ − 3 ∆sech2 ( 2∆ (x − V t)) . 2|a|
(3.7)
ip t
u(x, t) =
b+
(2) When a < 0 and h ∈ (h1 , h2 ) u(x, t) = z3 + (z2 − z3 )sn (
6|a|(z1 − z3 ) (x − V t), 6
√
√ 4 √ ∆ + 3 ∆sech2 ( 2∆ (x − V t)) . 2a
an
u(x, t) =
−b −
z2 − z3 ). z1 − z3
us
(3) When a > 0 and h = h2
√
(4) When a > 0 and h ∈ (h1 , h2 ) √ u(x, t) = z1 − (z1 − z2 )sn (
6|b|(z1 − z3 ) (x − V t), 6
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2
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(5) When b < 0 and h = h3 = 0
√
z1 − z2 ). z1 − z3
(3.9)
(3.10)
√
3b − 3btanh2 ( 2−b (x − V t)) . u(x, t) = − 2a
te
(3.8)
cr
√ 2
(3.11)
(6) When b > 0 and h = h4
ce p
√ b b 3 2 u(x, t) = − (1 − sech ( (x − V t))). a 2 2
(3.12)
Ac
In physics, the bright and dark soliton solutions are important traveling wave solutions which have important physical properties. Based on the above results, by using the numerical simulation method, we will simulate all the bright and dark soliton solutions of Eq.(1.2). With the aid of Maple software, the three-dimensional graphics and their orbits of bright and dark soliton solutions of Eq.(1.2) are given in Figs.7-18. Usually, a periodic wave solution of a travelling wave equation corresponds to a periodic orbit of a dynamical system. A solitary wave solution of a travelling wave equation corresponds to a homoclinic orbit of a dynamical system. From Figs.7-18, it’s easy to see that (3.7), (3.11)(a < 0, b < 0) and (3.12)(a < 0, b > 0) are dark soliton solutions. The solution (3.9), (3.11)(a > 0, b < 0) and (3.12)(a > 0, b > 0) are bright soliton solutions.
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cr
ip t
Fig. 7: The three-dimensional graphics of Fig. 8: The homoclinic orbit defined by solution (3.7) (α1 = 1, α3 = 5, α4 = 1, ε = H(P, y) = h2 corresponding to (3.7) (α1 = 3, V = 2, g = 3). 1, α3 = 5, α4 = 1, ε = 3, V = 2, g = 3).
Ac
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Fig. 9: The three-dimensional graphics of Fig. 10: The homoclinic orbit defined by solution (3.9) (α1 = −1, α3 = 5, α4 = H(P, y) = h2 corresponding to (3.9) (α1 = 1, ε = 1, V = 2, g = 1). −1, α3 = 5, α4 = 1, ε = 1, V = 2, g = 1).
Fig. 11: The three-dimensional graphics Fig. 12: The homoclinic orbit defined of solution (3.11) (α1 = 21 , α3 = 5, α4 = by H(P, y) = h3 corresponding to (3.11) (α1 = 12 , α3 = 5, α4 = 1, ε = −1, V = 1, ε = −1, V = 2, g = 0). 2, g = 0).
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ip t
Fig. 16: The homoclinic orbit defined by H(P, y) = h4 corresponding to (3.12) (α1 = 2, α3 = 5, α4 = 1, ε = 12 , V = √ 3, g = 0).
Ac
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te
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Fig. 15: The three-dimensional of solution (3.12) (α1 = 2, α3 = 5, α4 = 1, ε = 21 , V = √ 3, g = 0).
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Fig. 13: The three-dimensional graphics Fig. 14: The homoclinic orbit defined graphics of solution (3.11) (α1 = − 21 , α3 = by H(P, y) = h3 corresponding to (3.11) (α1 = − 12 , α3 = 5, α4 = 1, ε = −1, V = 5, α4 = 1, ε = −1, V = 2, g = 0). 2, g = 0).
Fig. 17: The three-dimensional graphics of Fig. 18: The homoclinic orbit defined solution (3.12) √ (α1 = −2, α3 = 5, α4 = by H(P, y) = h4 corresponding to1 (3.12) 1 1, ε = 2 , V = 3, g = 0). (α1 = −2, α3 = 5, α4 = 1, ε = 2 , V = √ 3, g = 0).
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4
Conclusion
In summary, by using the dynamical system method, the nonlinear dynamics of the strain wave equation in microstructured solids are studied and the periodic travelling wave solutions which are expressed by Jacobian elliptic functions and solitary wave solutions which are expressed by the hyperbolic functions are obtained. Among them, (3.8) and (3.10) are periodic travelling wave solutions. (3.7), (3.11)(a < 0, b < 0) and (3.12)(a < 0, b > 0) are dark soliton solutions. (3.9), (3.11)(a > 0, b < 0) and (3.12)(a > 0, b > 0) are bright soliton solutions.
ip t
Acknowledgements
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The authors gratefully acknowledge the support of the Intergovernmental Key International S&T Innovation Cooperation Program (No. 2016YFE0102400) and the Fundamental Research Funds for the Central Universities-Beijing Normal University Research Fund (No. 12500-310421103).
References
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[1] Hafez, M. G., and M. A. Akbar, Ain Shams Engineering Journal 6(2014)683-690. [2] Porubov AV, Pastrone F, Int J Non-Lin Mech 39(2004) 1289-1299.
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[5] H.Z. Liu, J.B. Li, Journal of Computational and Applied Mathematics 257(2014)144-156.
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[6] H. Wang, S.H. Zheng, Chaos Solitons and Fractals 82(2016)83-86.
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[7] H. Wang, S.H. Zheng, Analysis and Mathematical Physics (DOI 10.1007/s13324017-0178-4). [8] H. Wang, S.H. Zheng, Optik-International Journal for Light and Electron Optics. 140(2017)730-734.
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[9] J.B. Li, L. Zhang, Chaos Solitons Fractals 14(2002)581-593. [10] B. He, M. Qing, Commun.Theor.Phys. 65(2016)1-10. [11] H. Wang, L.W. Chen, H.J. Liu, et al, Mathematical Problems in Engineering 2016(2016)1-10. [12] H. Wang, S.H. Zheng, L.W. Chen, et al, Pramana. 87(2016)77.
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