Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2014, Article ID 381717, 9 pages http://dx.doi.org/10.1155/2014/381717
Research Article On Differential Equations Derived from the Pseudospherical Surfaces Hongwei Yang,1 Xiangrong Wang,1 and Baoshu Yin2,3 1
College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China Institute of Oceanology, Chinese Academy of Sciences, Qingdao 266071, China 3 Key Laboratory of Ocean Circulation and Wave, Chinese Academy of Sciences, Qingdao 266071, China 2
Correspondence should be addressed to Baoshu Yin;
[email protected] Received 2 January 2014; Revised 19 March 2014; Accepted 21 March 2014; Published 24 April 2014 Academic Editor: Weiguo Rui Copyright Β© 2014 Hongwei Yang 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. We construct two metric tensor fields; by means of these metric tensor fields, sinh-Gordon equation and elliptic sinh-Gordon equation are obtained, which describe pseudospherical surfaces of constant negative Riemann curvature scalar π = β2, π = β1, respectively. By employing the BΒ¨acklund transformation, nonlinear superposition formulas of sinh-Gordon equation and elliptic sinh-Gordon equation are derived; various new exact solutions of the equations are obtained.
1. Introduction The soliton equation [1] is related to several fields in mathematics [2] (such as differential geometry and nonlinear partial differential equation [3, 4]) and theoretical physics (such as Josephson transition line [5], solitary Rossby waves and internal solitary waves in the ocean [6β9], chain of coupled pendula [10], pulse propagation in two-level atomic system [11], and quantum field theory [12]). Soliton equation can be derived from pseudospherical surfaces. Extensions to other soliton equations are straightforward. Soliton equations have several remarkable properties in common. Firstly, the initial value problem can be solved exactly by means of the inverse scattering methods [13]. Secondly, they have an infinite number of conservation laws [14, 15]. Thirdly, they have BΒ¨acklund transformations [16, 17]. Fourthly, they pass the PainlevΒ΄e test [18]. Furthermore they describe pseudospherical surfaces, that is, surfaces of constant negative Gaussian curvature [19, 20]. Sinh-Gordon equation and elliptic sinh-Gordon equation are two important soliton equations in the field of soliton. From the model building perspective, there are various interesting examples making use of the sinh-Gordon equation and elliptic sinh-Gordon equation [21], such as the propagation of splay waves on a lipid membrane,
one-dimensional models for elementary particles, selfinduced transparency of short optical pulses, and domain walls in ferroelectric and ferromagnetic materials. The second point worth noting is the historical development of the equations. They first appeared in differential geometry, where they were used to describe surfaces with a constant negative Gaussian curvature, but the previous study mainly focuses on sine-Gordon equation [22β25]; there are few scholarstic research on sinh-Gordon equation and elliptic sinh-Gordon equation. In this paper, we will first construct two metric tensor fields; through these metric tensor fields, sinh-Gordon equations and elliptic sinh-Gordon equation are obtained. The method to derive soliton equations is greatly different from the previous papers [26]. Then, we will discuss analytic solutions of the sinh-Gordon equation and elliptic sinh-Gordon equation by using BΒ¨acklund transformation. On the basis of the BΒ¨acklund transformation, the formulas of nonlinear superposition of sinh-Gordon equation and elliptic sinhGordon equation are proposed in this paper, and the singlesoliton (breather) solution and double-soliton (breather) solution have been calculated. Finally, computer simulations of the single-soliton (breather) solution and double-soliton (breather) solution are presented by using the mathematical software Matlab.
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Abstract and Applied Analysis
2. General Method to Derive Soliton Equations from Pseudospherical Surfaces Metric tensor is used to study the invariant quantity of a surface [27, 28], such as the length of a curve drawn along the surface, the angle between a pair of curves drawn along the surface, and meeting at a common point, or tangent vectors at the same point of the surface, the area of a piece of the surface, and so on. However, many PDEs describe constant curvature surfaces. So, we can derive PDE via metric tensor. In this section, we introduce the general procedure for deriving soliton equations from pseudospherical surfaces. The metric tensor field for the PDE is given by π = π11 ππ₯ β ππ₯ + π12 ππ₯ β ππ‘ + π21 ππ‘ β ππ₯ + π22 ππ‘ β ππ‘, (1)
π = ππ₯ β ππ₯ + cosh (π’ (π₯, π‘)) ππ₯ β ππ‘ + cosh (π’ (π₯, π‘)) ππ‘ β ππ₯ + ππ‘ β ππ‘,
ππ₯ 2 ππ₯ ππ‘ ππ‘ 2 ππ 2 + π22 ( ) . ) = π11 ( ) + (π12 + π21 ) ππ ππ ππ ππ ππ (2)
The quantity π can be written in matrix form π π π = ( 11 12 ) , π21 π22
ππ₯ ππ‘ ππ₯ 2 ππ‘ 2 ππ 2 ) = ( ) + 2 cosh (π’ (π₯, π‘)) + ( ) , (12) ππ ππ ππ ππ ππ
where π’ is a smooth function of π₯ and π‘. Firstly, we will calculate the Riemann curvature scalar π from π. Then the sinh-Gordon equation follows when we impose the condition π = β2. We have π11 = π22 = 1,
(3)
πβ1
π π π = ( 11 12 ) , π21 π22 (4)
1 π := β πππ (πππ,π + πππ,π β πππ,π ) , πππ 2 π
(5)
πβ1 = (
ππππ , ππ₯
πππ,2 =
ππππ . ππ‘
:=
π πππ,π
β
π πππ ,π
+
π π πππ β (πππ π
β
π π πππ πππ ) .
ππ
= π πππ .
(7)
π12 = π21 =
cosh (π’) . sinh2 (π’)
(16)
π π ππ‘ = ( 11,2 12,2 ) , π21,2 π22,2
(17)
where ππ₯ =
(8)
ππ , ππ₯
π11,1 = π22,1 = 0,
π12,1 = π21,1 = sinh (π’) π’π₯ , ππ ππ‘ = , ππ‘
(9)
(18)
π11,2 = π22,2 = 0,
π12,2 = π21,2 = sinh (π’) π’π‘ .
Finally, the curvature scalar π is given by π π := ππ .
1 , sinh2 (π’)
π π ππ₯ = ( 11,1 12,1 ) , π21,1 π22,1
and is constructed by contraction. From πππ , we obtain πππ via πππ
(15)
Differentiating (14) with respect to π₯ and π‘, we obtain
The Ricci tensor follows as π π = βππππ , πππ := ππππ
π11 π12 ), π21 π22
(6)
Following, we calculate the Riemann curvature tensor which is given by π πππ π
(14)
where π11 = π22 = β
where
(13)
and the inverse of π is given by
Next we have to calculate the Christoffel symbols. They are defined as
πππ,1 :=
π21 = π12 = cosh (π’) .
The quantity π can be written in matrix form
and then the inverse of π is given by π11 π12 = ( 21 22 ) . π π
(11)
and the line element is (
and the line element is (
2.1. Sinh-Gordon Equation Derived from Pseudospherical Surfaces. Sinh-Gordon equation and elliptic sinh-Gordon equation appear in wide range of physical applications including integrable quantum field theory, kink dynamics, fluid dynamics, and nonlinear optics [29β31]. In this section, we will derive sinh-Gordon equation from pseudospherical surfaces following the method presented in the previous section. The metric tensor field for the sinh-Gordon equation is given by
(10)
If the given π is a constant, we will get a partial differential equation.
Since 1 π πππ = πππ (πππ,π + πππ,π β πππ,π ) , 2
(19)
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3
we obtain
By virtue of
1 = π11 1 π12
cosh (π’) π’π₯ , sinh (π’) 2 π12
= 0, 1 π22 =β
2 π11 =β 1 π21
= 0,
π’π‘ , sinh (π’)
2 π21
= 0,
2 π22 =
π π = βππππ , πππ := ππππ
π’π₯ , sinh (π’)
(24)
we get
= 0,
(20) π11 = β
cosh (π’) π’π‘ . sinh (π’)
π’π₯π‘ , sinh (π’)
π12 = β
cosh (π’) π’π₯π‘ , π21 = β sinh (π’)
Differentiating (20) with respect to π₯ and π‘, we obtain
cosh (π’) π’π₯π‘ sinh (π’)
π’π₯π‘ π22 = β . sinh (π’)
(25)
By virtue of 1 π11,1 = 1 π11,2
cosh (π’) sinh (π’) π’π₯π₯ β (π’π₯ ) sinh (π’)
π11 = β
2
cosh (π’) (π’π₯ ) β sinh (π’) π’π₯π₯ sinh2 (π’)
,
=
=
2 π21,1
=
2 π21,2
=
2 π22,1
cosh (π’) sinh (π’) π’π₯π‘ β π’π₯ π’π‘ = , sinh2 (π’)
sinh2 (π’)
sinh2 (π’)
(28)
π := β
2π’π₯π‘ . sinh (π’)
π’π₯π‘ = sinh (π’)
(29)
(30)
is obtained.
,
2.2. Elliptic Sinh-Gordon Equation Derived from Pseudospherical Surfaces. In this section, we will derive elliptic sinhGordon equation from pseudospherical surfaces. The metric tensor field for the elliptic sinh-Gordon equation is given by
2
cosh (π’) sinh (π’) π’π‘π‘ β (π’π‘ )
(27)
When given π = β2, the well-known sinh-Gordon equation
2
cosh (π’) (π’π‘ ) β sinh (π’) π’π‘π‘
1 π22,2
π’π₯π‘ . sinh (π’)
we get (21)
= 0,
π22 = β
π π := ππ ,
cosh (π’) π’π₯ π’π‘ β sinh (π’) π’π₯π‘ = , sinh2 (π’)
2 = π22,2
π’π₯π‘ , sinh (π’)
Finally, with the help of
cosh (π’) π’π‘ π’π₯ β sinh (π’) π’π₯π‘ = , sinh2 (π’)
1 π21,2
(26)
we get
1 1 2 2 1 = π12,2 = π12,1 = π12,2 = π21,1 π12,1
1 π22,1
πππ = πππ πππ ,
,
2
cosh (π’) sinh (π’) π’π₯π‘ β π’π₯ π’π‘ = , sinh2 (π’)
2 = π11,1 2 π11,2
2
.
π = cosh (π’ (π₯, π‘)) ππ₯ β ππ₯ + ππ₯ β ππ‘ + ππ‘ β ππ₯ + cosh (π’ (π₯, π‘)) ππ‘ β ππ‘,
By virtue of
(31)
and the line element is
π π π π π π π πππ π := πππ,π β πππ ,π + β (πππ πππ β πππ πππ ) , π
(22)
(
ππ₯ 2 ππ₯ ππ‘ ππ 2 ) = cosh (π’ (π₯, π‘)) ( ) + 2 ππ ππ ππ ππ ππ‘ 2 + cosh (π’ (π₯, π‘)) ( ) . ππ
we get 1 = 0, π111 1 = 0, π211 2 π121
1 π112 =β
1 π212 =β
π’π₯π‘ , =β sinh (π’)
2 =β π221
cosh (π’) π’π₯π‘ , sinh (π’)
cosh (π’) π’π₯π‘ , sinh (π’)
π’π₯π‘ sinh (π’) 2 π122
= 0,
2 = 0. π222
(32)
Firstly, we calculate the Riemann curvature scalar π from π. Then the elliptic sinh-Gordon equation follows when we impose the condition π = β1. We have (23)
π11 = π22 = cosh (π’) ,
π21 = π12 = 1.
(33)
The quantity π can be written in matrix form π π π = ( 11 12 ) , π21 π22
(34)
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Abstract and Applied Analysis
and the inverse of π is given by
2 =β π11,1
π11 π12 πβ1 = ( 21 22 ) , π π
(35)
π
11
=π
22
cosh (π’) = , sinh2 (π’)
π
12
=π
21
1 =β . (36) sinh2 (π’)
π π ππ₯ = ( 11,1 12,1 ) , π21,1 π22,1
π π ππ‘ = ( 11,2 12,2 ) , π21,2 π22,2
2 π12,1 =
1 = π12,2
Differentiate (34) with respect to π₯ and π‘, we have (37)
2sinh2 (π’)
2 =β π11,2
1 π12,1 =
where
2
cosh (π’) sinh (π’) π’π₯π‘ β cosh (π’) (π’π₯ ) β π’π₯ π’π‘ + sinh (π’) π’π₯π₯
2 = π12,2
1 = π21,1
2
cosh (π’) sinh (π’) π’π‘π‘ β cosh (π’) π’π₯ π’π‘ β (π’π‘ ) + sinh (π’) π’π₯π‘ 2sinh2 (π’)
ππ , ππ₯ = ππ₯
π11,1 = π22,1 = sinh (π’) π’π₯ ,
2 = π21,1
π12,1 = π21,1 = 0, ππ ππ‘ = , ππ‘
(38) 2 π21,2 =
π11,2 = π22,2 = sinh (π’) π’π‘ , π12,1 = π21,1 = 0.
1 = π22,1
Since
1 = π22,2
1 π = πππ (πππ,π + πππ,π β πππ,π ) , πππ 2
(39)
we obtain
2 = π22,2
1 π11 =
cosh (π’) π’π₯ + π’π‘ , 2 sinh (π’)
2 =β π11
1 = π12
cosh (π’) π’π‘ β π’π₯ , 2 sinh (π’)
2 π12 =
cosh (π’) π’π‘ β π’π₯ 1 π21 = , 2 sinh (π’) 1 =β π22
2sinh2 (π’)
cosh (π’) π’π₯ + π’π‘ , 2 sinh (π’)
2sinh2 (π’)
2
cosh (π’) π’π₯ π’π‘ + cosh (π’) sinh (π’) π’π₯π₯ β (π’π₯ ) β sinh (π’) π’π₯π‘ 2sinh2 (π’)
1 π11,2 =
2
cosh (π’) (π’π‘ ) + cosh (π’) sinh (π’) π’π₯π‘ β π’π₯ π’π‘ β sinh (π’) π’π‘π‘ 2sinh2 (π’)
,
2
cosh (π’) (π’π₯ ) + cosh (π’) sinh (π’) π’π₯π‘ β π’π₯ π’π‘ β sinh (π’) π’π₯π₯ 2sinh2 (π’)
,
2
cosh (π’) π’π₯ π’π‘ + cosh (π’) sinh (π’) π’π‘π‘ β (π’π‘ ) β sinh (π’) π’π₯π‘ 2sinh2 (π’)
,
2
cosh (π’) π’π₯ π’π‘ + cosh (π’) sinh (π’) π’π₯π₯ β (π’π₯ ) β sinh (π’) π’π₯π‘ 2sinh2 (π’)
,
2
cosh (π’) (π’π‘ ) + cosh (π’) sinh (π’) π’π₯π‘ β π’π₯ π’π‘ β sinh (π’) π’π‘π‘ 2sinh2 (π’)
,
2
cosh (π’) π’π₯ π’π‘ β cosh (π’) sinh (π’) π’π₯π₯ β sinh (π’) π’π₯π‘ + (π’π₯ )
,
2sinh2 (π’)
2
cosh (π’) (π’π‘ ) β cosh (π’) sinh (π’) π’π₯π‘ β sinh (π’) π’π‘π‘ + π’π₯ π’π‘ 2sinh2 (π’)
,
2
cosh (π’) sinh (π’) π’π₯π‘ β cosh (π’) (π’π₯ ) β π’π₯ π’π‘ + sinh (π’) π’π₯π₯ 2
2sinh (π’)
,
2
cosh (π’) sinh (π’) π’π‘π‘ β cosh (π’) π’π₯ π’π‘ β (π’π‘ ) + sinh (π’) π’π₯π‘ 2
2sinh (π’)
π π π π π π π := πππ,π β πππ ,π + β (πππ πππ β πππ πππ ) , πππ π π
π’π₯ + cosh (π’) π’π‘ . 2 sinh (π’) (40)
.
2
,
2
β cosh (π’) (π’π‘ ) + cosh (π’) sinh (π’) π’π₯π‘ + sinh (π’) π’π‘π‘ β π’π₯ π’π‘
,
(42)
we get 1 π111 = 0, 1 = 0, π211
2sinh2 (π’) 2sinh2 (π’)
,
By virtue of
βπ’π‘ + cosh (π’) π’π₯ 2 = π21 , 2 sinh (π’)
β cosh (π’) π’π₯ π’π‘ + cosh (π’) sinh (π’) π’π₯π₯ + sinh (π’) π’π₯π‘ β (π’π₯ )
,
(41)
βπ’π‘ + cosh (π’) π’π₯ , 2 sinh (π’)
2 π22 =
,
2
cosh (π’) π’π₯ π’π‘ + cosh (π’) sinh (π’) π’π‘π‘ β (π’π‘ ) β sinh (π’) π’π₯π‘
π’π₯ + cosh (π’) π’π‘ , 2 sinh (π’)
Differentiating (40) with respect to π₯ and π‘, we obtain 1 π11,1 =
2 = π22,1
,
2
cosh (π’) (π’π₯ ) + cosh (π’) sinh (π’) π’π₯π‘ β π’π₯ π’π‘ β sinh (π’) π’π₯π₯
where 1 = π21,2
,
2 π121
1 π112 =β 1 π212 =β
π’π₯π₯ + π’π‘π‘ , 2 sinh (π’)
(π’π‘π‘ + π’π₯π₯ ) cosh (π’) 2 sinh (π’)
(π’ + π’π₯π₯ ) cosh (π’) = β π‘π‘ , 2 sinh (π’) 2 π221 =β
π’π‘π‘ + π’π₯π₯ , 2 sinh (π’)
2 π122 2 = 0. π222
= 0,
(43)
Abstract and Applied Analysis
5
By virtue of π π πππ := ππππ = βππππ ,
(44)
we get π11 = β π21 = β
(π’π‘π‘ + π’π₯π₯ ) cosh (π’) , 2 sinh (π’)
π’π‘π‘ + π’π₯π₯ , 2 sinh (π’)
π12 = β
π22 = β
0
π’π‘π‘ + π’π₯π₯ 2 sinh (π’)
(π’π‘π‘ + π’π₯π₯ ) cosh (π’) . 2 sinh (π’)
β2
(45)
By virtue of πππ
ππ
= π πππ ,
10
β4
u
8
β6
6
β8
4
β10 β8
(46)
β6
β4
t
2 β2 x
0
2
4
6
0
Figure 1: Waterfall of (58), where πΌ = 1.5, πΆ = 0.
we get π11 = β
π’π₯π₯ + π’π‘π‘ , 2 sinh (π’)
π22 = β
π’π₯π₯ + π’π‘π‘ . 2 sinh (π’)
(47)
Finally, with the help of π π := ππ ,
(48)
So, if we get the solutions of the sinh-Gordon equation, it is very easy to get the solutions of the elliptic sinh-Gordon equation. The auto-BΒ¨acklund transformations for the sinh-Gordon equation π’π₯π‘ = sinh (π’)
we get π := β
π’π₯π₯ + π’π‘π‘ . sinh (π’)
(49)
is given by (
When given π = β1, the well-known elliptic sinh-Gordon equation π’π₯π₯ + π’π‘π‘ = sinh (π’)
3. Solutions to the Sinh-Gordon Equation and Elliptic Equation BΒ¨acklund transformations play an important role in finding solutions of a certain class of nonlinear partial differential equations [32, 33]. From a solution of a nonlinear partial differential equation, we can sometimes find a relationship that will generate the solution of a different partial differential equation, which is known as a BΒ¨acklund transformation, or of the same partial differential equation where such a relation is then known as an auto-BΒ¨acklund transformation. As to elliptic sinh-Gordon equation π’π₯π₯ + π’π‘π‘ = sinh (π’)
(51)
under the transformation π π₯ σ³¨σ³¨β (π₯ β ππ‘) , 2
1 π‘ σ³¨σ³¨β (π₯ + ππ‘) , 2π
(52)
π’σΈ β π’ π’σΈ + π’ ) = π sinh ( ), 2 2 π₯
1 π’σΈ + π’ π’σΈ β π’ ) = sinh ( ). ( 2 π 2 π‘
(50)
is obtained.
(54)
(55)
If π’ is a solution of the sinh-Gordon equation, π’σΈ is also a solution of the sinh-Gordon equation. Here we are looking for solutions of the sinh-Gordon equation by using the BΒ¨acklund transformations. Obviously π’(π₯, π‘) = 0 is a solution of the sinh-Gordon equation. This is known as the vacuum solution. We make use of the auto-BΒ¨acklund transformation to construct another solution of the sinh-Gordon equation from the vacuum solution. Inserting this solution into the given BΒ¨acklund transformation results in (
π’σΈ π’σΈ ) = π sinh ( ) , 2 π₯ 2
(
1 π’σΈ π’σΈ ) = sinh ( ) . (56) 2 π‘ π 2
Since β«
π’ ππ’ = β4tanhβ1 exp ( ) , sinh (π’/2) 2
(57)
we obtain a new solution of the sinh-Gordon equation; namely,
where π is a positive constant, (51) is transformed into the sinh-Gordon equation
π 1 π’σΈ = 2 ln tanh (β π₯ β π‘ + πΆ) , 2 2π
π’π₯π‘ = sinh (π’) .
where πΆ is a constant of integration, and the computer simulation of (58) is presented in Figures 1 and 2.
(53)
(58)
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Abstract and Applied Analysis 0
or
β1
π’3 β π’0 π’ β π’0 π’ β π’2 π’ β π’2 cosh 1 + sinh 1 cosh 3 ) 4 4 4 4 π’ β π’0 π’ β π’1 cosh 2 = π2 (sinh 3 4 4 π’ β π’0 π’ β π’1 + sinh 2 cosh 3 ). 4 4 (63)
π1 (sinh
β2
u, t = 5
β3 β4 β5 β6 β7 β8
After abbreviation, the following nonlinear superposition formula obtained
β9 β8
β6
β4
β2
0
2
6
4
x
tanh
Figure 2: Solution of (58), where πΌ = 1.5, πΆ = 0, and π‘ = 5.
π’3 β π’0 π2 + π1 π’ β π’2 tanh 1 = , 4 π2 β π1 4
(64)
or This solution may be used to determine another solution for the sinh-Gordon equation and so on. If we use this method to calculate other new solutions, it is very difficult to solve the first-order equation. However, we can get the nonlinear superposition formula via (55). From π’0 , first by employing π1 , π’1 is obtained; then by employing π2 , π’3 can be obtained: (
π’ β π’0 π’1 + π’0 ) = π1 sinh 1 , 2 2 π₯
π’ β π’1 π’ + π’1 ( 3 ) = π2 sinh 3 . 2 2 π₯
(59)
π’2 + π’0 π’ β π’0 ) = π2 sinh 2 , 2 2 π₯
π’ β π’2 π’ + π’2 ( 3 ) = π1 sinh 3 . 2 2 π₯
(66)
π1 1 π‘ + πΆ1 ) π₯β 2 2π1 π2 1 π‘ + πΆ2 )) Γ (2)β1 ) , π₯β 2 2π2 (67)
(61)
By simple calculation, (61) can be rewritten as π’1 β π’0 + π’3 β π’2 π’ β π’0 β π’3 + π’2 cosh 1 4 4 π’2 β π’0 + π’3 β π’1 π’2 β π’0 β π’3 + π’1 = 2π2 sinh cosh , 4 4 (62)
2π1 sinh
π1 1 π‘ + πΆ1 ) , π₯β 2 2π1
π2 + π1 π2 β π1
β ln tanh (β
π1 (sinh
(65)
by means of (65), we can easily get the fourth solution
Γ tanh ((ln tanh (β
From (59) and (60), we get π’1 β π’0 π’ β π’2 + sinh 3 ) 2 2 π’ β π’0 π’ β π’1 + sinh 3 ). = π2 (sinh 2 2 2
π’1 = 2 ln tanh (β
π 1 π‘ + πΆ2 ) , π’2 = 2 ln tanh (β 2 π₯ β 2 2π2
π’3 = 4tanhβ1 (60)
π2 + π1 π’ β π’2 tanh 1 . π2 β π1 4
If we are given
π’0 = 0,
Meanwhile by changing the using order of π1 and π2 , π’2 and π’4 are also obtained, respectively. If π’3 = π’4 , then (
π’3 = π’0 + 4tanhβ1
and the computer simulation of the solution is presented in Figures 3 and 4. In this way, by algebraic operation, a series of new solutions of sinh-Gordon equation can be easily obtained. Similarly, from (52), (58), and (67), we can get the single breather solution 1 1 1 π’σΈ = 2 ln tanh (β ((ππ + ) π₯ β π (ππ β ) π‘) + πΆ) 4 ππ ππ (68)
Abstract and Applied Analysis
7
8 6 4 10
20 u
8
u
β20 β10
4
β4
t
x
0
5
10
β6
2
β5
0 β2
6
0
2
β8 β3
0
5 β2
β1 x
Figure 3: Waterfall of (67), where πΆ1 = 1, πΆ2 = 0, πΌ1 = 1.3, and πΌ2 = 1.7.
0
1
2
t
0
3
Figure 5: Waterfall of (68), where πΌ = 1, π = 2, and πΆ = 0.
10 8 6
10
u, t = 5
4
5
2 u
0 β2
0
30
β5
β4
20
β10 β5
β6 β8
10 x
β10 β10
β5
0
5
x
Figure 4: Solution of (67), where πΆ1 = 1, πΆ2 = 0, πΌ1 = 1.3, πΌ2 = 1.7, and π‘ = 5.
0 5
t
0
Figure 6: Waterfall of (69), where πΆ1 = 0, πΆ2 = 0, πΌ1 = 1.3, πΌ2 = 1.7, and π = 1.
of the elliptic sinh-Gordon equation. The computer simulation of the solutions is presented in Figures 5 and 6.
and double breather solution π’3 = 4tanhβ1
4. Summary and Discussion
π2 + π1 π2 β π1
1 1 Γ tanh (ln tanh (β ((π1 π + )π₯ 4 π1 π β π (π1 π β β ln tanh ( β
1 ) π‘) + πΆ1 ) π1 π
1 1 ((π2 π + )π₯ 4 π2 π β π (π2 π β
1 ) π‘)+πΆ2 ) π2 π
Γ (2)β1 ) (69)
In this paper, we obtain sinh-Gordon equation and elliptic sinh-Gordon equation by means of pseudospherical surfaces. In addition, we give the B¨acklund transformations and nonlinear superposition formulas of sinh-Gordon equation and elliptic sinh-Gordon equation, which lead to new exact solutions of the sinh-Gordon equation and elliptic sinhGordon equation. On the basis of the B¨acklund transformations and nonlinear superposition formulas, the singlesoliton (breather) solution and double-soliton (breather) solution of the sinh-Gordon equation and elliptic sinhGordon equation have been calculated. Finally, computer simulations of the single-soliton (breather) solution and double-soliton (breather) solution are presented by using the mathematical software Matlab. In forthcoming days, we will further discuss the problem. It is also interesting for us to see how the metric tensor field will be for other soliton equations.
8
Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment This work was supported by the Strategic Pioneering Program of Chinese Academy of Sciences (no. XDA 10020104), the National Natural Science Foundation of China (no. 11271007), the Nature Science Foundation of Shandong Province of China (no. ZR2013AQ017), the Open Fund of the Key Laboratory of Data Analysis and Application, State Oceanic Administration (no. LDAA-2013-04), and the SDUST Research Fund (no. 2012KYTD105).
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