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On differential equations derived from the pseudospherical surfaces.

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 [6-9], 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 Backlund transformations [16, 17]. Fourthly, they pass the Painleve 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, self-induced 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 [22-25]; 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 Backlund transformation. On the basis of the Backlund transformation, the formulas of nonlinear superposition of sinh-Gordon equation and elliptic sinh-Gordon equation are proposed in this paper, and the single-soliton (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.

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

f - [f.sub.11] dx [cross product] dx + [f.sub.12] dx [cross product] dt + [f.sub.21] dt [cross product] dx + [f.sub.22] dt [cross product] dt, (1)

and the line element is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (2)

The quantity f can be written in matrix form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (3)

and then the inverse of f is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (4)

Next we have to calculate the Christoffel symbols. They are defined as

[[tau].sup.a.sub.mn] := [summation over (b)] [f.sup.ab] ([f.sub.bm,n] - [f.sub.mn,b]), (5)

where

[f.sub.bm,1] := [partial derivative][f.sub.bm]/[partial derivative]x, [f.sub.bm,2] = [partial derivative][f.sub.bm]/[partial derivative]t. (6)

Following, we calculate the Riemann curvature tensor which is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (7)

The Ricci tensor follows as

[[sigma].sub.mq] := [[sigma].sup.a.sub.maq] = [[sigma].sup.a.sub.mqa], (8)

and is constructed by contraction. From [[sigma].sub.nq], we obtain [[sigma].sup.m.sub.q] via

[[sigma].sup.m.sub.q] = [f.sup.mn] [[sigma].sub.mq]. (9)

Finally, the curvature scalar a is given by

[sigma] := [[sigma].sup.m.sub.m]. (10)

If the given [sigma] is a constant, we will get a partial differential equation.

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 [29-31]. 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

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (11)

and the line element is

[(ds/d[lambda]).sup.2] = [(dx/d[lambda]).sup.2] + 2 cosh (u(x,t)) dx/d[lambda] dt/d[lambda] + [(dt/d[lambda]).sup.2], (12)

where u is a smooth function of x and t. Firstly, we will calculate the Riemann curvature scalar a from f. Then the sinh-Gordon equation follows when we impose the condition [sigma] = -2. We have

[f.sub.11] = [f.sub.22] = 1 [f.sub.21] = [f.sub.12] = cosh(u). (13)

The quantity f can be written in matrix form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (14)

and the inverse of f is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (15)

where

[f.sup.11] = [f.sup.22] = - 1/[sinh.sup.2](u), [f.sup.12] = [f.sup.21] = cosh(u)/[sinh.sup.2](u). (16)

Differentiating (14) with respect to x and t, we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (17)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (18)

Since

[[tau].sup.a.sub.mn] = 1/2 [f.sup.ab]([f.sub.bm,n] + [f.sub.bn,m] - [f.sub.mn,b]), (19)

we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (20)

Differentiating (20) with respect to x and t, we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (21)

By virtue of

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (22)

we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (23)

By virtue of

[[sigma].sub.mq] := [[sigma].sup.a.sub.maq] = -[sigma].sup.a.sub.mqa], (24)

we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (25)

By virtue of

[[sigma].sup.m.sub.q] = [f.sup.mn][[sigma].sub.nq] (26)

we get

[[sigma].sup.1.sub.1] = -[u.sub.xt]/sinh(u), [[sigma].sup.2.sub.2] = -[u.sub.xt]/sinh(u). (27)

Finally, with the help of

[sigma] := [[sigma].sup.m.sub.m], (28)

we get

[sigma] := -[2u.sub.xt]/sinh(u). (29)

When given [sigma] = -2, the well-known sinh-Gordon equation

[u.sub.xt] = sinh(u) (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

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (31)

and the line element is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (32)

Firstly, we calculate the Riemann curvature scalar [sigma] from f. Then the elliptic sinh-Gordon equation follows when we impose the condition [sigma] = -1. We have

[f.sub.11] = [f.sub.22] = cosh(u), [f.sub.21] = [f.sub.12] = 1. (33)

The quantity f can be written in matrix form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (34)

and the inverse of f is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (35)

where

[f.sub.11] = [f.sup.22] = cosh(u)/[sinh.sup.2](u), [f.sup.12] = [f.sup.21] = - 1/[sinh.sup.2](u). (36)

sinh2 (u) sinh2 (u)

Differentiate (34) with respect to x and t, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (37)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (38)

Since

[[tau].sup.a.sub.mn] = 1/2 [f.sup.ab] ([f.sub.bm,n] - [f.sub.mn,b]), (39)

we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (40)

Differentiating (40) with respect to x and t, we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (41)

By virtue of

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (42)

we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (43)

By virtue of

[[sigma].sub.mq] := [f.sup.mn] [q.sub.nq], (44)

we get

[[sigma].sup.1.sub.1] = -[u.sub.xx] + [u.sub.tt]/2 sinh(u), [[sigma].sup.2.sub.2] = -[u.sub.xx] + [u.sub.tt]/2 sinh(u). (47)

Finally, with the help of

[sigma] := [[sigma].sup.m.sub.m], (48)

we get

[sigma] := [u.sub.xx] + [u.sub.tt]/sinh(u). (49)

When given [sigma] = -1, the well-known elliptic sinh-Gordon equation

[u.sub.xx] + [u.sub.tt] = sinh (u) (50)

is obtained.

3. Solutions to the Sinh-Gordon Equation and Elliptic Equation

Backlund 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 Baacklund transformation, or of the same partial differential equation where such a relation is then known as an auto-Backlund transformation. As to elliptic sinh-Gordon equation

[u.sub.xx] + [u.sub.tt] = sinh (u) (51)

under the transformation

x [??] b/2 (x- it), t [??] 1/2b (x + it), (52)

where b is a positive constant, (51) is transformed into the sinh-Gordon equation

[u.sub.xt] = sinh(u). (53)

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-Backlund transformations for the sinh-Gordon equation

[u.sub.xt] = sinh(u) (54)

is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (55)

If u is a solution of the sinh-Gordon equation, u' is also a solution of the sinh-Gordon equation. Here we are looking for solutions of the sinh-Gordon equation by using the Backlund transformations. Obviously u(x, t) = 0 is a solution of the sinh-Gordon equation. This is known as the vacuum solution. We make use of the auto-Backlund transformation to construct another solution of the sinh-Gordon equation from the vacuum solution. Inserting this solution into the given Baacklund transformation results in

[(u'/2).sub.x] = a sinh (u'/2), [(u'/2).sub.t] = 1/a sinh (u'/2). (56)

Since

[integral] du/sinh(u/2) = 4[tanh.sup.-1] exp (u/2), (57)

we obtain a new solution of the sinh-Gordon equation; namely,

u' =2 ln tanh (-a/2 x - 1/2a t + C), (58)

where C is a constant of integration, and the computer simulation of (58) is presented in Figures 1 and 2.

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 [u.sub.0], first by employing [a.sub.1], [u.sub.1] is obtained; then by employing [a.sub.2], [u.sub.3] can be obtained:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (59)

Meanwhile by changing the using order of [a.sub.1] and [a.sub.2], [u.sub.2] and [u.sub.4] are also obtained, respectively. If [u.sub.3] = [u.sub.4], then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (60)

From (59) and (60), we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (61)

By simple calculation, (61) can be rewritten as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (62)

or

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (63)

After abbreviation, the following nonlinear superposition formula obtained

tanh [u.sub.3] - [u.sub.0]/4 = [a.sub.2] + [a.sub.1]/[a.sub.2] = [a.sub.1] tanh [u.sub.1] - [u.sub.2]/4, (64)

or

[u.sub.3] = [u.sub.0] + 4tanh -1 [a.sup.2] + [a.sub.1]/[a.sup.2 - [a.sub.1] tanh [u.sub.1] - [u.sub.2]/4. (65)

If we are given

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (66)

by means of (65), we can easily get the fourth solution

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (67)

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

u' = 2 ln tanh (-1/4 ((ab + 1/ab) x - i (ab - 1/ab)t) + C) (68)

and double breather solution

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (69)

of the elliptic sinh-Gordon equation. The computer simulation of the solutions is presented in Figures 5 and 6.

4. Summary and Discussion

In this paper, we obtain sinh-Gordon equation and elliptic sinh-Gordon equation by means of pseudospherical surfaces. In addition, we give the Baacklund 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 sinh-Gordon equation. On the basis of the Baacklund transformations and nonlinear superposition formulas, the singlesoliton (breather) solution and double-soliton (breather) solution of the sinh-Gordon equation and elliptic sinh-Gordon 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.

http://dx.doi.org/10.1155/2014/381717

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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Hongwei Yang, (1) Xiangrong Wang, (1) and Baoshu Yin (2,3)

(1) College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China

(2) Institute of Oceanology, Chinese Academy of Sciences, Qingdao 266071, China

J Key Laboratory of Ocean Circulation and Wave, Chinese Academy of Sciences, Qingdao 266071, China

Correspondence should be addressed to Baoshu Yin; baoshuyin@126.com

Received 2 January 2014; Revised 19 March 2014; Accepted 21 March 2014; Published 24 April 2014

Academic Editor: Weiguo Rui
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Title Annotation:Research Article
Author:Yang, Hongwei; Wang, Xiangrong; Yin, Baoshu
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Date:Jan 1, 2014
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