Abstract and Applied Analysis

Volume 2014 (2014), Article ID 381717, 9 pages

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

## On Differential Equations Derived from the Pseudospherical Surfaces

^{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^{3}Key Laboratory of Ocean Circulation and Wave, Chinese Academy of Sciences, Qingdao 266071, China

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

Academic Editor: Weiguo Rui

Copyright © 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.

#### Abstract

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 *σ* = −2, *σ* = −1, respectively. By employing the Bäcklund 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 [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 Bäcklund transformations [16, 17]. Fourthly, they pass the Painlevé 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 Bäcklund transformation. On the basis of the Bäcklund 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 and the line element is The quantity can be written in matrix form and then the inverse of is given by Next we have to calculate the Christoffel symbols. They are defined as where Following, we calculate the Riemann curvature tensor which is given by The Ricci tensor follows as and is constructed by contraction. From , we obtain via Finally, the curvature scalar is given by If the given 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 and the line element is where is a smooth function of and . Firstly, we will calculate the Riemann curvature scalar from . Then the sinh-Gordon equation follows when we impose the condition . We have The quantity can be written in matrix form and the inverse of is given by where Differentiating (14) with respect to and , we obtain where Since we obtain Differentiating (20) with respect to and , we obtain By virtue of we get By virtue of we get By virtue of we get Finally, with the help of we get When given , the well-known sinh-Gordon equation is obtained.

##### 2.2. Elliptic Sinh-Gordon Equation Derived from Pseudospherical Surfaces

In this section, we will derive elliptic sinh-Gordon equation from pseudospherical surfaces. The metric tensor field for the elliptic sinh-Gordon equation is given by and the line element is Firstly, we calculate the Riemann curvature scalar from . Then the elliptic sinh-Gordon equation follows when we impose the condition . We have The quantity can be written in matrix form and the inverse of is given by where Differentiate (34) with respect to and , we have where Since we obtain Differentiating (40) with respect to and , we obtainBy virtue of we get By virtue of we get By virtue of we get Finally, with the help of we get When given , the well-known elliptic sinh-Gordon equation is obtained.

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

Bäcklund 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 Bäcklund transformation, or of the same partial differential equation where such a relation is then known as an auto-Bäcklund transformation.

As to elliptic sinh-Gordon equation under the transformation where is a positive constant, (51) is transformed into the sinh-Gordon equation 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-Bäcklund transformations for the sinh-Gordon equation is given by If is a solution of the sinh-Gordon equation, is also a solution of the sinh-Gordon equation. Here we are looking for solutions of the sinh-Gordon equation by using the Bäcklund transformations. Obviously is a solution of the sinh-Gordon equation. This is known as the vacuum solution. We make use of the auto-Bäcklund transformation to construct another solution of the sinh-Gordon equation from the vacuum solution. Inserting this solution into the given Bäcklund transformation results in Since we obtain a new solution of the sinh-Gordon equation; namely, where 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 , first by employing , is obtained; then by employing , can be obtained: Meanwhile by changing the using order of and , and are also obtained, respectively. If , then From (59) and (60), we get By simple calculation, (61) can be rewritten as or After abbreviation, the following nonlinear superposition formula obtained or If we are given by means of (65), we can easily get the fourth solution 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 and double breather solution 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 Bäcklund 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 Bäcklund transformations and nonlinear superposition formulas, the single-soliton (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.

#### 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).

#### References

- E. Infeld and G. Rowlands,
*Nonlinear Waves, Solitons and Chaos*, Cambridge University Press, Cambridge, UK, 2nd edition, 2000. View at Publisher · View at Google Scholar · View at MathSciNet - A. Dold and B. Eckmann,
*Bäcklund Transformations*, Lecture Note in Mathematics, Springer, New York, NY, USA, 1974. - Y. F. Zhang and H. W. Tam, “Discussion on integrable properties for higher-dimensional variable-coefficient nonlinear partial differential equations,”
*Journal of Mathematical Physics*, vol. 54, Article ID 013516, 2013. View at Google Scholar · View at Zentralblatt MATH - T. C. Xia, H. Q. Zhang, and Z. Y. Yan, “New explicit and exact travelling wave solutions for a class of nonlinear evolution equations,”
*Applied Mathematics and Mechanics*, vol. 22, no. 7, pp. 788–793, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. C. Scott, “Propagation of magnetic flux on a long Josephson tunnel junction,”
*Il Nuovo Cimento B Series 10*, vol. 69, no. 2, pp. 241–261, 1970. View at Publisher · View at Google Scholar · View at Scopus - H.-W. Yang, B.-S. Yin, D.-Z. Yang, and Z.-H. Xu, “Forced solitary Rossby waves under the influence of slowly varying topography with time,”
*Chinese Physics B*, vol. 20, no. 12, Article ID 120203, 2011. View at Publisher · View at Google Scholar · View at Scopus - H. W. Yang, B. S. Yin, and Y. L. Shi, “Forced dissipative Boussinesq equation for solitary waves excited by unstable topography,”
*Nonlinear Dynamics*, vol. 70, no. 2, pp. 1389–1396, 2012. View at Publisher · View at Google Scholar · View at MathSciNet - Z. H. Xu, B. S. Yin, Y. J. Hou et al., “Variability of internal tides and near-inertial waves on the continental slope of the northwestern South China Sea,”
*Journal of Geophysical Research*, vol. 118, pp. 197–211, 2013. View at Google Scholar - Z. Xu, B. Yin, Y. Hou, Z. Fan, and A. K. Liu, “A study of internal solitary waves observed on the continental shelf in the northwestern South China Sea,”
*Acta Oceanologica Sinica*, vol. 29, no. 3, pp. 18–25, 2010. View at Publisher · View at Google Scholar · View at Scopus - P. G. Drazin,
*Solitons*, vol. 85 of*London Mathematical Society Lecture Note Series*, Cambridge University Press, London, UK, 1983. View at Publisher · View at Google Scholar · View at MathSciNet - G. L. Lamb, Jr., “Analytical descriptions of ultrashort optical pulse propagation in a resonant medium,”
*Reviews of Modern Physics*, vol. 43, pp. 99–124, 1971. View at Publisher · View at Google Scholar · View at MathSciNet - G. Feverati, F. Ravanini, and G. Takács, “Non-linear integral equation and finite volume spectrum of sine-Gordon theory,”
*Nuclear Physics B*, vol. 540, no. 3, pp. 543–586, 1999. View at Publisher · View at Google Scholar · View at MathSciNet - M. J. Ablowitz and P. A. Clarkson,
*Solitons, Nonlinear Evolution Equations and Inverse Scattering*, vol. 149 of*London Mathematical Society Lecture Note Series*, Cambridge University Press, New York, NY, USA, 1991. View at Publisher · View at Google Scholar · View at MathSciNet - A. H. Khater, D. K. Callebaut, and S. M. Sayed, “Conservation laws for some nonlinear evolution equations which describe pseudo-spherical surfaces,”
*Journal of Geometry and Physics*, vol. 51, no. 3, pp. 332–352, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - W.-X. Ma, “Variational identities and applications to Hamiltonian structures of soliton equations,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 71, no. 12, pp. e1716–e1726, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Y. Zhang, Z. Han, and H.-W. Tam, “An integrable hierarchy and Darboux transformations, bilinear Bäcklund transformations of a reduced equation,”
*Applied Mathematics and Computation*, vol. 219, no. 11, pp. 5837–5848, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - T. Xia, X. Chen, and D. Chen, “Darboux transformation and soliton-like solutions of nonlinear Schrödinger equations,”
*Chaos, Solitons & Fractals*, vol. 26, no. 3, pp. 889–896, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - W. X. Ma, “Exact solutions to Tu system through Painlevé analysis,”
*Journal of Fudan University. Natural Science*, vol. 33, no. 3, pp. 319–326, 1994. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - E. G. Reyes, “Pseudo-spherical surfaces and integrability of evolution equations,”
*Journal of Differential Equations*, vol. 147, no. 1, pp. 195–230, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - G. M. Gharib, “Surfaces of a constant negative curvature,”
*International Journal of Differential Equations*, vol. 2012, Article ID 720687, 17 pages, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. J. Ablowitz and P. A. Clarkson,
*Solitons, Nonlinear Evolution Equations and Inverse Scattering*, vol. 149, Cambridge University Press, Cambridge, UK, 1991. View at Publisher · View at Google Scholar · View at MathSciNet - H. Eleuch and Y. V. Rostovtsev, “Analytical solution to sine-Gordon equation,”
*Journal of Mathematical Physics*, vol. 51, Article ID 093515, 2010. View at Publisher · View at Google Scholar · View at MathSciNet - Y. Matsuno, “A direct method for solving the generalized sine-Gordon equation,”
*Journal of Physics A: Mathematical and Theoretical*, vol. 43, Article ID 105204, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - T. Aktosun, F. Demontis, and C. van der Mee, “Exact solutions to the sine-Gordon equation,”
*Journal of Mathematical Physics*, vol. 51, Article ID 123521, 2010. View at Publisher · View at Google Scholar · View at MathSciNet - G. Hariharan, “Haar wavelet method for solving the Klein-Gordon and the Sine-Gordon equations,”
*International Journal of Nonlinear Science*, vol. 11, no. 2, pp. 180–189, 2011. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - N. Kamran and K. Tenenblat, “On differential equations describing pseudo-spherical surfaces,”
*Journal of Differential Equations*, vol. 115, no. 1, pp. 75–98, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. J. Klein, “Geometrical interpretation of the solutions of the sine-Gordon equation,”
*Journal of Mathematical Physics*, vol. 26, no. 9, pp. 2181–2185, 1985. View at Publisher · View at Google Scholar · View at MathSciNet - W.-H. Steeb,
*Continuous Symmetries, Lie Algebras, Differential Equations and Computer Algebra*, World Scientific, River Edge, NJ, USA, 1996. View at MathSciNet - J. K. Perring and T. H. R. Skyrme, “A model unified field equation,”
*Nuclear Physics B*, vol. 31, pp. 550–555, 1962. View at Publisher · View at Google Scholar · View at MathSciNet - A. M. Wazwaz, “Exact solutions to the double sinh-Gordon equation by the tanh method and a variable separated ODE method,”
*Computers & Mathematics with Applications*, vol. 50, no. 10-12, pp. 1685–1696, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Tang and W. Huang, “Bifurcations of travelling wave solutions for the generalized double sinh-Gordon equation,”
*Applied Mathematics and Computation*, vol. 189, no. 2, pp. 1774–1781, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - R. K. Dodd and R. K. Bullough, “Bäcklund transformations for the sine-Gordon equations,”
*Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences*, vol. 351, no. 1667, pp. 499–523, 1976. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. Rogers and W. F. Shadwick,
*Bäcklund Transformations and Their Applications*, vol. 161, Academic Press, New York, NY, USA, 1982. View at MathSciNet