Surfaces of a Constant Negative Curvature

Gharib, G. M.

doi:https://doi.org/10.1155/2012/720687

International Journal of Differential Equations

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Research Article | Open Access

Volume 2012 | Article ID 720687 | https://doi.org/10.1155/2012/720687

Surfaces of a Constant Negative Curvature

G. M. Gharib^1,2

Academic Editor: Wenming Zou

Received10 Mar 2012

Revised30 May 2012

Accepted13 Aug 2012

Published16 Oct 2012

Abstract

I study the geometric notion of a differential system describing surfaces of a constant negative curvature and describe a family of pseudospherical surfaces for the nonlinear partial differential equations with constant Gaussian curvature .

1. Introduction

In recent decades, a class of transformations having their origin in the work by Bäcklund in the late nineteenth century has provided a basis for remarkable advances in the study of nonlinear partial differential equations (NLPDEs) [1]. The importance of Bäcklund transformations (BTs) and their generalizations is basically twofold. Thus, on one hand, invariance under a BT may be used to generate an infinite sequence of solutions for certain NLPDEs by purely algebraic superposition principles. On the other hand, BTs may also be used to link certain NLPDEs (particularly nonlinear evolution equations (NLEEs) modelling nonlinear waves) to canonical forms whose properties are well known [2, 3]. Nonlinear wave phenomena have attracted the attention of physicists for a long time. Investigation of a certain kind of NLPDEs has made great progress in the last decades. These equations have a wide range of physical applications and share several remarkable properties [4–6]: (i) the initial value problem can be solved exactly in terms of linear procedures, the so-called “inverse scattering method (ISM);” (ii) they have an infinite number of “conservation laws;” (iii) they have “BTs;” (iv) they describe pseudo-spherical surfaces (pss), and hence one may interpret the other properties (i)–(iii) from a geometrical point of view; (v) they are completely integrable [1, 3]. This geometrical interpretation is a natural generalization of a classical example given by Chern and Tenenblat [2] who introduced the notion of a differential equation (DE) for a function that describes a pss, and they obtained a classification for such equations of type (). These results provide a systematic procedure to obtain a linear eigenvalue problem associated to any NLPDE of this type [7].

Sasaki [6] gave a geometrical interpretation for inverse scattering problem (ISP), considered by Ablowitz et al. [4], in terms of pss. Based on this interpretation, one may consider the following definition.

Let be a two-dimensional differentiable manifold with coordinates (). A DE for a real function describes a pss if it is a necessary and sufficient condition for the existence of differentiable functions: depending on and its derivatives such that the one-forms satisfy the structure equations of a pss, that is,

This structure was considered for the first time by Chern and Tenenblat [2], motivated by Sasaki’s observation [6] that the equations which are the necessary and sufficient condition for the integrability of a linear problem of Ablowitz, Kaup, Newell, Segur-(AKNS-) type [4, 7–12] do describe pss. Its importance, in the present context, arises from the fact that the connection between pss and integrability of DEs goes well beyond the AKNS framework, as will be explained in Section 2. A DE for a real-valued function is kinematically integrable if it is the integrability condition of a one-parameter family of linear problems [13–20]: in which and are -valued functions of , and ( and its derivatives) up to a finite order. Thus, an equation is kinematically integrable if it is equivalent to the zero curvature condition: where , for each (spectral parameter or eigenvalue). In addition, a DE will be said to strictly kinematically integrable if it’s kinematically integrable and diagonal entries of the matrix introduced above are and .

The main aim of this paper is to use the geometric properties and differentiable functions in the construction of BTs for some NLEEs which describe pss.

The paper is organized as follows. In Section 2 we summarize the AKNS formulation of the ISM using the language of exterior differential forms; this language is very useful for geometry. The correspondence between NLEEs and their families of pss is established in Section 3. In Section 4 we find the BTs for some NLEEs (Liouville, Burgers, and sinh-Gordon equations, a third-order evolution equation (TOEE), a modified Korteweg-de Vries (mKdV) equation, and both families of equations I and II) which describe pss. Finally, we give some conclusions in Section 5.

2. The AKNS System for Some NLEEs

The ISM was first devised for the Korteweg-de Vries (KdV) equation [5]. Later, it was extended by Zakharov and Shabat [21] to a scattering problem for the nonlinear Schrödinger equation (NLSE) and subsequently generalized by Ablowitz et al. [4] to include a variety of NLEEs. The AKNS method consists of the following steps: (i) set up an appropriate, linear scattering (eigenvalue) problem in the “space” variable in which the solution of the NLEE plays the role of the potential; (ii) choose the “time” dependence of the eigenfunctions in such a way that the eigenvalues remain invariant as the potential evolves according to the NLEEs; (iii) solve the direct scattering problem at the initial “time” and determine the “time” dependence of the scattering data; (iv) do the ISP at later “times,” namely, reconstruct the potential from the scattering data. In this section, we concentrate on the first step of the AKNS method. As a consequence, each solution of the DE provides a metric on , whose Gaussian curvature is constant, equal to . Moreover, the above definition of a DE is equivalent to saying that the DE for is the integrability condition for the problem: where denotes exterior differentiation, is a vector, and the matrix is traceless: and consists of a one-parameter , family of one-forms in the independent variables , the dependent variable and its derivatives. Equation (1.2) has three one-forms , , and consisting of independent and dependent variables and their derivatives, such that the NLPDE is given by which is, by construction, the original NLPDE to be solved. We illustrate here the following examples given by AKNS system.(a) Liouville’s equation: (b) Burgers’ equation: (c) sinh-Gordon equation: (d) A TOEE [22]: where , .(e) A mKdV equation: where is a constant, where , .(f) A family of equations I [23]: where is a differentiable function of which satisfies , with , and are real constants, such that , (g) A family of equations II [23] similar to the family I, but with some signs changed: where is a differentiable function of which satisfies , with , and are real constants, such that , keeping in mind that the parameter plays the role of the eigenvalue for the scattering problem in (2.1). Note that the one-form, , is not unique for a given NLPDE, for the scattering equations (2.1), (2.2), and (2.3) are form invariant under the “gauge” transformation: where is an arbitrary matrix with determinant unity, Integrability of (2.1) is, requires the vanishing of the two form It should be noted that the solution of these equations is of a very special kind. In general, (2.21) gives three different equations, which cannot be satisfied simultaneously by one-dependent variable . It has been pointed out [17, 24] that can be interpreted as a connection one form for the principle bundle on and as its curvature two form. The geometrical explanation of the is given in [6, 16].

3. The NLEEs Which Describe pss

Whenever the functions are real, Sasaki [6] gave a geometrical interpretation for the problem. Consider the one-forms defined by where .

Let be a two-dimensional differentiable manifold parametrized by coordinates , . We consider a metric on defined by . The first two equations in (1.3) are the structure equations which determine the connection form , and the last equation in (1.3), the Gauss equation, determines that the Gaussian curvature of is −1, that is, is a pss. Moreover, an EE must be satisfied for the existence of forms (3.1) satisfying (1.3). This justifies the definition of a DE which describes a pss that we considered in the introduction.

We will restrict ourselves to the case where . More precisely, we say that a DE for describes a pss if it is a necessary and sufficient condition for the existence of functions , , , depending on and its derivatives, , such that the one-forms in (1.2), satisfy the structure equations (1.3) of a pss. It follows from this definition that for each nontrivial solution of the DE, one gets a metric defined on , whose Gaussian curvature is .

It has been known, for along time, that the sinh-Gordon (SG) equation describes a pss. In this paper, we extend the same analysis to include the Liouville, Burgers, sinh-Gordon equations, a TOEE, a mKdV equation, and both families of equations I and II.

Example 3.1. Let be a differentiable surface, parametrized by coordinates , . (a) Liouville’s equation. Consider Then is a pss if and only if satisfies Liouville’s equation (2.4). (b) Burgers’ equation. Consider Then is a pss if and only if satisfies the Burgers’ equation (2.6). (c) sinh-Gordon equation. Consider Then is a pss if and only if satisfies the sinh-Gordon equation (2.8). (d) A TOEE. Consider Then is a pss iff satisfies a TOEE (2.10). (e) A mKdV equation. Consider Then is a pss iff satisfies a mKdV equation (2.12).(f) A family of equations I. Consider Then is a pss if and only if satisfies the family of equations I (2.14). (g) A family of equations II. Consider Then is a pss if and only if satisfies the family of equations II (2.16).

4. A Geometric Method Which Provides BTs

In this section, we show how the geometric properties of a pss may be applied to obtain analytic results for some NLEEs which describe pss.

The classical Bäcklund theorem originated in the study of pss, relating solutions of the SG equation. Other transformations have been found relating solutions of specific equations in [15, 17, 24, 25]. Such transformations are called BTs after the classical one. A BT which relates solutions of the same equation is called a self-Bäcklund transformation (sBT). An interesting fact which has been observed is that DEs which have sBT also admit a superposition formula. The importance of such formulas is due to the following: if is a solution of the NLEE and are solutions of the same equation obtained by the sBT, then the superposition formula provides a new solution algebraically. By this procedure one obtains the soliton solutions of a NLEE. In what follows we show that geometrical properties of pss provide a systematic method to obtain the BTs for some NLEEs which describe pss.

Proposition 4.1. Given a coframe and corresponding connection one-form on a smooth Riemannian surfaces , there exists a new coframe and new connection one-form satisfying the following: if and only if the surface is pss. For the sake of clarity, we give a revised proof of [26].

Proof. Assume that the orthonormal dual to the coframes and possesses the same orientation. The one-forms and are connected by means of [27–31]: It follows that satisfying (4.1) exist if and only if the Pfaffian system, on the space of coordinates is completely integrable for , and this happens if and only if is pss.
Geometrically, (4.1) and (4.3) determine geodesic coordinates on . Now, if describes pss with associated one-forms , (4.1) and (4.3) imply that the Pfaffian system, is completely integrable for whenever is a local solution of [2, 32].

Proposition 4.2. Let be a NLEE which describe a pss with associated one-forms (1.2). Then, for each solution of , the system of equations for , is completely integrable. Moreover, for each solution of of and corresponding solution , is a closed one-form [2].

Eliminating from (4.5), by using the substitution where then (4.5) is reduced to the Riccati equations: The procedure in the following is that one constructs a transformation satisfying the same equation as (4.10) with a potential where Thus, eliminating in (4.9), (4.10) and (4.11), we have a BT to a desired NLEE. We consider the following examples [33].(a) BT for Liouville’s equation. For (2.4) we consider the functions defined by for any solution of (2.4), the above functions satisfy (2.21). Then (4.9) becomes If we choose and as then and satisfy (4.13). If we eliminate in (4.13) and (4.10) with (4.14), we get the BT: Equation (4.15) is the BT for Liouville’s equation (2.4) with , , and given in (4.12). (b) BT for Burgers’ equation. For any solution of the Burgers’ equation (2.6), the functions The above functions satisfy (2.21). Then (4.9) becomes [27] If we choose and as then and satisfy (4.17). If we eliminate in (4.17) and (4.10) with (4.18), we get the BT: where we put and . Equation (4.19) is the BT for the Burgers’ equation (2.6) with , and given in (4.16).(c) BT for sinh-Gordon equation. For (2.8) we consider the following functions of defined by [28] for any solution of (2.8), the above functions satisfy (2.21). Then (4.9) becomes [29] If we choose and as then and satisfy (4.21). If we eliminate in (4.21) and (4.10) with (4.22), we get the BT: Equation (4.23) is the BT for the sinh-Gordon equation (2.8) with , , and given in (4.20). (d) BT for a TOEE. For (2.10) we consider the functions defined by The above functions satisfy (2.21). Then (4.9) becomes [30] If we choose and as then and satisfy (4.25). If we eliminate in (4.25) and (4.10) with (4.26), we get the BT: Equation (4.27) is the BT for a TOEE (2.10) with , , and given in (4.24).(e) BT for a mKdV equation. For (2.12) we consider the following functions of defined by for any solution of (2.12), the above functions satisfy (2.21). Then (4.9) becomes [31] If we choose and as then and satisfy (4.29). If we eliminate in (4.29) and (4.10) with (4.30), we get the BT: where we put and . Equation (4.31) is the BT for an mKdV equation (2.12) with , , and given in (4.28).(f) BT for the family of equations I. For any solution of the family of equations I (2.14), the functions The above functions satisfy (2.21). Then (4.9) becomes [34] If we choose and as then and satisfy (4.33). If we eliminate in (4.33) and (4.10) with (4.34), we get the BT: Equation (4.35) is the BT for the family of equations I (2.14) with , , and given in (4.32).(g) BT for the family of equations II. For (2.16) we consider the functions of defined by for any solution of (2.16), the above functions satisfy (2.21). Then (4.9) becomes If we choose and as then and satisfy (4.37). If we eliminate in (4.37) and (4.10) with (4.38), we get the BT:

Equation (4.39) is the BT for the family of equations II (2.16) with , , and given in (4.36).

We have previously discussed the relationships among the geometrical properties and the BT. There, we restrict our discussion to the NLEEs which can be reduced to the Liouville’s form of the geometrical properties such as the Burgers, the sinh-Gordon equations, a TOEE, a mKdV, and the two family of equations I and II.

5. Conclusions

We may hope to find some relationships among various soliton equations which describe pss. The latter yields directly the curvature condition (Gaussian curvature equal to −1, corresponding to pseudo-spherical surfaces). This geometrical method is considered for several NLPDEs which describe pss: Liouville, Burgers, sinh-Gordon equations, a TOEE, a mKdV, and the two families of equations I and II. We show how the geometric properties of a pss may be applied to obtain analytic results for some NLEEs which describe pss. This geometrical method allows some further generalization of the work on Bäcklund transformations given by Wadati et al. [3]. The Bäcklund transformations for all seven NLPDEs mentioned above are derived in this way.

References

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 Site | Zentralblatt MATH
S. S. Chern and K. Tenenblat, “Pseudospherical surfaces and evolution equations,” Studies in Applied Mathematics, vol. 74, no. 1, pp. 55–83, 1986.
View at: Google Scholar | Zentralblatt MATH
M. Wadati, H. Sanuki, and K. Konno, “Relationships among inverse method, Bäcklund transformation and an infinite number of conservation laws,” Progress of Theoretical Physics, vol. 53, pp. 419–436, 1975.
View at: Publisher Site | Google Scholar | Zentralblatt MATH
M. J. Ablowitz, D. J. Kaup, A. C. Newell, and H. Segur, “The inverse scattering transform-Fourier analysis for nonlinear problems,” Studies in Applied Mathematics, vol. 53, no. 4, pp. 249–315, 1974.
View at: Google Scholar | Zentralblatt MATH
C. S. Gardner, J. M. Greene, M. D. Kruskal, and R. M. Miura, “Method for solving the korteweg-de Vries equation,” Physics Letters, vol. 19, pp. 1095–1097, 1967.
View at: Google Scholar
R. Sasaki, “Soliton equations and pseudospherical surfaces,” Nuclear Physics B, vol. 154, no. 2, pp. 343–357, 1979.
View at: Publisher Site | Google Scholar
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 Site | Google Scholar | Zentralblatt MATH
A. H. Khater, A. A. Abdalla, A. M. Shehatah, D. K. Callebaut, and S. M. Sayed, “Bäcklund transformations and exact solutions for self-dual $SU (3)$ Yang-Mills fields,” Il Nuovo Cimento della Società Italiana di Fisica B, vol. 114, no. 1, pp. 1–10, 1999.
View at: Google Scholar
A. H. Khater, D. K. Callebaut, and O. H. El-Kalaawy, “Bäcklund transformations and exact soliton solutions for nonlinear Schrödinger-type equations,” Il Nuovo Cimento della Società Italiana di Fisica B, vol. 113, no. 9, pp. 1121–1136, 1998.
View at: Google Scholar
A. H. Khater, D. K. Callebaut, and R. S. Ibrahim, “Bäcklund transformations and Painlevé analysis: exact soliton solutions for the unstable nonlinear Schrödinger equation modeling electron beam plasma,” Physics of Plasmas, vol. 5, no. 2, pp. 395–400, 1998.
View at: Publisher Site | Google Scholar
A. H. Khater, D. K. Callebaut, and A. R. Seadawy, “Nonlinear dispersive instabilities in Kelvin-Helmholtz magnetohydrodynamic flow,” Physica Scripta, vol. 67, pp. 340–349, 2003.
View at: Google Scholar
A. H. Khater, O. H. El-Kalaawy, and D. K. Callebaut, “Bäcklund transformations for Alfvén solitons in a relativistic electron-positron plasma,” Physica Scripta, vol. 58, pp. 545–548, 1998.
View at: Google Scholar
K. Chadan and P. C. Sabatier, Inverse Problems in Quantum Scattering Theory, Springer, New York, NY, USA, 1977.
A. H. Khater, W. Malfliet, D. K. Callebaut, and E. S. Kamel, “Travelling wave solutions of some classes of nonlinear evolution equations in $(1 + 1)$ and $(2 + 1)$ dimensions,” Journal of Computational and Applied Mathematics, vol. 140, no. 1-2, pp. 469–477, 2002.
View at: Publisher Site | Google Scholar | Zentralblatt MATH
K. Konno and M. Wadati, “Simple derivation of Bäcklund transformation from Riccati form of inverse method,” Progress of Theoretical Physics, vol. 53, no. 6, pp. 1652–1656, 1975.
View at: Publisher Site | Google Scholar | Zentralblatt MATH
M. Marvan, “Scalar second-order evolution equations possessing an irreducible ${sl}_{2}$ -valued zero-curvature representation,” Journal of Physics A, vol. 35, no. 44, pp. 9431–9439, 2002.
View at: Publisher Site | Google Scholar
R. M. Miura, Bäcklund Transformations, the Inverse Scattering Method, Solitons, and Their Applications, vol. 515 of Lecture Notes in Mathematics, Springer, New York, NY, USA, 1976.
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 Site | Google Scholar | Zentralblatt MATH
E. G. Reyes, “Conservation laws and Calapso-Guichard deformations of equations describing pseudo-spherical surfaces,” Journal of Mathematical Physics, vol. 41, no. 5, pp. 2968–2989, 2000.
View at: Publisher Site | Google Scholar | Zentralblatt MATH
A. V. Shchepetilov, “The geometric sense of the Sasaki connection,” Journal of Physics A, vol. 36, no. 13, pp. 3893–3898, 2003.
View at: Publisher Site | Google Scholar | Zentralblatt MATH
V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self–modulation of waves in nonlinear Media,” Soviet Physics, vol. 34, pp. 62–69, 1972.
View at: Google Scholar
J. A. Cavalcante and K. Tenenblat, “Conservation laws for nonlinear evolution equations,” Journal of Mathematical Physics, vol. 29, no. 4, pp. 1044–1049, 1988.
View at: Publisher Site | Google Scholar
R. Beals, M. Rabelo, and K. Tenenblat, “Bäcklund transformations and inverse scattering solutions for some pseudospherical surface equations,” Studies in Applied Mathematics, vol. 81, no. 2, pp. 125–151, 1989.
View at: Google Scholar | Zentralblatt MATH
M. Crampin, “Solitons and SL(2,R),” Physics Letters A, vol. 66, no. 3, pp. 170–172, 1978.
View at: Publisher Site | Google Scholar
A. C. Scott, F. Y. F. Chu, and D. W. McLaughlin, “The soliton: a new concept in applied science,” Proceedings of the IEEE, vol. 61, pp. 1443–1483, 1973.
View at: Google Scholar
E. G. Reyes, “On Geometrically integrable equations and Hierarchies of pseudospherical type,” Contemporary Mathematics, vol. 285, pp. 145–155, 2001.
View at: Google Scholar
M. L. Rabelo and K. Tenenblat, “A classification of pseudospherical surface equations of type $u_{t} = u_{x x x} + G (u, u_{x}, u_{x x})$ ,” Journal of Mathematical Physics, vol. 33, no. 2, pp. 537–549, 1992.
View at: Publisher Site | Google Scholar | Zentralblatt MATH
A. Sakovich and S. Sakovich, “On transformations of the Rabelo equations,” SIGMA. Symmetry, Integrability and Geometry, vol. 3, pp. 1–8, 2007.
View at: Publisher Site | Google Scholar | Zentralblatt MATH
M. J. Ablowitz, R. Beals, and K. Tenenblat, “On the solution of the generalized wave and generalized sine-Gordon equations,” Studies in Applied Mathematics, vol. 74, no. 3, pp. 177–203, 1986.
View at: Google Scholar | Zentralblatt MATH
S. M. Sayed, O. O. Elhamahmy, and G. M. Gharib, “Travelling wave solutions for the KdV-Burgers-Kuramoto and nonlinear Schrödinger equations which describe pseudospherical surfaces,” Journal of Applied Mathematics, vol. 2008, Article ID 576783, 10 pages, 2008.
View at: Publisher Site | Google Scholar | Zentralblatt MATH
S. M. Sayed, A. M. Elkholy, and G. M. Gharib, “Exact solutions and conservation laws for Ibragimov-Shabat equation which describe pseudo-spherical surface,” Computational & Applied Mathematics, vol. 27, no. 3, pp. 305–318, 2008.
View at: Publisher Site | Google Scholar | Zentralblatt MATH
K. Tenenblat, Transformations of Manifolds and Applications to Differential Equations, vol. 93 of Pitman Monographs and Surveys in Pure and Applied Mathematics, Addison Wesley Longman, Harlow, UK, 1998.
M. C. Nucci, “Pseudopotentials, Lax equations and Bäcklund transformations for nonlinear evolution equations,” Journal of Physics A, vol. 21, no. 1, pp. 73–79, 1988.
View at: Publisher Site | Google Scholar | Zentralblatt MATH
S. M. Sayed and G. M. Gharib, “Canonical reduction of self-dual Yang-Mills equations to Fitzhugh-Nagumo equation and exact solutions,” Chaos, Solitons and Fractals, vol. 39, no. 2, pp. 492–498, 2009.
View at: Publisher Site | Google Scholar | Zentralblatt MATH

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Copyright © 2012 G. M. Gharib. 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.

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