On generalized Bäcklund transformations for equations describing pseudo-spherical surfaces

Reyes, Enríque G.

doi:10.1016/s0393-0440(01)00077-8

Cited by 13 publications

(49 citation statements)

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“…These observations were then developed in several papers by Prof. Tenenblat and her coworkers, see [4,10,51] and the monograph [78], and by the present author in the later papers [31,55,56,60,61]. Also, the authors of [12], and then several other researchers, see [20,37,38,51,54,57], obtained characterization results for equations of pseudo-spherical type, essentially "unwrapping" the zero curvature condition arising from the aforementioned fact that equations in this class admit associated sl(2, R)-valued linear problems. These characterization theorems are quite technical and not always of straightforward applicability, but they have proven useful in obtaining actual explicit classifications of equations describing pseudo-spherical surfaces, see [30,52,53], and as a tool for the comparison of different types of integrability for evolution equations [25,54].…”

Section: Introductionmentioning

confidence: 77%

“…As a rule, these theorems are quite technical (see [12] and the later papers [20,37,38,51,54,57]) but, as stated in Sect. 1, they have proven to be very useful in classifying PSS equations [20,30,52,53] and in comparing different types of integrability [25,54].…”

Section: Characterization Resultsmentioning

confidence: 99%

“…Vol. 60 (2011) Equations of Pseudo-Spherical Type 61 The characterization theorems considered here are valid only for evolution equations of strictly pseudo-spherical type, see [12,38,54,57]. They are consequences of the fact that Definition 2.10 implies that the dependence of its associated functions f αβ on derivatives of u is very restricted, and that these functions satisfy several identities which allow one to "reconstruct" the right hand side of the equation u t = F : Lemma 2.11.…”

Section: Characterization Resultsmentioning

confidence: 99%

“…It has been also realized that, independently of their integrability properties, new analytic results about equations of pseudo-spherical type can be found motivated by geometric considerations: inspired by the fact that two surfaces of constant Gaussian curvature equal to −1 are locally indistinguishable, Kamran and Tenenblat [38]-and then the present author [57]-have showed that if Ξ = 0 and Ξ = 0 describe pseudo-spherical surfaces, there exists a smooth mapping transforming a (generic) local solution u(x, t) of Ξ = 0 into a (generic) local solution u( x, t ) of Ξ = 0. In some instances one can find an explicit formula for u( x, t ) in terms of u(x, t), see [38,57,58].…”

Section: Introductionmentioning

confidence: 97%

“…In some instances one can find an explicit formula for u( x, t ) in terms of u(x, t), see [38,57,58]. Also, motivated by the observation that evolution equations u t = F which are the integrability condition of a non-trivial one-parameter family of linear problems v x = X v, v t = T v (in which X and T are, say, sl(2, R)-valued functions of u…”

Section: Introductionmentioning

confidence: 99%

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Equations of Pseudo-Spherical Type (After S.S. Chern and K. Tenenblat)

Reyes

2011

Results. Math.

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This paper surveys some recent developments around the notion of a scalar partial differential equation describing pseudo-spherical surfaces due to Chern and Tenenblat. It is shown how conservation laws, pseudo-potentials, and linear problems arise naturally from geometric considerations, and it is also explained how Darboux and Bäcklund transformations can be constructed starting from geometric data. Classification results for equations in this class are stated, and hierarchies of equations of pseudo-spherical type are introduced, providing a connection between differential geometry and the study of hierarchies of equations which are the integrability condition of sl(2, R)-valued linear problems. Furthermore, the existence of correspondences between any two solutions to equations of pseudo-spherical type is reviewed, and a correspondence theorem for hierarchies is also mentioned. As applications, an elementary immersion result for pseudo-spherical metrics arising from the Chern-Tenenblat construction is proven, and non-local symmetries of the Kaup-Kupershmidt, Sawada-Kotera, fifth order Korteweg-de Vries and Camassa-Holm (CH) equation with non-zero critical wave speed are considered. It is shown that the existence of a non-local symmetry of a particular type is enough to single the first three equations out from a whole family of equations describing pseudo-spherical surfaces while, in the CH case, it is shown that it admits an infinite-dimensional Lie algebra of non-local symmetries which includes the Virasoro algebra.Mathematics Subject Classification (2010). Primary 53B20; Secondary 37K10, 76M60. Equations of Pseudo-Spherical Type 55and a finite number of its derivatives with respect to x) are usually members of infinite hierarchies of evolution equations u τn = F n such that (see for instance [19]): (a) the flows generated by the equations u τn = F n commute, and (b) each equation u τn = F n is the integrability condition of a linear problem of the form v x = X v, v τn = T n v, hierarchies of evolution equations describing pseudo-spherical surfaces were defined in [59]; it was proven that they do possess characteristics (a) and (b), and moreover, that there exist correspondences between (suitably generic) solutions of any two hierarchies of equations of pseudo-spherical type. Thus, the Chern-Tenenblat structure is of interest for integrability, analysis and certainly geometry, since it is quite natural to ask which equations (besides the classical sine-Gordon equation [22]) allow one to construct pseudospherical metrics. With respect to analysis, one can trace back the relevance of the class of equations of pseudo-spherical type to the local uniqueness of surfaces of constant Gaussian curvature; with respect to integrability, one would say that the relevance of the Chern-Tenenblat structure arises from the fact that the Chern-Tenenblat construction [12] encodes in differential geometric terms the idea of a Wahlquist-Eastbrook quadratic pseudopotential [80]. This is the reason behind the fact that one can unc...

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