TY - JOUR
T1 - Local isometric immersions of pseudo-spherical surfaces and evolution equations
AU - Kahouadji, Nabil
AU - Kamran, Niky
AU - Tenenblat, Keti
N1 - Funding Information:
Research partially supported by NSERC Grant RGPIN 105490-2011 and by the Ministério de Ciência e Tecnologia, Brazil, CNPq Proc. No. 303774/2009-6.
Publisher Copyright:
© Springer Science+Business Media New York 2015
PY - 2015
Y1 - 2015
N2 - The class of differential equations describing pseudo-spherical surfaces, first introduced by Chern and Tenenblat (Stud. Appl. Math. 74, 55–83, 1986), is characterized by the property that to each solution of a differential equation within the class, there corresponds a two-dimensional Riemannian metric of curvature equal to [1]1. The class of differential equations describing pseudo-spherical surfaces carries close ties to the property of complete integrability, as manifested by the existence of infinite hierarchies of conservation laws and associated linear problems. As such, it contains many important known examples of integrable equations, like the sine-Gordon, Liouville and KdV equations. It also gives rise to many new families of integrable equations. The question we address in this paper concerns the local isometric immersion of pseudo-spherical surfaces in E3 from the perspective of the differential equations that give rise to the metrics. Indeed, a classical theorem in the differential geometry of surfaces states that any pseudo-spherical surface can be locally isometrically immersed in E3. In the case of the sine-Gordon equation, one can derive an expression for the second fundamental form of the immersion that depends only on a jet of finite order of the solution of the pde. A natural question is to know if this remarkable property extends to equations other than the sine-Gordon equation within the class of differential equations describing pseudospherical surfaces. In an earlier paper (Kahouadji et al., Second-order equations and local isometric immersions of pseudo-spherical surfaces, 25 pp. [arXiv:1308.6545], to appear in Comm. Analysis and Geometry (2015), we have shown that this property fails to hold for all other second order equations, except for those belonging to a very special class of evolution equations. In the present paper, we consider a class of evolution equations for u.x;
AB - The class of differential equations describing pseudo-spherical surfaces, first introduced by Chern and Tenenblat (Stud. Appl. Math. 74, 55–83, 1986), is characterized by the property that to each solution of a differential equation within the class, there corresponds a two-dimensional Riemannian metric of curvature equal to [1]1. The class of differential equations describing pseudo-spherical surfaces carries close ties to the property of complete integrability, as manifested by the existence of infinite hierarchies of conservation laws and associated linear problems. As such, it contains many important known examples of integrable equations, like the sine-Gordon, Liouville and KdV equations. It also gives rise to many new families of integrable equations. The question we address in this paper concerns the local isometric immersion of pseudo-spherical surfaces in E3 from the perspective of the differential equations that give rise to the metrics. Indeed, a classical theorem in the differential geometry of surfaces states that any pseudo-spherical surface can be locally isometrically immersed in E3. In the case of the sine-Gordon equation, one can derive an expression for the second fundamental form of the immersion that depends only on a jet of finite order of the solution of the pde. A natural question is to know if this remarkable property extends to equations other than the sine-Gordon equation within the class of differential equations describing pseudospherical surfaces. In an earlier paper (Kahouadji et al., Second-order equations and local isometric immersions of pseudo-spherical surfaces, 25 pp. [arXiv:1308.6545], to appear in Comm. Analysis and Geometry (2015), we have shown that this property fails to hold for all other second order equations, except for those belonging to a very special class of evolution equations. In the present paper, we consider a class of evolution equations for u.x;
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U2 - 10.1007/978-1-4939-2950-4_13
DO - 10.1007/978-1-4939-2950-4_13
M3 - Article
AN - SCOPUS:84941626946
SN - 1069-5265
VL - 75
SP - 369
EP - 381
JO - Fields Institute Communications
JF - Fields Institute Communications
ER -