# Local Isometric Immersions of Pseudo-Spherical Surfaces and Evolution Equations

- 3 Citations
- 1.2k Downloads

## Abstract

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. 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 **E**^{3} 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 **E**^{3}. 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 pseudo-spherical 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*, *t*) of order *k* ≥ 3 describing pseudo-spherical surfaces. We show that whenever an isometric immersion in **E**^{3} exists, depending on a jet of finite order of *u*, then the coefficients of the second fundamental form are universal, that is they are functions of the independent variables *x* and *t* only.

## Keywords

Local Isometric Immersion Pseudo-spherical Surfaces Tenenblat sine-Gordon Equation Fundamental Form## Notes

### Acknowledgements

Research partially supported by NSERC Grant RGPIN 105490-2011 and by the Ministério de Ciência e Tecnologia, Brazil, CNPq Proc. No. 303774/2009-6.

## References

- 1.Beals, R., Rabelo, M., Tenenblat, K.: Bäcklund transformations and inverse scattering for some pseudospherical surface equations. Stud. Appl. Math.
**81**, 121–151 (1989)MathSciNetCrossRefGoogle Scholar - 2.Cavalcante, D., Tenenblat, K.: Conservation laws for nonlinear evolution equations. J. Math. Phys.
**29**, 1044–1049 (1988)zbMATHMathSciNetCrossRefGoogle Scholar - 3.Chern, S.-S., Tenenblat, K.: Pseudospherical surfaces and evolution equations. Stud. Appl. Math.
**74**, 55–83 (1986)zbMATHMathSciNetCrossRefGoogle Scholar - 4.Ding, Q., Tenenblat, K.: On differential equations describing surfaces of constant curvature. J. Diff. Equ.
**184**, 185–214 (2002)zbMATHMathSciNetCrossRefGoogle Scholar - 5.Ferraioli, D.C., Tenenblat, K.: Fourth order evolution equations which describe pseudospherical surfaces. J. Differ. Equ.
**257**, 3165–3199 (2014)zbMATHMathSciNetCrossRefGoogle Scholar - 6.Foursov, V., Olver, P.J., Reyes, E.: On formal integrability of evolution equations and local geometry of surfaces. Differ. Geom. Appl.
**15**, 183–199 (2001)zbMATHMathSciNetCrossRefGoogle Scholar - 7.Gomes Neto, V.P.: Fifth-order evolution equations describing pseudospherical surfaces. J. Differ. Equ.
**249**, 2822–2865 (2010)zbMATHMathSciNetCrossRefGoogle Scholar - 8.Gorka, P., Reyes, E.: The modified Hunter-Saxton equation. J. Geom. Phys.
**62**, 1793–1809 (2012)zbMATHMathSciNetCrossRefGoogle Scholar - 9.Huber, A.: The Cavalcante–Tenenblat equation—does the equation admit physical significance? Appl. Math. Comput.
**212**, 14–22 (2009)zbMATHMathSciNetCrossRefGoogle Scholar - 10.Jorge, L., Tenenblat, K.: Linear problems associated to evolution equations of type \(u_{tt} = F(u,u_{x},\ldots,u_{x^{k}})\). Stud. Appl. Math.
**77**, 103–117 (1987)zbMATHMathSciNetCrossRefGoogle Scholar - 11.Kahouadji, N., Kamran, N., Tenenblat, K.: Second-order equations and local isometric immersions of pseudo-spherical surfaces, 25 pp. [arXiv:1308.6545], to appear in Comm. Analysis and Geometry (2015)Google Scholar
- 12.Kamran, N., Tenenblat, K.: On differential equations describing pseudo-spherical surfaces. J. Differ. Equ.
**115**, 75–98 (1995)zbMATHMathSciNetCrossRefGoogle Scholar - 13.Rabelo, M.: A characterization of differential equations of type \(u_{xt} = F(u,u_{x},\ldots,u_{x^{k}})\) which describe pseudo-spherical surfaces. An. Acad. Bras. Cienc.
**60**, 119–126 (1988)zbMATHMathSciNetGoogle Scholar - 14.Rabelo, M.: On equations which describe pseudo-spherical surfaces. Stud. Appl. Math.
**81**, 221–148 (1989)zbMATHMathSciNetCrossRefGoogle Scholar - 15.Rabelo, M., Tenenblat, K.: On equations of the type
*u*_{xt}=*F*(*u*,*u*_{x}) which describe pseudo-spherical surfaces. J. Math. Phys.**29**, 1400–1407 (1990)MathSciNetCrossRefGoogle Scholar - 16.Rabelo, M., Tenenblat, K.: A classification of equations of the type \(u_{t} = u_{xxx} + G(u,u_{x},u_{xx})\) which describe pseudo-spherical surfaces. J. Math. Phys.
**33**, 1044–149 (1992)MathSciNetCrossRefGoogle Scholar - 17.Reyes, E.: Pseudospherical surfaces and integrability of evolution equations. J. Differ. Equ.
**147**, 195–230 (1998)zbMATHMathSciNetCrossRefGoogle Scholar - 18.Reyes, E.: Pseudopotentials, nonlocal symmetries and integrability of some shallow water wave equations. Sel. Math. N. Ser.
**12**, 241–270 (2006)zbMATHMathSciNetCrossRefGoogle Scholar - 19.Sakovich, A., Sakovich, S.: Solitary wave solutions of the short pulse equation. J. Phys. A
**39**, L361-L367 (2006)zbMATHMathSciNetCrossRefGoogle Scholar - 20.Sakovich, A., Sakovich, S.: On transformations of the Rabelo equations. SIGMA Symmetry Integrability Geom. Methods Appl.
**3**, 8 p., paper 086 (2007)Google Scholar