Local Isometric Immersions of Pseudo-Spherical Surfaces and Evolution Equations

Local isometric immersions of pseudo-spherical surfaces and kth order evolution equations

Communications in Contemporary Mathematics ◽

10.1142/s0219199718500256 ◽

2019 ◽

Vol 21 (04) ◽

pp. 1850025

Author(s):

Nabil Kahouadji ◽

Niky Kamran ◽

Keti Tenenblat

Keyword(s):

Isometric Immersion ◽

Evolution Equations ◽

Second Fundamental Form ◽

Second Order ◽

Isometric Immersions ◽

Spherical Surfaces ◽

Pseudospherical Surfaces ◽

The Mean ◽

Local Isometric Immersion ◽

Straight Lines

We consider the class of evolution equations of the form [Formula: see text], [Formula: see text], that describe pseudo-spherical surfaces. These were classified by Chern and Tenenblat in [Pseudospherical surfaces and evolution equations, Stud. Appl. Math 74 (1986) 55–83.]. This class of equations is characterized by the property that to each solution of such an equation, there corresponds a 2-dimensional Riemannian metric of constant curvature [Formula: see text]. Motivated by the special properties of the sine-Gordon equation, we investigate the following problem: given such a metric, is there a local isometric immersion in [Formula: see text] such that the coefficients of the second fundamental form of the immersed surface depend on a jet of finite order of [Formula: see text]? We extend our earlier results for second-order evolution equations [N. Kahouadji, N. Kamran and K. Tenenblat, Local isometric immersions of pseudo-spherical surfaces and evolution equations, Fields Inst. Commun. 75 (2015) 369–381; N. Kahouadji, N. Kamran and K. Tenenblat, Second-order equations and local isometric immersions of pseudo-spherical surfaces, Comm. Anal. Geom. 24(3) (2016) 605–643.] to [Formula: see text]th order equations by proving that there is only one type of equation that admit such an isometric immersion. More precisely, we prove under the condition of finite jet dependency that the coefficients of the second fundamental forms of the local isometric immersion determined by the solutions [Formula: see text] are universal, i.e. they are independent of [Formula: see text]. Moreover, we show that there exists a foliation of the domain of the parameters of the surface by straight lines with the property that the mean curvature of the surface is constant along the images of these straight lines under the isometric immersion.

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A Novikov equation describing pseudo‐spherical surfaces, its pseudo‐potentials, and local isometric immersions

Studies in Applied Mathematics ◽

10.1111/sapm.12457 ◽

2021 ◽

Author(s):

Igor Leite Freire ◽

Rodrigo da Silva Tito

Keyword(s):

Isometric Immersions ◽

Spherical Surfaces ◽

Novikov Equation

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Third-order differential equations and local isometric immersions of pseudospherical surfaces

Communications in Contemporary Mathematics ◽

10.1142/s0219199716500218 ◽

2016 ◽

Vol 18 (06) ◽

pp. 1650021 ◽

Cited By ~ 2

Author(s):

Tarcísio Castro Silva ◽

Niky Kamran

Keyword(s):

Differential Equations ◽

Fundamental Form ◽

Evolution Equations ◽

Gordon Equation ◽

Second Fundamental Form ◽

Isometric Immersions ◽

Sine Gordon Equation ◽

Third Order ◽

The Second Fundamental Form ◽

Pseudospherical Surfaces

The class of differential equations describing pseudospherical surfaces enjoys important integrability properties which manifest themselves by the existence of infinite hierarchies of conservation laws (both local and nonlocal) and the presence of associated linear problems. It thus contains many important known examples of integrable equations, like the sine-Gordon, Liouville, KdV, mKdV, Camassa–Holm and Degasperis–Procesi equations, and is also home to many new families of integrable equations. Our paper is concerned with the question of the local isometric immersion in E3 of the pseudospherical surfaces defined by the solutions of equations belonging to the class introduced by Chern and Tenenblat [Pseudospherical surfaces and evolution equations, Stud. Appl. Math. 74 (1986) 55–83]. In the case of the sine-Gordon equation, it is a classical result that the second fundamental form of the immersion depends only on a jet of finite order of the solution of the partial differential equation. A natural question is therefore 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 a pair of earlier papers [N. Kahouadji, N. Kamran and K. Tenenblat, Second-order equations and local isometric immersions of pseudo-spherical surfaces, to appear in Comm. Anal. Geom., arXiv: 1308.6545; Local isometric immersions of pseudo-spherical surfaces and evolution equations, in Hamiltonian Partial Differential Equations and Applications, eds. P. Guyenne, D. Nichols and C. Sulem, Fields Institute Communications, Vol. 75 (Springer-Verlag, 2015), pp. 369–381], it was shown that this property fails to hold for all [Formula: see text]th-order evolution equations [Formula: see text] and all other second-order equations of the form [Formula: see text], except for the sine-Gordon equation and a special class of equations for which the coefficients of the second fundamental form are universal, that is functions of [Formula: see text] and [Formula: see text] which are independent of the choice of solution [Formula: see text]. In this paper, we consider third-order equations of the form [Formula: see text], [Formula: see text], which describe pseudospherical surfaces with the Riemannian metric given in [T. Castro Silva and K. Tenenblat, Third order differential equations describing pseudospherical surfaces, J. Differential Equations 259 (2015) 4897–4923]. This class contains the Camassa–Holm and Degasperis–Procesi equations as special cases. We show that whenever there exists a local isometric immersion in E3 for which the coefficients of the second fundamental form depend on a jet of finite order of [Formula: see text], then these coefficients are universal in the sense of being independent on the choice of solution [Formula: see text]. This result further underscores the special place that the sine-Gordon equations seem to occupy amongst integrable partial differential equations in one space variable.

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