On Differential Equations Describing Pseudo-Spherical Surfaces

Kamran, Niky; Tenenblat, Keti

doi:10.1006/jdeq.1995.1005

Cited by 62 publications

(89 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…But, a, b and c constants imply from (178) that a − c = 0 and b = 0, which contradicts the Gauss equation (12). Therefore m 1 c t − m 2 c x = 0 and, thus, from (184) and (185) Therefore, for arbitrary m 2 ψ we have shown that…”

Section: Ormentioning

confidence: 87%

“…However, φ 2 + 4b 2 = 0 if, and only if, φ = b = 0 which implies by (144) that a = c. But, a = c and b = 0 contradict the Gauss equation (12). Therefore, the coefficient of b in the equation (147) does not vanish in a non-empty open set.…”

Section: Ormentioning

confidence: 95%

“…If (99) holds then it follows from (58) that a vn = 0 and a wn = 0 and, consequently, by (56) we obtain b vn = c vn = 0 and b wn = c wn = 0. Thus, successive differentiation of (54), (55) and (12) …”

Section: Necessary Conditions For the Existence Of Second Fundamentalmentioning

confidence: 99%

“…For the sine-Gordon equation, with the choice of 1-forms ω 1 , ω 2 and ω 3 = ω 12 given by (8), it is easily verified that the 1-forms ω 13 , ω 23 are given by ω 13 = sin u 2 (dx + dt) = tan u 2 ω 1 ,…”

Section: Introductionmentioning

confidence: 99%

“…It is also shown in [5] that one can generate infinite sequences of conservation laws for the class of differential equations describing η pseudospherical surfaces by making use of the structure equations (3), although some of these conservation laws may end up being non-local. Important further developments of these ideas around this theme can be found in [15], [16], [17], [18], [12], [19], [20], [7], [9], [10], [11], [8], [21]. One may also consider the class of differential equations describing pseudospherical surfaces from an extrinsic point of view, motivated by the classical result which says that every pseudospherical surface can be locally isometrically immersed in E 3 .…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

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

Silva

Kamran

2016

Commun. Contemp. Math.

View full text Add to dashboard Cite

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 non-local) and the presence 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 E 3 of the pseudospherical surfaces defined by the solutions of equations belonging to the class of Chern and Tenenblat [3]. 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 pde. 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 [13], [14] we have shown that this property fails to hold for all k-th order evolution equations ut = F (u, ux, ..., u x k ) and all other second order equations of the form uxt = F (u, ux), 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 x and t which are independent of the choice of solution u. In the present paper, we consider thirdorder equations of the form ut − uxxt = λuuxxx + G(u, ux, uxx), λ ∈ R, which describe pseudospherical surfaces. This class contains the Camassa-Holm and Degasperis-Procesi equations as special cases. We show that whenever there exists a local isometric immersion in E 3 for which the coefficients of the second fundamental form depend on a jet of finite order of u, then these coefficients are universal in the sense of being independent on the choice of solution u. This result further underscores the special place that the sine-Gordon equations seems to occupy amongst integrable partial differential equations in one space variable.

show abstract

Section: Ormentioning

confidence: 87%

Section: Ormentioning

confidence: 95%

Section: Necessary Conditions For the Existence Of Second Fundamentalmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

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

Silva

Kamran

2016

Commun. Contemp. Math.

View full text Add to dashboard Cite

show abstract

Induced Surfaces and Their Integrable Dynamics

1996

View full text Add to dashboard Cite

A method is considered to induce surfaces in three-dimensional (pseudo) Euclidean space via the solutions to two-dimensional linear problems (20 LPs) and their integrable dynamics (deformations) via the 2 + I-dimensional nonlinear integrable equations associated with these 20 LPs. Coordinates Xi of the induced surfaces are defined as integrals over certain bilinear combinations of the wave functions l/f of these 20 LPs. General formulation as well as three concrete examples are considered. Some properties and features of such induction are discussed. Three-dimensional Riemann spaces associated with 2 + I-dimensional nonlinear integrable equations are considered also.

show abstract

Pseudo-spherical Surfaces and Integrability of Evolution Equations

Reyes¹

1998

Journal of Differential Equations

View full text Add to dashboard Cite

A complete classification of evolution equations u t =F(x, t, u, u x , ..., u x k ) which describe pseudo-spherical surfaces, is given, thus providing a systematic procedure to determine a one-parameter family of linear problems for which the given equation is the integrability condition. It is shown that for every second-order equation which admits a formal symmetry of infinite rank ( formal integrability) such a family exists (kinematic integrability). It is also shown that this result cannot be extended as proven to third-order formally integrable equations. This fact notwithstanding, a special case is proven, and moreover, several equations of interest, including the Harry Dym, cylindrical KdV, and a family of equations solved by inverse scattering by Calogero and Degasperis, are shown to be kinematically integrable. Conservation laws of equations describing pseudo-spherical surfaces are studied, and several examples are given. Academic Press

show abstract

On Differential Equations Describing Pseudo-Spherical Surfaces

Cited by 62 publications

References 0 publications

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

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

Induced Surfaces and Their Integrable Dynamics

Pseudo-spherical Surfaces and Integrability of Evolution Equations

Contact Info

Product

Resources

About