On Equations Which Describe Pseudospherical Surfaces

Rabelo, Mauro Luiz

doi:10.1002/sapm1989813221

Cited by 171 publications

(129 citation statements)

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“…If f ij are given by (33) with µ = 0 is easy to see that ∆ 13 = −ληz 1 ( = 0) and ∆ 23 = −λm 1 z 1 ( = 0). Therefore, it follows from c = ∓2∆ 13 /∆ 23 that c = ∓2η/m 1 is a constant and, from (48) and (49), we obtain ∆ 13 = ∆ 23 = 0, which is a contradiction with (16).…”

Section: Necessary Conditions For the Existence Of Second Fundamentalmentioning

confidence: 83%

“…which holds if, and only if, ∆ 23 = 0, which implies ∆ 13 = 0 and, thus, a contradiction by (16). (13) and (14) are equivalent to…”

Section: Necessary Conditions For the Existence Of Second Fundamentalmentioning

confidence: 89%

“…If φ 12 = 0 then from (47) we conclude that a is a constant and (46) reduces to ∓2∆ 13 + a∆ 23 = 0. This equation with (45) implies that ∆ 13 = ∆ 23 = 0 which contradicts (16).…”

Section: Necessary Conditions For the Existence Of Second Fundamentalmentioning

confidence: 91%

“…Looking at (51) is easy to see that, if f 21 = 0 and λz 0 f 21 + f 22 = 0 then c is a constant and from (48) and (49) we obtain ∆ 13 = ∆ 23 = 0, which is a contradiction with (16) If f ij are given by (35) with µ = m 1 = 0 and m 2 = 0 then ∆ 13 = 0. However, ∆ 13 = 0 implies c = 0 which is a contradiction, because we firstly showed that c = 0.…”

Section: Necessary Conditions For the Existence Of Second Fundamentalmentioning

confidence: 94%

“…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%

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

Silva

Kamran

2016

Commun. Contemp. Math.

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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.

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Section: Necessary Conditions For the Existence Of Second Fundamentalmentioning

confidence: 83%

“…which holds if, and only if, ∆ 23 = 0, which implies ∆ 13 = 0 and, thus, a contradiction by (16). (13) and (14) are equivalent to…”

Section: Necessary Conditions For the Existence Of Second Fundamentalmentioning

confidence: 89%

“…If φ 12 = 0 then from (47) we conclude that a is a constant and (46) reduces to ∓2∆ 13 + a∆ 23 = 0. This equation with (45) implies that ∆ 13 = ∆ 23 = 0 which contradicts (16).…”

Section: Necessary Conditions For the Existence Of Second Fundamentalmentioning

confidence: 91%

Section: Necessary Conditions For the Existence Of Second Fundamentalmentioning

confidence: 94%