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

Abstract

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.

A note on isometric immersions and differential equations which describe pseudospherical surfaces

“…For recent developments on the study and classification of pss equations the reader is referred to [2,3,4,5,6,7,10,11,12,13,15].…”

“…Hence, in view of the Bonnet theorem, to any generic solution z of E it is associated a pair (I[z], II[z]) of first and second fundamental forms, which satisfy the Gauss-Codazzi equations and describes a local isometric immersion into E 3 of an associated pseudospherical surface. Recall that (see [12] and also [7,11,13,2]), by introducing the 1-forms…”

“…In a series of recent papers [2,7,11,12,13] the existence of finite-order local isometric immersions has been investigated for the families of pss equations previously studied in the papers [3,6,8,10,15]. Surprisingly, those papers showed that, except for the sine-Gordon equation, all local isometric immersions admitted by pss equations described in these papers have a triple {a, b, c} which is "universal", in the sense that it only depends on x and t. Hence, contrary to the case of sine-Gordon equation, for any pss equation E of the type described in [3,6,8,10,15] the principal curvatures (and hence the mean curvature) of the immersion is a function of x and t which does not depend on the generic solutions.…”

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In this paper, we provide families of second order non-linear partial differential equations, describing pseudospherical surfaces (pss equations), with the property of having local isometric immersions in E 3 , with principal curvatures depending on finite-order jets of solutions of the differential equation. These equations occupy a particularly special place amongst pss equations, since a series of recent papers [2,7,11,12,13], on several classes of pss equations, seemed to suggest that only the sine-Gordon equation had the above property. Explicit examples are given, which include the short pulse equation and some generalizations.

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