Third-order differential equations and local isometric immersions of 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 E-3 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 kth-order evolution equations ut = F(u, u(x),..., u(x)k) and all other second-order equations of the form u(xt) = F(u, u(x)), 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 this paper, we consider third-order equations of the form u(t) - u(xxt) = lambda uu(xxx) + G(u, u(x), u(xx)), lambda is an element of R, 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 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 seem to occupy amongst integrable partial differential equations in one space variable.