Local isometric immersions of pseudospherical surfaces described by evolution equations in conservation law form

For the pseudospherical surfaces described by a class of second order evolution equations, of the form zt=A(x,t,z)z2+B(x,t,z,z1)zt=A(x,t,z)z2+B(x,t,z,z1), we consider the problem of local isometric immersion into the 3-dimensional Euclidean space E3E3 with a second fundamental form depending on finite-order jets of solutions z of the considered equations. We also provide an extension of our analysis to the case of k-th order evolution equations in conservation law form. Examples of equations admitting such local isometric immersions, are provided by equations like Burgers, Murray, Svinolupov–Sokolov, Kuramoto–Sivashinsky, Sawada–Kotera, Kaup–Kupershmidt, as well as hierarchies of evolution equations in conservation law form like Burgers, mKdV and KdV.