Systems of differential equations of higher order describing pseudo-spherical or spherical surfaces

Abstract

We consider systems of partial differential equations of the form$\{\begin{array}{c}{u}_{xt}=F(u,u_x,...,{\partial }^{n}u/\partial x^n,v,v_x,...,{\partial }^{m}v/\partial x^m),\\ v_{xt}=G(u,u_x,...,{\partial }^{n}u/\partial x^n,v,v_x,...,{\partial }^{m}v\partial x^m),\end{array}$ $n,m\ge 2$, describing pseudospherical (pss) or spherical surfaces (ss). Generic solutions of such a system define metrics on open subsets of the plane, with coordinates $(x,t)$, whose Gaussian curvature is $K=-1$ or $K=1$. These systems are the integrability conditions of $g$-valued linear problems, with $g=\mathrm{sl}(2,R)$ or $g=\mathrm{su}\left(2\right)$, when $K=-1$, $K=1$, respectively. We obtain classification results for classes of such systems of differential equations of order n and m, in terms of four arbitrary smooth functions satisfying certain generic conditions. We also provide classification results for special type of second and third order systems. We include several explicit examples. Applications of the theory provide new examples and new families of systems of differential equations which contain the vector short-pulse and its generalizations. We provide the first conservation laws, from an infinite sequence, for some of the systems that describe pss.