Foliations on constant curvature surfaces and nonlinear partial differential equations (Q2705779)

Let \(M\subset\mathbb{R}^3\) be a surface of constant Gaussian curvature \(K\). The structural equations \(d\omega_1= \omega_2\wedge \omega_{12}\), \(d\omega_2= \omega_{12}\wedge \omega_1\), \(d\omega_{12}= K\omega_1\wedge \omega_2\) can be represented as the compatibility conditions for the linear system \(dV= \Omega V\), where \(\Omega= {1\over 2}\left(\begin{smallmatrix} \omega_2 & \omega_1+\omega_{12}\\ \omega_1- \omega_{12} & \omega_2\end{smallmatrix}\right)\) and either \(V: M\to\text{SL}(2,\mathbb{R})\) (if \(K<0)\) or \(V: M\to\text{SU}(2)\) (if \(K>0\)). Then, assuming \(\omega_{12}= p\omega_1+ q\omega_2\), \(u= f(p,q,\partial_1 p,\partial_2p,\dots, \partial^l_1\cdots\partial^k_2 q)\), the author determines necessary and sufficient conditions for the existence of foliations \(\omega_2= 0\) (equivalently, \(t\)=const. where \(x\), \(t\) are certain coordinates) such that the functions \(u\) (expressed in terms of \(t\) and \(x\)) are described by certain differential equations \(F(u,u_x,u_t,\dots, u_{t\cdots tx\cdots x})= 0\). The problem is resolved for the particular cases of the differential equations \(u_t= G(u,u_x,\dots, u_{x\cdots x})\) or \(u_{xt}= G(u,u_x,\dots, u_{x\cdots x})\) or \(u_{xt}= G(u, u_x,u_t)\). The KdV, MKdV, sinh-Gordon, Calogero-Degasperis-Fokas, Sawada-Kotera, and Kamp-Kupershmidt equations are involved.