Geometric method of exact integrability for elliptic Liouville equation \(\Delta u=e^u\) (Q1358556)

The author solves the elliptic Liouville equation \(\Delta u=e^u\) in \(\mathbb{R}^n \) using the geometric concept of pseudospherical metrics. This is a metric \(E(u)dx^2 +2F(u) dxdt +g(u)dt^2\) defined in terms of solutions \(u\) of the equation such that the Gaussian curvature \(K\) is constant \(K= -1\). By a coordinate transformation it can be shown that the simple ordinary differential equation \(y''- y=0\) defines a pseudospherical metric by the same expression as the Liouville equation. This is used to derive solutions of the Liouville equation from the well-known solutions of the ordinary differential equation.