Nonlinear Cauchy-Riemann Equations and Liouville Equation For Conformal Metrics

We introduce the Nonlinear Cauchy-Riemann equations as B"{a}cklund transformations for several nonlinear and linear partial differential equations. From these equations we treat in details the Laplace and the Liouville equations by deriving general solution for the nonlinear Liouville equation. By M"{o}bius transformation we relate solutions for the Poincare model of hyperbolic geometry, the Klein model in half-plane and the pseudo-sphere. Conformal form of the constant curvature metrics in these geometries, stereographic projections and special solutions are discussed. Then we introduce the hyperbolic analog of the Riemann sphere, which we call the Riemann pseudosphere. We identify point at infinity on this pseudosphere and show that it can be used in complex analysis as an alternative to usual Riemann sphere to extend the complex plane. Interpretation of symmetric and antipodal points on both, the Riemann sphere and the Riemann pseudo-sphere, are given. By M"{o}bius transformation and homogenous coordinates, the most general solution of Liouville equation as discussed by Crowdy is derived.

💡 Research Summary

The paper introduces a nonlinear extension of the classical Cauchy‑Riemann (CR) system, called the Nonlinear Cauchy‑Riemann (NCR) equations, and demonstrates that these equations serve as a Bäcklund transformation linking solutions of various linear and nonlinear partial differential equations. The authors begin by recalling that the standard CR equations are a Bäcklund transformation for the Laplace equation, then generalize the system to
 uₓ = v_y + g(u,v), u_y = –vₓ + f(u,v)
with the auxiliary functions f and g required to satisfy the CR relations f_u = g_v, f_v = –g_u. Under this condition the real components u and v each satisfy a nonlinear Laplace‑type equation:
 Δu = ½∂_u(f²+g²), Δv = ½∂_v(f²+g²).
In complex notation, setting w = u + i v and G(w) = f + i g, the system collapses to a single first‑order equation
 ∂̄ w = (i/2) G(w)
and a second‑order nonlinear Laplace equation
 Δw = G′(w) G(w).
Choosing different analytic forms for G produces familiar equations: G = 0 recovers the linear CR system; G = w yields the Helmholtz equation; G = w² gives a cubic nonlinear Schrödinger‑type equation; and the crucial choice G(w) = e^w leads to the Liouville equation Δu = e^{2u} together with the linear Laplace equation Δv = 0 for the imaginary part.

The core of the paper is the derivation of the general solution of the Liouville equation using the NCR framework. By solving the associated Laplace equation for the harmonic conjugate v = F(z) + \overline{F(z)} and introducing φ = e^u, the authors perform a sequence of substitutions that culminate in the explicit formula
 u(x,y) = ½ ln