Cauchy Problem for a Darboux Integrable Wave Map System and Equations of Lie Type

TL;DR

This paper investigates the Cauchy problem for harmonic maps into a Darboux integrable Lorentzian manifold, reducing the evolution to Lie type equations and deriving explicit solutions and a Weierstrass-type formula.

Contribution

It introduces a novel reduction of the harmonic map Cauchy problem to Lie systems, including explicit solution formulas and a new hyperbolic Weierstrass representation.

Findings

Reduction to Riccati equation enables explicit solutions.

Derivation of a hyperbolic Weierstrass-type formula.

Framing of open problems for future research.

Abstract

The Cauchy problem for harmonic maps from Minkowski space with its standard flat metric to a certain non-constant curvature Lorentzian 2-metric is studied. The target manifold is distinguished by the fact that the Euler-Lagrange equation for the energy functional is Darboux integrable. The time evolution of the Cauchy data is reduced to an ordinary differential equation of Lie type associated to SL(2) acting on a manifold of dimension 4. This is further reduced to the simplest Lie system: the Riccati equation. Lie reduction permits explicit representation formulas for various initial value problems. Additionally, a concise (hyperbolic) Weierstrass-type representation formula is derived. Finally, a number of open problems are framed.