Harmonic diffeomorphisms between pseudo-riemannian surfaces

TL;DR

This paper unifies the study of harmonic maps between pseudo-Riemannian surfaces, classifies them via a generalized sine-Gordon equation, and explores soliton solutions and transformations connecting different solutions.

Contribution

It introduces a unified formalism for harmonic maps between various pseudo-Riemannian surfaces and analyzes their solutions through a generalized sine-Gordon equation, including solitons and Bäcklund transformations.

Findings

Classification of harmonic maps via sine-Gordon solutions

Explicit construction of one-soliton harmonic maps

Bäcklund transformation connecting solutions

Abstract

We study locally harmonic maps between a Riemann surface or Lorentz surface $M$ and a Riemann surface or Lorentz surface $N$. {All four cases are studied in a unified way}. All four cases are written using a unified formalism. Therefore solutions to the harmonic map problem can be studied in a unified way. Harmonic maps between pseudo-Riemannian surfaces are classified by the classification of the solutions of a generalized sine-Gordon equation. We then study the one-soliton solutions of this equation and we find the corresponding harmonic maps in a unified way. Next, we discuss a B\"acklund transformation of the harmonic map equations that provides a connection between the solutions of two sine-Gordon type equations. Finally, we give an example of a harmonic map that is constructed by the use of a B\"acklund transformation.