2023/08/12 by Anestis Fotiadis, Fotiadis, A., C. Daskaloyannis +1 · 1 citation
Physics and Astronomy · Mathematics · #Advanced Differential Geometry Research #Geometric Analysis and Curvature Flows #Nonlinear Waves and Solitons
paper · pdf · doi:10.48550/arxiv.2308.06625
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 Bäcklund 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 Bäcklund transformation.