How can we solve a complex mathematical equation with far-reaching implications in physics? This paper addresses the nonlocal sine-Gordon equation, deriving it from the AKNS system using Parity and Time symmetries. The core contribution lies in constructing the Darboux transformation for this equation, enabling the generation of various types of solutions. By employing a seed solution, the researchers obtain soliton solutions, kink solutions, and mixed solutions. These findings offer a deeper understanding of the mathematical properties of the nonlocal sine-Gordon equation and its potential applications in various physical systems. The main work is to construct the Darboux transformation for the nonlocal sine-Gordon equation. In essence, this research provides a powerful mathematical framework for analyzing and solving the nonlocal sine-Gordon equation, opening avenues for further exploration of its physical significance and applications.
Published in The European Physical Journal C, this article aligns with the journal’s focus on particle physics, field theory, and mathematical physics. By presenting a mathematical analysis of the nonlocal sine-Gordon equation, the paper contributes to the journal's mission of advancing theoretical physics and its applications.