Algebraic construction of exact travelling wave solutions for the complex Gerdjikov-Ivanov equation

This study employs birational transformations to derive explicit exact solutions of the complex Gerdjikov--Ivanov (GI) equation. Through travelling wave reduction and singularity analysis, we establish the system's complete integrability and its algebraic-geometric structure via elliptic curves. We construct a comprehensive set of solutions, including elliptic functions, bright and dark solitons, and rational solutions, with detailed analysis of their physical relevance to nonlinear optics, Bose–Einstein condensates, and wave focusing phenomena. Our approach highlights the power of algebraic geometry in solving nonlinear integrable PDEs and offers a general framework applicable to similar complex systems.