Bifurcation and meromorphic exact solutions of five nonlinear partial differential equations

Differential equations model how scientific systems change over time and space. They are essential tools across physics, engineering, biology, and economics. Searching for solutions to these equations is of great significance, and simultaneously seeking meromorphic exact solutions of multiple nonlinear partial differential equations (PDEs) would be innovative. In view of this, we transform five nonlinear PDEs into ordinary differential equations (ODEs) through appropriate transformations and summarize these ODEs into one equation, then explore the bifurcation of this equation with commonalities. Subsequently, we utilize the complex method and the exp $(-\vartheta (z))$-expansion approach to derive meromorphic exact solutions for five nonlinear PDEs at once, including the Tzizeica equation, the Zhiber–Shabat (ZS) equation, the modified Kortweg de Vries (mKdV) equation, the Tzitzeica–Dodd–Bullough (TDB) equation, and the modified double sine-Gordon (mDSG) equation. To better understand the dynamic structure of the results, two-dimensional, three-dimensional, and contour graphs are given to illustrate the solutions. The conceptual framework developed here could be extended to analyze additional nonlinear physical equations in scientific research.