Three-way conflict analysis typically investigates conflict situations by analyzing the trisection of agent pairs, agents, and issues from the perspectives of either conflict or alliance. However, in fuzzy conflict situations, most existing studies rely on linear conflict functions to measure conflict degrees, which fail to accurately depict the intricate and dynamic trends of conflict intensity. In this paper, we aim to conduct three-way conflict analysis by utilizing a nonlinear conflict function within a fuzzy situation table. First, we propose a nonlinear conflict function to address the limitations of traditional linear conflict functions, providing a more precise and flexible representation of conflict intensity. Second, using the nonlinear conflict function, we analyze the relations between agent pairs, categorizing them into alliance, neutrality, and conflict relations. An algorithm is developed to identify the optimal trisection. Finally, based on a case study of the Middle East conflict, we implement case and comparison analyses to demonstrate the practical effectiveness and superiority of our proposed model in real-world fuzzy scenarios.