Hirota Bilinear Analysis and Physics-Informed Neural Network Simulation for a Sixth-Order (2+1)-Dimensional Generalized Calogero–Bogoyavlenskii–Schiff Equation

This paper introduces a new sixth-order (2+1)-dimensional generalized Calogero–Bogoyavl enskii–Schiff (S2GCBS) equation and provides exact analytical solutions by using Physics-Informed Neural Network simulation (PINNs). By applying an appropriate traveling wave transformation, the governing nonlinear partial differential equation is reduced to an ordinary differential equation. A systematic solution procedure is then employed to derive several classes of exact solutions. As a result, different types of wave structures are obtained, including hyperbolic function solutions (soliton solutions), trigonometric function solutions (periodic wave solutions), and rational solutions. In particular, the obtained results contain bright soliton solutions, dark soliton solutions, singular soliton solutions, and periodic wave solutions expressed in compact analytical forms. These solutions reveal the rich dynamical behavior of the gCBS model and demonstrate the effectiveness of the proposed method in constructing exact solutions for nonlinear evolution equations arising in mathematical physics. The physical characteristics of the obtained solutions are illustrated through several graphical representations.