Determinantal solutions to the (3+1)-dimensional Painlevé-type evolution equation: Higher-order rogue and soliton waves

We derive and characterize general rogue wave solutions (RWSs) of the (3+1)-dimensional Painlevé-type (P-type) integrable nonlinear evolution equation using the Hirota bilinear method in conjunction with the Kadomtsev–Petviashvili hierarchy reduction method (KPHRM). These solutions arise from intricate nonlinear interactions and exhibit diverse dynamical patterns, such as bright and dark triangular, pentagonal, and other structures, governed by key free parameters and the signs of system coefficients. Additionally, we address new nonlinear soliton solutions using the KPHRM in a determinantal framework. To further generalize the model, we incorporate spatiotemporal coefficients, which introduce additional nonlinear modulation. Using the Wronskian approach, another determinant-based technique, we construct $N$-soliton solutions for the variable-coefficient equation and analyze their nonlinear dynamics, demonstrating how parameter variation influences wave evolution and interactions.