This paper focuses on the investigation of the Yu–Toda–Sasa–Fukuyama (YTSF) equation in its three-dimensional form. Based on the well-known Euler operator, a set of seven non-singular local multipliers is explored. Using these seven non-singular multipliers, the corresponding local conservation laws are derived. Additionally, an auxiliary potential-related system of partial differential equations (PDEs) is constructed. This study delves into nonlocal systems, which reveal numerous intriguing exact solutions of the YTSF equation. The nonlinear systems exhibit stable structures such as kink solitons, representing transitions, and breather or multi-solitons, modeling localized energy packets and complex interactions. These are employed in materials science, optics, communications, and plasma. Additionally, patterns such as parabolic backgrounds with ripples inform designs involving structured or varying media such as waveguides.