Analyzing breather waves and soliton solutions to the <i>P</i>-type (3+1)-dimensional evolution equation arising in mathematical physics

This study investigates the Painlevé integrable generalized P-type [Formula: see text]-dimensional evolution equation. We apply the [Formula: see text]–expansion method and Nucci’s reduction approach to derive diverse traveling wave solutions of the generalized P-type [Formula: see text]-dimensional evolution equation. These techniques, along with wave transformation and homogeneous balancing methods, allow us to obtain concise and general traveling wave solutions. The obtained solutions include rational, hyperbolic trigonometric, and circular trigonometric functions. We plot two- and three-dimensional graphs for the obtained solutions to get a deeper comprehension of the dynamical properties of the P-type model. The plots show a variety of wave profiles, including singular, bright, dark, bright-dark solitons, and breather waves. These solutions will improve the exact wave types found in the P-type model’s currently available literature. They might also be useful in explaining the potential physical mechanisms behind the wave phenomenon in plasma physics and nonlinear wave theory.