Exploring the exact soliton solutions and modulation instability analysis of the (3+1)-dimensional oceanic wave model

The oceanic wave equation plays an important role in modeling wave propagation phenomena related to weather forecasting and coastal dynamics. In this work, the Hirota trilinear scheme is employed to investigate the nonlinear (3+1)-dimensional fifth-order dynamical ocean equation. By applying this analytical technique, breather-wave and two-soliton solutions of the considered model are successfully derived and verified symbolically using Maple computational software. To illustrate the physical characteristics and propagation behavior of the obtained solutions, several structures are presented through two-dimensional, three-dimensional, and contour plots. The obtained wave structures represent new exact solutions that have not been reported in previous studies of this model. Furthermore, modulation instability analysis is performed to examine the stability of the steady-state solutions of the governing equation. The results provide deeper insight into nonlinear ocean wave dynamics and may contribute to applications in ocean engineering, coastal wave analysis, and related nonlinear physical systems. The effectiveness and simplicity of the Hirota trilinear scheme demonstrated in this study also suggest its applicability to other higher-dimensional nonlinear evolution equations arising in applied mathematics and fluid dynamics.