Generation of Grey and Straddled Soliton for the Stochastic Kakutani-Matsuuchi Model under Multiplicative Itô Noise

In this study, we investigate the stochastic Kakutani-Matsuuchi model (SKMM) with multiplicative noise in the It & ocirc; sense, which describes the propagation of internal gravity waves in stratified fluids such as the Earth's atmosphere and ocean. These waves, generated by density or temperature variations, play a fundamental role in transferring energy and momentum across the system. The multiplicative noise term accounts for random fluctuations whose intensity depends on wave amplitude, thereby providing a realistic description of noise-wave interactions in geophysical environments, while the It & ocirc; framework ensures a rigorous mathematical treatment of such randomness and its cumulative effect on system evolution. By applying the Sub-ODE method, we derive a broad spectrum of exact analytical solutions, including bright soliton, periodic wave, rational-type, hyperbolic-type, and singular structures. Their geometrical characteristics are explored through 3D graphical representations obtained under different values of the random noise parameter, which reveal distinctive behaviors such as localization, periodic modulation, algebraic decay, and blow-up dynamics. These findings deepen the understanding of nonlinear wave phenomena governed by the SKMM and demonstrate the versatility of the Sub-ODE approach in capturing the impact of stochastic influences. The results are expected to provide a valuable reference for modeling wave propagation in oceanic and atmospheric systems where stochastic effects cannot be ignored.