Soliton, Modulation Instability and Interaction Dynamics for a (3+1)-Dimensional Ito Equation Applied to Coastal Hydrodynamics

The present paper aims to find the integrability characteristics and solutions of the (3+1)-dimensional Ito equation, which is a nonlinear evolutionary partial differential equation playing an important role in the theory of wave dynamics. The application of the Hirota bilinear technique allowed different kinds of solutions, such as breather solutions, two-wave solutions, and interaction solutions, to be obtained. It should be emphasized that breather solutions can be used to describe wave propagation with oscillations and are applied to model rogue waves, energy localization in optical fibers, as well as excitations in Bose–Einstein condensates. Two-wave solutions are used to analyze energy distribution and resonance effects in fluid mechanics, plasma physics, and nonlinear optics. The soliton solutions, which are stable and interact elastically, were obtained using the extended tanh-function and the [Formula: see text]-expansion methods. The obtained results provide further insight into integrable models in higher dimensions, while the wide range of applications of the Ito equation becomes more evident in both mathematics and physics. A review of the literature shows that the Hirota bilinear, the extended tanh-function, and the [Formula: see text] methods have been employed to obtain solitons and interactions of the Ito equation for the first time. Moreover, the Hirota bilinear method allows the construction of solutions expressed as combinations of trigonometric, hyperbolic, and exponential functions. In addition, rational, hyperbolic, and trigonometric solutions are derived by utilzing the extended tanh-function and [Formula: see text] methods.