The Impacts of Brownian Motion on the Solutions of Stochastic Kakutani-Matsuuchi Equation

Abstract

In this study, the stochastic Kakutani--Matsuuchi equation (SKME) forced by a multiplicative stochastic term is considered. The Kakutani-Matsuuchi equation is a fundamental tool for analyzing internal gravity waves in fluid dynamics, particularly in geophysical contexts such as oceanography and meteorology. Internal gravity waves occur in stratified fluids where density variations are influenced by gravity, leading to fascinating phenomena among different layers of fluid. Due to the importance of the Kakutani-Matsuuchi equation in examining the waves of internal gravity in the oceans and atmosphere, the solutions of the SKME can help us comprehend a variety of exciting scientific phenomena. Utilizing two distinct techniques, namely the Sardar subequation technique and the Jacobi elliptic function method, we derive novel bright and dark solitons, periodic solitons, as well as Kink and anti-Kink soliton solutions for the SKME. Moreover, we show many 3D and 2D graphs illustrating the influence of noise on SKME solutions.