Exploring the impact of Brownian motion on novel closed-form solutions of the extended Kairat-II equation.

This work considers a stochastic form of an extended version of the Kairat-II equation by adding Browning motion into the deterministic equation. Two analytical approaches are utilized to derive analytical solutions of the modified equation. The first method is the modified Tanh technique linked with the Riccati equation, which is implemented to extract some closed-form solutions in the form of tangent and cotangent functions. The second technique is the Sardar sub-equation method (SSEM) which is used to attain several analytical solutions in the form of trigonometric and hyperbolic functions. Solutions selected randomly from the large families of solutions with suggested techniques are visualized in 3D and 2D scenarios. From the simulations an intriguing observation is made: the solutions generated through the modified tanh method exhibit a singular nature, with some of hybrid waves among them. On contrary to this, solutions derived through the SSEM, tend to be mostly non-singular in nature. The varying influence of the noise intensity revealed that the high amplitude and high energy regions of the waves are more vulnerable to the induced noise as compared to lower energy regions, which are relatively robust. This study introduces novel approaches by incorporating Brownian motion into the extended Kairat-II equation, providing new insights into the behavior of stochastic integrable systems that have not been previously explored.