Modulation instability and dynamical behavior of solitary wave solution of time M-fractional clannish random Walker's Parabolic equation via two analytic techniques

The theory of solitary waves or solitons is crucial in nonlinear models due to their ability to propagate without distortion, making them essential in fields like physics, biology, engineering, and mathematics. This study explores solitary waves in the time fractional Clannish Random Walker's Parabolic equation using two effective approaches: polynomial expansion technique (PET) and unified technique (UT). This model is particularly significant in areas such as ecology, sociology, and urban planning, where understanding individual interactions within spatial contexts is vital. By solving this equation, we gain insights into group formation and navigation, enhancing our comprehension of emergent patterns and system design. The application of PET and UT allows us to derive various solutions, including exponential, hyperbolic, and trigonometric forms. Utilizing the Maple programming language, we visualize novel phenomena, such as kink waves, anti-kink waves, and periodic lump waves. This research demonstrates that the proposed methodologies can yield precise soliton solutions, contributing valuable insights to nonlinear science and engineering applications.