Novel soliton solutions of liquid drop model appear in fluid dynamics and modulation instability of dynamical system

The main concern of this manuscript is to investigate the function of dispersion within the formation of patterns in liquid drops described in the generalized (2+1)-dimensional Camassa–Holm–Kadomtsev–Petviashvili equation. The study incorporates the dynamics of molecular movement in liquids and the displacement of particles due to evaporation, leading to diverse cracking patterns observed in dried colloidal deposits. The considered model is particularly significant due to its ability to describe complex wave phenomena in multidimensional systems, making it highly relevant for understanding pattern formation in liquid systems. The technique includes the movement of molecules within a liquid and the displacement of particles brought on by evaporation. A method known as the Sardar sub-equation method is applied to find a new soliton solution of the Camassa–Holm–Kadomtsev–Petviashvili model. The obtained results are in the form of bright (non-topological) solitary or bell-shaped and dark (topological/smooth) solitary or anti-bell-shape soliton solutions. The newly derived soliton solutions offer valuable insights into the behavior of solitary waves under dispersion effects and contribute to a deeper understanding of the pattern formation process. To see the physical insights of the obtained solutions, various graphical representations are utilized, as well as three-dimensional, two-dimensional, and contour plots plus density plots. Further, the outcomes obtained are compared with the existing literature for a better understanding. The importance of these solutions lies in their potential to enhance the accuracy of predictive models for liquid drop dynamics and colloidal deposition processes. Additionally, modulational instability is also discussed, for small versions, a set of resonances occurs in the gain spectrum. It is a crucial concept in the study of nonlinear wave dynamics, offering insights into wave behavior, pattern formation, and energy localization. Lastly, the proposed method is expected to provide a significant enhancement.