Investigating traveling wave structures in the van der Waals normal form for fluidized granular matter through the modified S-expansion method

This research discovers traveling wave solutions (TWSs) of the van der Waals normal form for fluidized granular matter using the modified S-expansion (MS-E) method. The model captures key behaviors such as phase transitions, clustering, and shock structures in granular flows. Applying a traveling wave transformation reduces the governing equation to a nonlinear ordinary differential equation (NODE), enabling the construction of TWSs relevant to geophysical and industrial applications. The MS-E technique is implemented to systematically derive TWSs—such as kink, bright, and dark solitons—that model density waves, shock fronts, and clustering in granular media. Comprehensive 2D, 3D, and contour plots are presented to validate and visualize the results, offering insights into wave behavior and soliton stability. This work highlights the MS-E method as a powerful tool for solving nonlinear integral and fractional partial differential equations (NLIFPDEs), with broad applications in granular physics, fluid mechanics, plasma waves, and nonlinear optics. This experiment offers a novel procedure to explore additional compound nonlinear wave phenomena by integrating the MS-E method, opening novel opportunities for additional expansions in soliton-driven knowledge. This method offers a promising pathway for future researchers to explore closed-form traveling wave solutions of other NLIFPDEs.