Graphene has emerged as a highly attractive nanomaterial, characterized by its distinctive thermodynamic, electrical, and mechanical features. These attributes render it ideal for a wide array of purposes across materials research, storing energy, computing, filtering water, and medication. In this study, we investigate the nonlinear thermophoretic wrinkle motion equation describing wave propagation and thermal-driven deformation in graphene sheets. Governing model is \((2+1)\) -dimensional graphene sheets (GS) equation, focusing on the component heat transfer during thermodynamic motion. The study employs modified generalized exponential differential function method and the improved Cham method. These techniques are employed to analyze the nonlinear behavior of wave propagation in GS, leading to exact wave solutions such as bright, bell, periodic, and twofold waves. The creative application of these analytical techniques provides efficient structures for tackling complex nonlinear theories in mathematical physics, thereby enhancing solution strategies for these equations. This study makes a notable contribution to computational mathematics, material research, and nanotechnology by providing exact solutions and deepening our comprehension of graphene non linear properties. The findings present significant implications, suggesting potential applications in the design of novel materials with customized properties to facilitate technological progress, thereby expanding the frontiers of nanotechnology and materials research.
