<p>The thermophoretic motion equation governing wrinkle propagation in substrate-supported graphene sheets is investigated. As one of the thinnest yet strongest nanoscale elastic materials, graphene has revolutionized material science with its diverse applications. Its exceptional physical and chemical properties, developed through extensive theoretical and experimental research, have made it a fundamental component in numerous electronic devices. Particularly intriguing is the study of graphene-based thermal energy conversion systems, complemented by the material’s remarkable optical characteristics. This research employs the Hirota bilinear method to derive multiple soliton solutions and M-lump-like waves, while the long-wave method is utilized to examine hybrid solutions. The physical relevance of these solutions is thoroughly explored through graphical representations that depict their behavior under specific parameter conditions. The findings confirm the reliability of the applied techniques and suggest their potential for uncovering novel solutions to nonlinear evolution equations in various scientific domains.</p>