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Manually tailored wrinkled graphene sheets hold great promise in fabricatingsmart solid-state devices. In this paper, we employ an energy method totransform the original third-order partial differential equation (pde), i.e.Eq. (1) into the first-order pde, i.e. Eq. (8) for the thermophoretic motion ofsubstrate-supported graphene sheets, which can be solved in terms of semi-groupand transcendental solutions. Unlike soliton solutions derived using other moresophisticated techniques [9, 23], the present transcendental solution can beeasily solved numerically and provides physical insights. Most importantly, weverify that the formation of various forms for wrinkling wave solutions can bedetermined by the evolution of equilibrium points for Eq. (1). This sheds alight on modifying the heat sources in order to control the configuration ofwrinkle waves that has not been previously addressed.

A simple transcendental travelling wave solution and stability study for  the thermophoretic motion with variable heat transmission factors on  substrate-supported grapheme sheet