Abstract
.Graphene is one of the thinnest and hardest elastic nanoscale materials and has opened new horizons in the field of material science for its versatile applications. The thermophoretic motion system is under investigation which describes the propagation of solitons in substrate-supported graphene sheets. A test function of the extended three-soliton method is used to construct the new soliton solutions. The one-order and mixed-order solutions involving solitons and lump waves are constructed. The dynamical behaviour of solitons under reflection, periodic distribution and interaction is depicted. Moreover, the bright and mixed type lump wave soliton propagation and interaction are discussed.