Small amplitude approximation and stabilities for dislocation motion in a superlattice

Starting from the traveling wave solution, in small amplitude approximation, the Sine–Gordon equation can be re- duced to a generalized Duffing equation to describe the dislocation motion in a superlattice, and the phase plane properties of the system phase plane are described in the absence of an applied field. The stabilities are also discussed in the presence of an applied field. It is pointed out that the separatrix orbit describing the dislocation motion as the kink wave may transfer the energy along the dislocation line, keep its form unchanged, and reveal the soliton wave properties of the dislocation motion. It is stressed that the dislocation motion process is the energy transfer and release process, and the system is stable when its energy is minimum.

A study on the compatibility of the generalized Kudryashov method to determine wave solutions

In this article,we establish solitary wave solutions to the Estevez-MansfieldClarkson(EMC)equation and the coupled sine-Gordon equations which are model equations to analyze the formation of shapes in liquid drops,surfaces of negative constant curvature,etc.through contriving the generalized Kudryashov method.The extracted results introduce several types’solitary waves,such as the kink soliton,bell-shape soliton,compacton,singular soliton,peakon and other sort of soliton for distinct valuation of the unknown parameters.The achieved analytic solutions are interpreted in details and their 2D and 3D graphs are sketched.The obtained solutions and the physical structures explain the soliton phenomenon and reproduce the dynamic properties of the front of the travelling wave deformation generated in the dispersive media.It shows that the generalized Kudryashov method is powerful,compatible and might be used in further works to found novel solutions for other types of nonlinear evolution equations ascending in physical science and engineering.