Nonlinear self-adjointness, conserved vectors, and traveling wave structures for the kinetics of phase separation dependent on ternary alloys in iron (Fe-Cr-Y (Y = Mo, Cu))

Abstract

The present exploration is concerned with fundamental elements corresponding to the phase decomposition in (Fe-Cr-Mo) and (Fe-Cr-Cu) ternary composites. For the ternary composites of iron, we examine the dynamical behavior of the phase separation. The dynamic of this separation is depicted by a model known as the Cahn-Hilliard equation. The nonlinear self-adjointness for the model under consideration is taken into account. The conserved quantities are calculated with the help of the direct method. For each symmetry generator, we have reduced the considered equation into non-linear ordinary differential equations (ODEs). Also, we have computed the optimal system of the equation under study to find the similarity reduction. Also, the traveling wave structures of the Cahn-Hilliard equation are obtained with the modified simple equation (MSE) technique. Moreover, solitary wave structures is exhibited graphically in the form of 3D, 2D and contour plots.