Propagation of diverse exact solitary wave solutions in separation phase of iron (Fe-Cr - X(X = Mo, Cu)) for the ternary alloys

In this paper, a specific consideration is paid to the nonlinear dynamics of solitary waves to kinetics of phase separation in iron (Fe-Cr - X(X = Mo, Cu)) based on ternary alloys. The convective-diffusive Cahn-Hilliard (CH) equation is used as a mathematical model to describe the dynamics of the separation phase for the ternary alloys of iron. A variety of solitary wave solutions with unknown parameters are extracted in different shapes like kink-type, bell-shape, shock-type, combine soliton, trigonometric, hyperbolic and Jacobi's elliptic function solutions with the assistant of recently computational tools, namely, extended Fan-sub equation method (EFSEM) and extended auxiliary equation method (EAEM). In addition, 3D, 2D, and their corresponding contour profiles of earned results are sketched in order to observe their dynamics with the choices of involved parameters. On the bases of achieved results, we may claim that the proposed computational methods are direct, dynamics, well organized, and will be useful for solving the more complicated nonlinear problems in diverse areas together with symbolic computations.

Automated summary

This paper focuses on the nonlinear dynamics of solitary waves in the phase separation kinetics of iron (Fe-Cr-X(X = Mo, Cu)) based on ternary alloys. The convective-diffusive Cahn-Hilliard equation is used as a mathematical model to describe the dynamics of phase separation. Various solitary wave solutions with unknown parameters are obtained using computational tools such as the extended Fan-sub equation method and extended auxiliary equation method. The results demonstrate that these computational methods are direct, dynamic, and well organized, making them useful for solving complex nonlinear problems in different areas.