Exact solutions for the Cahn–Hilliard equation in terms of Weierstrass‑elliptic and Jacobi‑elliptic functions

Abstract

Despite the historical position of the F‑expansion method as a method for acquiring exact solutions to nonlinear partial differential equations (PDEs), this study highlights its superiority over alternative auxiliary equation methods. The efficacy of this method is demonstrated through its application to solve the convective–diffusive Cahn–Hilliard (cdCH) equation, describing the dynamic of the separation phase for ternary iron alloys (Fe–Cr–Mo) and (Fe–X–Cu). Significantly, this research introduces an extensive collection of exact solutions by the auxiliary equation, comprising fifty‑ two distinct types. Six of these are associated with Weierstrass‑elliptic function solutions, while the remaining solutions are expressed in Jacobi‑elliptic functions. I think it is important to emphasize that, exercising caution regarding the statement of the term ’new,’ the solutions presented in this context are not entirely unprecedented. The paper examines numerous examples to substantiate this perspective. Furthermore, the study broadens its scope to include soliton‑like and trigonometric‑ function solutions as special cases. This underscores that the antecedently obtained outcomes through the recently specific cases encompassed within the more comprehensive scope of the present findings.