Solutions of Cauchy Problems for the Caudrey–Dodd–Gibbon–Kotera–Sawada Equation in Three Spatial and Two Temporal Dimensions

A.S. Fokas has obtained integrable nonlinear partial differential equations (PDEs) in 4 + 2 dimensions by complexifying the independent variables. In this work, the complexification of the independent variables of the 2 + 1-dimensional Caudrey–Dodd–Gibbon–Kotera–Sawada (CDGKS) equation yields the 4 + 2 integrable extension of the CDGKS equation. Then, by transforming two temporal variables, the CDGKS equation in three dimensions is reduced, and the Lax pairs of the corresponding equations are given. Finally, the solutions of Cauchy problems for the CDGKS equation in three spatial and two temporal dimensions are constructed by introducing a novel nonlocal d-bar formalism, in which several new long derivative operators, Dx, Dy, and Dt, are constructed for the study of the initial value problem for the CDGKS equation. Some significant propositions and results are presented in this paper.