Nonlocal ∂̄ formalism for the three-spatial-dimensions Kaup–Kuperschmidt equation with two temporal variables

Abstract

By complexifying the independent variables of the Kaup–Kuperschmidt (KK) equation, we derive the 4+2 integrable extension of the KK equation and its Lax pair. The construction of the associated nonlinear Fourier transform pair comprising both direct and inverse transforms is accomplished by conducting a spectral analysis of the $t$-independent part of the Lax pair. In the final section, the nonlinear Fourier transform pair will be used, after also taking into account the appropriate time evolution, for solving the Cauchy initial value problem of the three-spatial-dimensions KK equation with two temporal variables.