Nonlocal ∂‾ Formalism for the (2+1)‐Dimensional Combined Kdv‐Mkdv Equation

ABSTRACT The content of this paper includes two parts: (i) We elaborate on a new methodology, which starting with an integrable evolution equation in two spatial dimensions, constructs a 4+2 integrable generalization of the combined KdV‐mKdV system of two equations involving two scalar functions by complexifying the independent variables. Through a series of transformations, the (4+2)‐dimensional equation can be reduced to two three‐spatial‐dimensions equations. The integrability of these equations is guaranteed, as their Lax pairs are provided. By conducting a spectral analysis of the ‐independent part of the Lax pair, the nonlinear Fourier transform pair comprising both direct and inverse transforms is worked out. With the help of the nonlocal formalism, the solution to the Cauchy problem for the three‐spatial‐dimensions combined KdV‐mKdV equation with two temporal variables is formulated and presented in the final section. (ii) A novel category of exact solutions of the combined KdV‐mKdV system are constructed exclusively by making use of new ‐dressing method. In Section 3, as a particular cases of solutions with functional parameters, the line soliton and rational solutions with a dependence of kernel on space and time variables are calculated.