The $\bar{\partial}$-dressing Method for Two (2+1)-dimensional Equations and Combinatorics

Abstract: The soliton solutions of two different (2+1)-dimensional equations with the third order and second order spatial spectral problems are studied by the $\bar{{\partial}}$-dressing method. The 3-reduced and 5-reduced equations of CKP equation are obtained for the first time by the binary Bell polynomial approach. The $\bar{{\partial}}$-dressing method is a very important method to explore the solution of a nonlinear soliton equation without analytical regions. Two different types of equations, the (2+1)-dimensional Kaup-Kuperschmidt equation named CKP equation and a generalized (2+1)-dimensional nonlinear wave equation, are studied by analyzing the eigenfunctions and Green's functions of their Lax representations as well as the inverse spectral transformations, then the new $\bar{{\partial}}$ problems are deduced to construct soliton solutions by choosing proper spectral transformations. Furthermore, once the time evolutions of the spectral data are determined, we will be able to completely obtain the solutions formally of the CKP equation and the generalized (2+1)-dimensional nonlinear wave equation. The reduced problem of the (2+1)-dimensional Kaup-Kuperschmidt is discussed. The main method is binary Bell polynomials related about the $\mathcal{Y}$-constraints and $P$-conditions which turns out to be an effective tool to represent the bilinear form. On the basis of this method, the bilinear form is determined under the scaling transformation. Starting from the bilinear representation, the CKP equation is reduced to Kaup-Kuperschmidt~equation and bidirectional Kaup-Kuperschmidt equation under the reduction of $t_3$ and $t_5$, respectively.