  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.
