  We consider the Kadomtsev-Petviashvili II (KP) model placed in $\mathbb R_t
\times \mathbb R_{x,y}^2$, in the case of smooth data that are not necessarily
in a Sobolev space. In this paper, the subclass of smooth solutions we study is
of ``soliton type'', characterized by a phase $\Theta=\Theta(t,x,y)$ and a
unidimensional profile $F$. In particular, every classical KP soliton and
multi-soliton falls into this category with suitable $\Theta$ and $F$. We
establish concrete characterizations of KP solitons by means of a natural set
of nonlinear differential equations and inclusions of functionals of Wronskian,
Airy and Heat types, among others. These functional equations only depend on
the new variables $\Theta$ and $F$. A distinct characteristic of this set of
functionals is its special and rigid structure tailored to the considered
soliton. By analyzing $\Theta$ and $F$, we establish the uniqueness of
line-solitons, multi-solitons, and other degenerate solutions among a large
class of KP solutions. Our results are also valid for other 2D dispersive
models such as the quadratic and cubic Zakharov-Kuznetsov equations.
