  Consider the generalized Korteweg-de Vries (gKdV) equations with power
nonlinearities $q=2,3,4\ldots$ in dimension $N=1$, and the Zakharov-Kuznetsov
(ZK) model with integer power nonlinearities $q$ in higher dimensions $N\geq
2$. Among these power-type models, the only conjectured equation with space
localized time periodic breathers is the modified KdV (mKdV), corresponding to
the case $q=3$ and $N=1$. Quasimonochromatic solutions were introduced by
Mandel to show that sine-Gordon is the only scalar field model with breather
solutions among this class. In this paper we consider smooth generalized
quasimonochromatic solutions of arbitrary size for gKdV and ZK models and
provide a rigorous proof that mKdV is the unique power-like model among them
with spatially localized breathers of this type. In particular, we show the
nonexistence of breathers of this class in the ZK models. The method of proof
involves the use of the naturally coherent algebra of Bell's polynomials to
obtain particularly distinctive structural elliptic PDEs satisfied by
breather-like quasimonochromatic solutions. A reduction of the problem to the
classification of solutions of these elliptic PDEs in the entire space is
performed, and de Giorgi type uniqueness results are proved in this particular
case, concluding the uniqueness of the mKdV breather, and the nonexistence of
localized smooth breathers in the ZK case. No assumption on well-posedness is
made, and the power of the nonlinearity is arbitrary.
