  In this paper, we propose a class of adaptive multiresolution (also called
adaptive sparse grid) ultra-weak discontinuous Galerkin (UWDG) methods for
solving some nonlinear dispersive wave equations including the Korteweg-de
Vries (KdV) equation and its two dimensional generalization, the
Zakharov-Kuznetsov (ZK) equation. The UWDG formulation, which relies on
repeated integration by parts, was proposed for KdV equation in
\cite{cheng2008discontinuous}. For the ZK equation which contains mixed
derivative terms, we develop a new UWDG formulation. The $L^2$ stability and
the optimal error estimate with a novel local projection are established for
this new scheme on regular meshes. Adaptivity is achieved based on
multiresolution and is particularly effective for capturing solitary wave
structures. Various numerical examples are presented to demonstrate the
accuracy and capability of our methods.
