Decomposition of the (2+1)-dimensional Gardner equation and its quasi-periodic solutions

To decompose the (2 + I)-dimensional Gardner equation, an isospectral probl em and a corresponding hierarchy of (I + 1)-dimensional soliton equations a re proposed. The (2 + 1)-dimensional Gardner equation is separated into the first two non-trivial (I + I)-dimensional soliton systems in the hierarchy , and in turn into two new compatible Hamiltonian systems of ordinary diffe rential equations. Using the generating function flow method, the involutiv ity and the functional independence of the integrals are proved. The Abel-J acobi coordinates are introduced to straighten out the associated flows. Th e Riemann-Jacobi inversion problem is discussed, from which quasi-periodic solutions of the (2 + I)-dimensional Gardner equation are obtained by resor ting to the Riemann theta functions.