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

Abstract

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