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

Abstract

To decompose the (2 + 1)-dimensional Gardner equation, an isospectralproblem and a corresponding hierarchy of (1 + 1)-dimensional solitonequations are proposed. The (2 + 1)-dimensional Gardner equation isseparated into the first two non-trivial (1 + 1)-dimensional solitonsystems in the hierarchy, and in turn into two new compatible Hamiltoniansystems of ordinary differential equations. Using the generating functionflow method, the involutivity and the functional independence of theintegrals are proved. The Abel-Jacobi coordinates are introduced tostraighten out the associated flows. The Riemann-Jacobi inversionproblem is discussed, from which quasi-periodic solutions of the (2 + 1)-dimensional Gardner equation are obtained by resorting to theRiemann theta functions.