Finite Genus Solutions Of 2+1 Dimensional Integrable Equations

Two KN spectral problems (positive , negative) and a discrete spectral problem are investigated in this thesis. Our discussion is mainly focus on the integrable decomposition of the soliton equations derived from the spectral problems. Soliton equations of the KN isospectral hierarchy, such as dNS, dmKdV, and 2+1mKP are obtained. Meanwhile, equations with a discrete variable are presented, including 1+1 dToda, 2+1 dToda. Furthermore, all equations are given finite parameter solutions, and finite genus solutions of two 2+1 dimensional equations are gotten.Based on the fundamental identity, the eigenvalue problem is nonlinearized through the Lax pair nonlinearization technique, Bargmann mapping, symplectic mapping and Moser matrix are obtained. Under the help of power series, conserved integrables are obtained, then continuous and discrete flows are straightened, the involutivity and independence of conserved integrables are proved, thus the nonlinear systems we get are integrable. And the nonlinear evolutionary equations are decomposed, their Abel-Jacobi solutions are obtained. In the end, finite genus solutions of 2+1 dimensional soliton equations are given.