Two (2Â +Â 1)-dimensional soliton equations and their quasi-periodic solutions

Two (2 + 1)-dimensional soliton equations and their decomposition into the mixed (1 + 1)-dimensional soliton equations are proposed. With the help of nonlinearization approach, the Lenard spectral problem related to the mixed soliton hierarchy is turned into a completely integrable Hamiltonian system with a Lie-Poisson structure on the Poisson manifold R3N. The Abel-Jacobi coordinates are introduced to straighten out the Hamiltonian flows. Based on the decomposition and the theory of algebra curve, the explicit quasi-periodic solutions for the (1 + 1)- and (2 + 1)-dimensional soliton equations are obtained.