Two (2 + 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.