Family Of 1 +1 Dimensional Soliton Equations And Their Quasi-periodic Solutions

A 3×3 spectral problem is proposed, from which a hierarchy of 1+1 dimensional soliton equations is derived. With the help of nonlinearization approach, the soliton systems in the hierarchy are decomposed into two new compatible Hamiltonian systems of ordinary differential equations with a Lie-Poisson structure on the Poisson manifold R^3N . The generating function flow method is used to prove the involutivity and the functional independence of the conserved integrals. The Abel-Jacobi coordinates are introduced to straighten out the associated flows. Using the Riemann-Jacobi inversion technique, the explicit quasi-periodic solutions for the 1+1 dimensional soliton equations are obtained.