The Application of Neumann Type Systems for Solving Integrable Nonlinear Evolution Equations

An algorithm to obtain finite-gap solutions of integrable nonlinear evolution equations (INLEEs) is provided by using the Neumann type systems in the framework of algebraic geometry. From the nonlinearization of Lax pairs, some INLEEs in 1+1 and 2+1 dimensions are reduced into a class of new Neumann type systems separating the spatial and temporal variables of INLEEs over a symplectic submanifold (M, omega(2)). Based on the Lax representations of INLEEs, we deduce the Lax-Moser matrix for those Neumann type systems that yield the integrals of motion, elliptic variables, and a hyperelliptic curve of Riemann surface. Then, we attain the Liouville integrability for a hierarchy of Neumann type systems in view of a Lax equation on (M, omega(2)) and a set of quasi-Abel-Jacobi variables. We also specify the relationship between Neumann type systems and INLEEs, where the involutive solutions of Neumann type systems give rise to the finite parametric solutions of INLEEs and the Neumann map cuts out a finite dimensional invariant subspace for INLEEs. Under the Abel-Jacobi variables, the Neumann type flows, the 1+1, and 2+1 dimensional flows are integrated with Abel-Jacobi solutions; as a result, the finite-gap solutions expressed by Riemann theta functions for some 1+1 and 2+1 dimensional INLEEs are achieved through the Jacobi inversion with the aid of the Riemann theorem.