Neumann type integrable reduction for nonlinear evolution equations in 1+1 and 2+1 dimensions

Abstract

A family of new Neumann type systems is given in view of the nonlinearization technique, realizing the variable separation of the modified Jaulent–Miodek hierarchy and a new coupled modified Kadomtsev–Petviashvili equation on the symplectic submanifold TSN−1. By two Casimir functions and a special solution of the Lenard eigenvalue equation, we deduce the Lax–Moser matrix of the Neumann type systems that yields integrals of motion and the constrained Hamiltonians whose vector fields are tangent to TSN−1. Based on the Dirac–Poisson bracket and a Lax equation on TSN−1, a new systematic way is proposed to prove the Liouville integrability of a family of Neumann type systems synchronously. The Dirac–Poisson bracket and the generating function are used to reveal the explicit relation between infinite dimensional integrable systems and Neumann type systems, and then we point out that compatible solutions of Neumann type systems yield the finite parametric solutions of 1+1 and 2+1 dimensional integrable nonlinear evolution equations and the finite-gap solutions of the Novikov equation.