Finite-gap solutions of 2+1 dimensional integrable nonlinear evolution equations generated by the Neumann systems

Abstract

Each soliton equation in the Korteweg–de Vries (KdV) hierarchy, the 2+1 dimensional breaking soliton equation, and the 2+1 dimensional Caudrey–Dodd–Gibbon–Kotera–Sawada (CDGKS) equation are reduced to two or three Neumann systems on the tangent bundle TSN−1 of the unit sphere SN−1. The Lax–Moser matrix for the Neumann systems of degree N−1 is deduced in view of the Mckean–Trubowitz identity and a bilinear generating function, whose favorite characteristic accounts for the problem of the genus of Riemann surface matching to the number of elliptic variables. From the Lax–Moser matrix, the constrained Hamiltonians in the sense of Dirac–Poisson bracket for all the Neumann systems are written down in a uniform recursively determined by integrals of motion. The involution of integrals of motion and constrained Hamiltonians is completed on TSN−1 by using a Lax equation and their functional independence is displayed over a dense open subset of TSN−1 by a direct calculation, which contribute to the Liouville integrability of a family of Neumann systems in a new systematical way. We also construct the hyperelliptic curve of Riemann surface and the Abel map straightening out the restricted Neumann flows that naturally leads to the Jacobi inversion problem on the Jacobian with the aid of the holomorphic differentials, from which some finite-gap solutions expressed by Riemann theta functions for the 2+1 dimensional breaking soliton equation, the 2+1 dimensional CDGKS equation, the KdV, and the fifth-order KdV equations are presented by means of the Riemann theorem.