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For finite dimensional Hamiltonian systems derived from 1+1 dimensional integrable systems, if they have Lax representations, then the Lax operator creates a set of conserved integrals. When these conserved integrals are in involution, it is believed quite popularly that there will be enough functionally independent ones among them to guarantee the Liouville integrability of the Hamiltonian systems, at least for those derived from physical problems. In this paper, we give a counterexample based on the U(2) principal chiral field. It is proved that the finite dimensional Hamiltonian systems derived from the U(2) principal chiral field are Liouville integrable. Moreover, their Lax operator gives a set of involutive conserved integrals, but they are not enough to guarantee the integrability of the Hamiltonian systems., Comment: LaTeX, 11pages

The Yang-Mills-Higgs-Bogomolny equations in both 2+1 dimensional Minkowski space-time and 2+1 dimensional anti-de Sitter space-time are known to be integrable and their soliton solutions have already been obtained. In this paper we show that there is a natural relation between the Lax pairs and soliton solutions in these two space-times when the curvature changes from 0 to -1. The change of the asymptotic behaviors of the solitons are also discussed., Comment: LaTeX, 11 pages, 8 figures

Binary symmetry constraints of the N-wave interaction equations in 1+1 and 2+1 dimensions are proposed to reduce the N-wave interaction equations into finite-dimensional Liouville integrable systems. A new involutive and functionally independent system of polynomial functions is generated from an arbitrary order square matrix Lax operator and used to show the Liouville integrability of the constrained flows of the N-wave interaction equations. The constraints on the potentials resulting from the symmetry constraints give rise to involutive solutions to the N-wave interaction equations, and thus the integrability by quadratures are shown for the N-wave interaction equations by the constrained flows., Comment: 41 pages, Latex

For the Davey-Stewartson I equation, which is an integrable equation in 1+2 dimensions, we have already found its Lax pair in 1+1 dimensional form by nonlinear constraints. This paper deals with the second nonlinearization of this 1+1 dimensional system to get three 1+0 dimensional Hamiltonian systems with a constraint of Neumann type. The full set of involutive conserved integrals is obtained and their functional independence is proved. Therefore, the Hamiltonian systems are completely integrable in Liouville sense. A periodic solution of the Davey-Stewartson I equation is obtained by solving these classical Hamiltonian systems as an example., Comment: 18 pages, LaTeX

The Davey-Stewartson I equation is a typical integrable equation in 2+1 dimensions. Its Lax system being essentially in 1+1 dimensional form has been found through nonlinearization from 2+1 dimensions to 1+1 dimensions. In the present paper, this essentially 1+1 dimensional Lax system is further nonlinearized into 1+0 dimensional Hamiltonian systems by taking the Bargmann constraints. It is shown that the resulting 1+0 dimensional Hamiltonian systems are completely integrable in Liouville sense by finding a full set of integrals of motion and proving their functional independence., Comment: 10 pages, in LaTeX, to be published in J. Phys. Soc. Jpn. 70 (2001)

The solutions of the Bogomolny equation in anti-de Sitter space-time are obtained by using Darboux transformations with both constant spectral parameters and variable "spectral parameters". These solutions give the Yang-Mills-Higgs fields in anti-de Sitter space-time. Some examples in SU(2) case are considered and qualitative asymptotic behaviors of the solutions as t tends to infinity are discussed in detail., Comment: LaTeX, 18 pages, 11 PS figures

Using the nonlinear constraint and Darboux transformation methods, the (m_1,...,m_N) localized solitons of the hyperbolic su(N) AKNS system are constructed. Here "hyperbolic su(N)" means that the first part of the Lax pair is F_y=JF_x+U(x,y,t)F where J is constant real diagonal and U^*=-U. When different solitons move in different velocities, each component U_{ij} of the solution U has at most m_i m_j peaks as t tends to infinity. This corresponds to the (M,N) solitons for the DSI equation. When all the solitons move in the same velocity, U_{ij} still has at most m_i m_j peaks if the phase differences are large enough., Comment: 15 pages, 5 figures, to appear in Inverse Problems

For the n-dimensional integrable system with a twisted so(p,q) reduction, Darboux transformations given by Darboux matrices of degree 2 are constructed explicitly. These Darboux transformations are applied to the local isometric immersion of space forms with flat normal bundle and linearly independent curvature normals to give the explicit expression of the position vector. Some examples are given from the trivial solutions and standard imbedding T^n\to R^{2n}., Comment: LaTeX, 21 pages, 5 Postscript figures, to appear in Inverse Problems (1998)