The monograph explicitly computes the Hitchin

integrable system on the moduli space of Higgs bundles, compares the Hitchin Hamiltonians with those found by van Geemen-Previato, and prove the transversality of the induced flow with the locus of unstable bundles.

For a two-dimensional quantum

integrable system with Hamiltonian H, there is always one operator like [A.sub.1] which commutes with Hamiltonian of the system, that is, [H, [A.sub.1]] = 0.

As a result, the nonlinear ODEs presented by the new units of variables turns out to be a completely

integrable system; that is, the ODEs can be solved explicitly, at least in principle.

Later, many researchers have studied the Rossby wave equation in many aspects [7, 8], such as

integrable system [9, 10], the integrable coupling of equations [11], and Hamiltonian structures [12].

By the theory of planar dynamical systems, we know that for an equilibrium point of a planar

integrable system, if J < 0, then the equilibrium point is a saddle point; if J > 0 and Trace (M([w.sub.e])) = 0, then it is a center point; if J > 0 and [(Trace(M([w.sub.e],[z.sub.e]))).sup.2] - 4J([w.sub.e],[z.sub.e]) > 0, then it is a node; if J = 0 and the index of the equilibrium point is 0, then it is a cusp; otherwise, it is a high order equilibrium point.

It is well known that BS equation is the reduction of the self-dual Yang-Mills equation; it is an

integrable system and has an infinite number of conservation laws and N-soliton solutions [30].

An

integrable system on a symplectic manifold (M, [omega]) of dimension 2N is a set of N functions which are functionally independent and mutually Poisson-commutative.

Therefore our method is a pure algebraic algorithm which can be applied to

integrable system and non-integrable system.

Is it possible to formulate a new effective numerical algorithm in terms of some discrete-time

integrable system? The answer is yes.

So the

integrable system (4) defines a Backlund transformation v [??]v' for the potential KdV equation (3), and it also gives a Backlund transformation u [??] u' for the KdV equation (1) which is defined by

As we all know, the generation of

integrable system, determination of exact solution, and the properties of the conservation laws are becoming more and more rich [1-5]; in particular, the discrete

integrable systems have many applications in statistical physics, quantum physics, and mathematical physics [6-11].

The KP equation is a worldwide integrable structure in two spatial dimensions in the similar line of attack that the KdV equation can be looked upon as a widespread

integrable system in one spatial dimension, since many other

integrable systems can be obtained as reductions [5].