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 , and Hamiltonian structures .
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 .
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 .