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Soliton formulation by Moyal algebra

We formulate the soliton equations on the lattice in terms of the reduced Moyal algebra which includes one parameter. The vanishing limit of the parameter leads to the continuous soliton equations.

It is well-known that the soliton equations are expressed as a zero-curvature equation emerging as a compatibility condition of the scattering problem equation and the time evolution equation in the inverse scattering problem. The potentials are sl(N, C) valued. Many of the familiar soliton equations fall into N = 2 case. In this letter we shall consider a formulation of soliton equation by using the Moyal algebra [1] which is identical to su(N) algebra for some value of parameter in it [2,3,4]. In order to realize this, we replace the commutation relation in the zero-curvature equation by the Moyal algebra, which introduces two new variables into the equations. We assume that the potentials are expanded by powers of e p where the variable p is one of the newly introduced variables. Specification of the expansion determines the type of soliton equation. Substituting the potentials into the zero-curvature equation, we obtain various soliton equations. Taking out coefficients at each order of power of e p in the zero-curvature equation, we obtain soliton equations on the lattice, one of which is the well-known Toda lattice equation. In general, the equations thus obtained include one parameter originating from the Moyal algebra. It is interesting to note that the parameter of the Moyal bracket has a physical meaning in the soliton equation as spacing between particles on the lattice [5]. We then naturally obtain the continuous correspondence of the discrete soliton equations by taking the vanishing limit of the parameter.
We first recapitulate the notations of the Moyal algebra following Strachan [6]. Define the star product by and the brackets by where x = (x 0 , x 1 ) andx = (x 0 ,x 1 ). Here the "−" bracket is the Moyal bracket. We also introduce D x operator by The D x is the Hirota operator. So far we have not yet assumed any x dependence of the functions in the Moyal algebra. Here we assume that the functions are the product of a function of x and the power of e p . It is interesting to note that the star product for those functions are related to the Hirota operator; In more general cases with other e p dependences, we have where m and n are integers. The brackets are explicitly obtained as We can naturally obtain the κ → ∞ limit of the brackets; lim κ→0 κ{e mp f, e np g} + = 2f g.
We shall next consider the soliton equation. Of the various formulations of soliton theory, we consider the inverse scattering method where the soliton equation is obtained as a compatibility condition of the scattering problem equation and time evolution equation. The soliton equation has a geometrical meaning as the zero-curvature equation: where the potentials A µ = A µ (x 0 , x 1 )(µ = 0, 1) are sl(N, C) valued fields and include a spectral parameter in the soliton theory. The expansion of potentials in terms of the spectral parameter implies the equations at each order of the parameter which leads to various soliton equations.
We shall discuss the Moyal version of the above equation. Our discussion is based on the fact that the Moyal algebra and the su(N) algebra are identical by some assumption on the parameter in the Moyal algebra [2,3,4]. This means that we might replace the commutation relation by the Moyal algebra. The equation reads In general, the replacement of the commutation relation by the Moyal bracket introduces two new variables, say, x and p and so transfers the original equation to the 2 dimensional higher equation in compensation of the loss of matrix. Here we assume that the functions constituting the above equation (10) are written as the product of function of induced variable x and a function of another induced variable p as Here k and l run over positive and/or non-positive integers of which the choice determines the soliton equation. The specification of the range of k and l together with the coefficients leads to various soliton equations. The expansions are to be compared with the inverse scattering method where the potentials are expanded in terms of spectral parameter. Here e p plays the role of spectral parameter. The zero-curvature equation (10) implies a set of equations taken out at each order of power of e p . Therefore, in the transition to the Moyal version of soliton equation, we do not add two dimensions but one dimension to the original equation.
In our previous paper [5], we obtained the algebra composed of brackets between operators which are the exponentiation of the operators satisfying conformal algebra. The brackets are found to be identical to those derived from the Moyal algebra by the reduction assuming that functions of x and p are expanded by e 0 and e ±p . The restriction of functions to those e p dependences implies the limitation on the potentials that they should take value only on the Cartan and simple algebra. In the following examples (i) and (ii) fall into this case. But we can allow for other e p dependences. The example could be found in (iii).
We shall put an ansatz that A µ = A µ (r; x, p) or A µ = A µ (t; x, p) where x 0 = 0 and x 1 = r just to identify our examples with the well-known equations. This leads to either or respectively. We shall consider several examples below.
By using (6), these are These equations lead to where ρ(t; x) = −2 log a(t; x). On the other hand, it is known that the κ → 0 limit in the Moyal algebra leads to the Poisson bracket. In identifying the Moyal algebra with su(N) algebra one sets the relation κ ∝ 1 N , which means that finite κ corresponds to finite N i.e.finite number of particles. This shows that the Moyal version of soliton equation derives the soliton equation with periodic boundary condition of which the algebraic specification is A n,1 . We can take κ → 0 limit of Eqs.(17) and (18) by using Eq. (7); ∂ ∂t a(t; x)e ±p = 2a ∂b ∂x e ±p .
We thus obtain the continuous counterpart of the Toda lattice equation quite naturally: (ii)Bogomolny equation [7,8,9] We assume that The Eq.(13) implies [6] e 0 dψ dr Equivalently, these are written as These lead to where By taking κ → 0 limit in the Moyal brackets, we can rewrite the Eqs.(28)-(30) by using the derivative operators instead of the differences. We thus obtain the continuous limit of the Bogomolny equation [5,10,11] as which can be interpreted as su(∞) Bogomolny equation.

(iii)KM equation
This example is the case where the expansion of potentials include e ±2p terms besides e ±p terms, which means that these expansion does not have an algebra counterpart taking value only on the Cartan sub algebra and simple roots in contrast with the previous examples. Assuming that This can be written as where This is the equation discussed by Kac and van Maerbeke(KM) [12]. We can also obtain the continuous limit; In this letter we showed the formulation of soliton equations in terms of the Moyal algebra and exemplified it by well-known equations. The specification how the potentials are expanded in terms of e p leads to the various soliton equations on one-dimensional link lattice of which the algebraic specification is A n,1 . We found that, in the present formulation the role of e p is the spectral parameter in the soliton equation in the inverse scattering method, and the parameter κ in the Moyal algebra has a physical meaning of spacing on the lattice. In KM equation we have an anti-commutation bracket expression for one of the potentials, which might be related to the super-symmetric formulation of the Moyal version of zero-curvature equation. We shall leave this for the following work.
The author would like to express his gratitude to Prof.S.Saito for discussion.