Notes on the Boussinesq integrable hierarchy
O. Dafounansou1, D. C. Mbah2, A. Boulahoual3, M. B. Sedra3, 4
1Department of Physics, Faculty of Science, Douala University, Douala, Cameroun
2CEPAMOQ, Douala University, Douala, Cameroun
3LHESIR, Faculty of Science of Kenitra, Ibn Toufail University, Kenitra, Morocco
4ENSAH, Mohammed First University, Al Hoceima, Morocco
To cite this article:
O. Dafounansou, D. C. Mbah, A. Boulahoual, M. B. Sedra. Notes on the Boussinesq Integrable Hierarchy. International Journal of Sustainable and Green Energy. Special Issue: Wind-Generated Waves, 2D Integrable KdV Hierarchies and Solitons. Vol. 4, No. 3-2, 2014, pp. 17-22. doi: 10.11648/j.ijrse.s.2015040302.14
Abstract: This work is dedicated to some notes on the Moyal momentum algebras applied to the Boussinesq integrable hierarchy. Starting from a brief review of the Moyal momentum algebra structures, we establish in detail the Non-commutative Boussinesq hierarchy by using the Lax pair Generating Technique. Then we shows that these equations can be obtained as 3-reduction of Non-commutative KP hierarchy in a similarly form via some conformal realizations.
The origin of integrable system dates back to the 19th century with the KdV equation, which describe the long solitary wave in the shallow water . Since the study of integrability of nonlinear system, has taken more consideration . For such systems integrability means the existence of an infinite number of conserved quantities in involutions. A definition given by Ward is that such system, more precisely few of them, can be derived from the anti-self-dual Yang Mills equations by reduction with gauge groups [3,4].
These studies yield exacts solutions in many problem in theoretical high energy physics and mathematics. It appears that the geometry of integrable system is crucial for understanding many aspects of field theories . E. Witten has conjectured that the energy of 2-dimensional gravity coincide with the Tau-function of KdV hierarchy .In addition the integrable systems can be linked to the infinite-dimensional conformal algebra and its extensions. From their Poisson bracket structure it turns out that Boussinesq and KP hierarchy are respectively isomorphic to and algebra. In current days, there are deep interest in the non-commutative aspect of different soliton equations [7, 8, 9], with successful applications to string theories ... It appears that the Moyal momentum algebra via its momentum Lax operators provides an interesting tools for the study of deformed integrable systems.
We will study integrable systems of (1 + 1) and (2 + 1) dimensional evolution equation namely the Boussinesq and KP equation respectively. We starts with some basic properties of the Moyal⋆ Product, introducing the Moyal Momentum Algebra . Then we adopt the Lax Pair Generating Technique to study the evolution equations of Non-commutative Boussinesq hierarchy. By the way we establish the Non-commutative KP hierarchy before discussing the 3-reduction of NC KP hierarchy and the link with the previous Boussinesq hierarchy.
2. Moyal Product ⋆ and Operators Algebra
Our formulation will be based on star product ⋆ called the Moyal product. Given a smooth manifold M with coordinates system. This manifold will be endowed with the skew-symmetric bilinear bracket defined on by [11, 12]:
verifies the Jacobi identity, if is a non-degenerate skew-symmetric matrix, hence M is symplectic manifold with and even dimension. We consider extend tensorial manifold with ) denoting the extra coordinates system of T. The Moyal product will not affect t and it is given by [2, 13]:
Expanding this equation we find:
The Moyal bracket is defined as follow:
If we consider the 2d-phase space , with coordinate, the matrix becomes:
hence expression (3) can be written as :
and the Moyal bracket :
The point of introducing the above properties is to define the Moyal momentum algebra. The Moyal momentum was introduced first by authors , and systematically studied later with some applications to conformal field theory and -deformed integrable models . This algebra is a pseudo momentum operators algebra denoted by .
consist of the object of the form where is polynomial in momentum The Moyal momentum algebra is isomorphic to the ordinary pseudo differential operator The construction of consist of replacing the ordinary pseudo differential lax operators by the Lax momentum operators:
is a function of ordinary spin living in a non-commutative space parameterized by . The conformal dimensions are given as follow:
can be decomposed as:
where denotes the space of momentum lax operators of conformal spin and degrees start from to :
involving zero value term belong to the space
is the space of operators of degree denoting function coefficient of conformal spin :
The Moyal bracket of two operators gives rise to an operator . To perform all the forthcoming calculations, the formulae (3) will be use its more simplified way. This has be done in several papers [8, 15].
3. Moyal Boussinesq Hierarchy
The moyal hierarchy is defined by the lax equation 
It follows that the coefficient in order vanishes, we have the special form of called hierarchy.
is the root of L. Thus the -Boussinesq moyal momentum Lax operator we will deal with is :
The explicit expression of and the straightforward calculations gives the Boussinesq hierrarchy. This has been done by many authors [8, 15].
Instead of the above approach in this section, we will adopt the Lax pair generating technique to determine the Non-commutative Boussinesq hierarchy . Briery, the Lax pair generating technique consist of finding for a given , the operator T such that:
The equation (19) is equivalent to (15) where T is the analogue of [4, 8]. This technique is based on the following ansatz:
with and .
where is a monome of momentum operator. Actually the clue of the problem is to determine the expression of the operator ; keeping in mind that T and have the same degree.
• The flow :
The equation (19) leads to trivial equations with :
where we denote by and We can obtain the ordinary form of the Boussinesq hierarchy via the correspondence
• The flow :
We consider the ansatz:
Considering the differential part of , we get:
By identifying with:
one gets :
taking , then:
Substitute A in:
We recognize the pair of equations (27) and (28) is nothing but the non-commutative Boussinesq equation.
• The ow
Here we consider the following ansatz:
where coefficients of polynome in p belong to . To find the Lax pair of equation
we start by calculating the following terms: and :
Then by identifying the order 4, 3, 2 in , we obtain:
With these values, the identification in order 1 leads to:
Finally, the term of the order fields:
Equations (34) and (35) correspond to the evolution equations of the non-commutative Boussinesq hierarchy.
4. Moyal KP Hierarchy
In this section, we drop the time derivation we start with a more familiar notations similar to Lax representation for a hierarchy in Sato's framework. We consider the KP Lax operator:
Then the non-commutative KP evolution equations take the lax form:
We use the later to determine KP hierarchy in a simpler way by using the Moyal ⋆ product and recover the hierarchy similar to the one found by using the supershmidt-Manin ⋆ product, just by a conformal realization of fields . It turns out that the KP hierarchy consists of an infinite set of differential equation for each time [13, 17].
** The flow :
** The flow :
we keep the terms in and then we get the following equations:
by a conformal realization the field is expressed in term of :
we find the previous hierarchy in the following form:
** The flow :
keeping the term up to , we find:
using the conformal realization (41), we get:
It appears that if one takes the first two equations of (42) and eliminating and in the fisrt equation of (44) we get the non-commutative KP equation where and .
5. Boussinesq Hierarchy as 3-Reduction of Moyal KP Hierarchy
This approach pictures the link between the KP Lax operator and others integrables models. Let's rewrite the KP Lax operator or in the form . For Boussinesq equation, we denote the Lax operators by .Then the Boussinesq hierarchy obtained by 3-reduction is given by the following Lax equation.
Where with the contrain . The flows are trivial.
For The flows we have:
and the contrain we find:
The term yields:
Finally with the lax equation (47) we obtain:
it turns out that the time derivation of equation (50) yields:
Therfore we get the Non-commutative Boussinesq equation:
Taking classical limit we obtain the Boussinesq equation in the ordinary form . Notice that the map (41) doesn't change equation (52).
The flows are given as follow: we start by calculating:
The condition yields :
then the equation :
gives rise to:
Notice that the presence of the term in the last equation doesn't matter, since when applying the conformal map (41) to the terms that coming from the KP Lax operator we recover the following equations.
We have presented by two different methods how to obtain the deformed Boussinesq hierarchy. Of course there are several versions of theory and each has its advantages and flaws. In this work, the results found in the first method show the consistency of Lax pair generating technique. Where by rescaling time derivation we recover the ordinary form of Boussinesq hierarchy. We also got a look to the KP hierarchy which has been simplified by using a conformal realization that shows the equivalence between the Moyal ⋆ product and the Kupershmidt-Manin ⋆ product. We have also shown that the Boussinesq hierarchy obtained by the 3-reduction of KP hierarchy using the same conformal map gives rise to equations similar to that obtained by Lax Pair Generating Technique. We hope our discussion will make the Moyal momentum be more accessible in the study of some integrable models.