Moyal noncommutative integrability and the Burgers–KdV mapping

Abstract

The Moyal momentum algebra, studied in [20,21], is once again used to discuss some important aspects of NC integrable models and 2d conformal field theories. Among the results presented, we setup algebraic structures and makes useful convention notations leading to extract non trivial properties of the Moyal momentum algebra. We study also the Lax pair building mechanism for particular examples namely, the noncommutative KdV and Burgers systems. We show in a crucial step that these two systems are mapped to each others through the following crucialx . This makes a strong constraint on the NC Burgers system which corresponds to linearizing its associated differential equation. From the CFT's point of view, this constraint equation is nothing but the analogue of the conservation law of the conformal current. We believe that the considered mapping might help to bring new insights towards understanding the integrability of noncommutative 2d-systems.

The Hirota–Satsuma coupled KdV hierarchy on noncommutative phase-space

“…In commutative case, it is well known that the existence of Lax pair operators (L, T ) for nonlinear differential equations is a strong indication of integrability, that is, the existence of a complete set of conservation laws, of multisoliton solutions and so on. In this context, many efforts have been devoted to extend the Lax pair operators for several (1 + 1) and (2 + 1) -dimensional integrable systems on noncommutative spaces [24][25][26][27][28].…”

“…On the other hand, the multiplication of two arbitrary functions f (x, p) and g(x, p) on the two dimensional NC phase-space has been shown to satisfy the star product law defined as follows: [20][21][22][23][24][25] f…”

“…. In this context, on NC spaces we define the Moyal bracket given by the following expression [20][21][22][23][24][25]:…”

“…The Lax representation of this system has been constructed in [3] by Dodd and Fordy. Different properties of the HS-coupled KdV system have been constructed such as Backlund transformation [4], Darboux transformation [5][6][7], bilinear form [1,8], and Painleve property [9,10] More recently, the noncommutative geometry (NCG) [11,12] has attracted attention from many researchers in physics [13][14][15][16][17][18][19][20][21][22][23][24][25]. On noncommutative (NC) spaces, the noncommutativity of the coordinates is defined as follows…”