Factorization methods for noncommutative KP and Toda hierarchy

Abstract

We consider the solution space of the noncommutative KP hierarchy with the noncommutative extension of the Sato theory. The noncommutative deformation of the Toda hierarchy is introduced. We derive the bilinear identities for the BakerAkhiezer functions and calculate the N -soliton solutions of the noncommutative Toda hierarchy. Noncommutative KP and Toda hierarchyRecently, the classical and quantum field theory over the noncommutative (NC) space times is extensively studied. The NC gauge theory has a great success in the string theory. In particular, the NC deformation of the ADHM construction of the (anti)SDYM equation [9] and the Nahm construction [1] were shown. It seems to imply the significance of the NC deformation of the integrable systems. Some authors have discussed the NC integrable systems on lower dimension [2,3,6,13,17] (See also [5,15,16]), especially the NC KP hierarchy. In this letter, we study the moduli space of the NC KP hierarchy throughout the Birkhoff decomposition of a certain formal group over a NC algebra. We also introduce the NC Toda hierarchy and derive the bilinear identities and the N -soliton solutions.Let (t 1 ,t 2 , · · · ) be coordinates of NC plane R 2∞ which satisfy [t i ,t j ] = iθ ij , and A θ a set of functions on it. The deformation parameters θ ij ∈ R are non-zero constants and we assume that a matrix (θ ij ) is invertible. By an orthogonal change of coordinates as [t 2n−1 ,t 2n ] = iθ n (n ≥ 1), the algebra is realized as operators over the Fock spaceThe NC KP hierarchy of the operator form is defined as follows [2,3,17]. We consider an operator valued monic pseudo differential operator (PDO)

Quasideterminant solutions of a non-Abelian Toda lattice and kink solutions of a matrix sine-Gordon equation

“…There has been great interest in non-commutative versions of some well-known soliton equations, such as the KP equation, the KdV equation and the HirotaMiwa equation (Kupershmidt 2000;Paniak 2001;Hamanaka 2003;Hamanaka & Toda 2003;Wang & Wadati 2003a,b;Sakakibara 2004;Wang et al 2004;Dimakis & Müller-Hoissen 2005;Nimmo 2006;. Often, these non-commutative versions are obtained simply by removing the assumption that the coefficients in the Lax pair of the commutative equation commute.…”

Bright and dark pulses in optical fibres in the vicinity of the zero-dispersion wavelength

“…An obvious generalization of the above Sato-Wilson equations is given by whereξ = n 1 t n ∂ n . Following [45] (see also [7,46]), this leads to the Birkhoff factorization (generalized Riemann-Hilbert problem, see [25,47] for example)…”

“…Hence, the Birkhoff factorization (8.50) is equivalent to the XncKP hierarchy equations(8.8). Via (8.50) the space of solutions of the XncKP hierarchy is determined from the same initial data W (0) as in the KP case[46].…”

A Non-Commutative Semi-Discrete Toda Equation and Its Quasi-Determinant Solutions

“…Recently, there has been a lot of interest in non-commutative versions of some well-known soliton equations, such as the KP equation, the KdV equation, the Hirota-Miwa equation, the modified KP equation and the twodimensional Toda lattice [1,[5][6][7][8][9]12,14,16,18,19,[21][22][23]. There are a number of reasons for this lack of commutativity.…”