Soliton-Antisoliton Scattering Configurations in a Noncommutative Sigma Model in 2+1 Dimensions

Abstract

In this paper we study the noncommutative extension of a modified U (n) sigma model in 2 + 1 dimensions. Using the method of dressing transformations, an iterative approach for the construction of solutions from a given seed solution, we demonstrate the construction of multi-soliton and soliton-antisoliton configurations for general n. As illustrative examples we discuss U (3) solitons and consider the head-on collision of a U (2) soliton and an antisoliton explicitly, which will result in a 90 o angle scattering. Further we discuss the head-on collision of one U (2) soliton with two antisolitons. This results in a 60 o angle scattering.

Sigma-Model Solitons in the Noncommutative Plane: Construction and Stability Analysis

“…For diagonal BPS solutions (2.33) 13) this strategy turns out to be successful at all k ≥ 1 but breaks down at k = 0. We note that now J = {r−1}, which implies a distinction of cases: "very off-diagonal" perturbations have k > r, "slightly off-diagonal" perturbations occur for 1 ≤ k ≤ r, and diagonal perturbations mean k = 0, to be discussed last.…”

“…These solutions have been discussed as static solutions of a 2+1 dimensional noncommutative sigma model [5,12,13,14] which results from Moyal deforming the WZWmodified integrable sigma model [15,16,17]. Here we review the results pertaining to the static situation.…”

Abelian Solitons in the Noncommutative Ward Model

“…4) In fact, this procedure generates genuine scattering solutions in the nonabelian case but turns out to fail here. The encountered solutions instead exhibit a breather like behavior.…”

Noncommutative KP Hierarchy and Hirota Triple-Product Relations

“…Various methods, for instance, bicomplex method, [3][4][5][6][7][8] Lax-pair generating technique 9,10) (see also refs. 11 and 12), NC (Dbar-) dressing method, [13][14][15][16] NC zero curvature representation 17,19,20) and reductions of NC (anti-) SDYM equation, 9,21) have been proposed to construct NC soliton equations such as NC KdV, NC NLS and NC KP equations. Many interesting aspects have been reported including integrability in the sense that there exist an infinite number of conserved quantities or commutating flows, [6][7][8][9][10]17,19,20,22) formalism in the framework of the Sato theory, 22,23) Moyal deformation of dispersionless systems in the Lax operator approach 24,25) or in the classical R-matrix formalism, 26) and constructions of multi-soliton solutions.…”

“…Many interesting aspects have been reported including integrability in the sense that there exist an infinite number of conserved quantities or commutating flows, [6][7][8][9][10]17,19,20,22) formalism in the framework of the Sato theory, 22,23) Moyal deformation of dispersionless systems in the Lax operator approach 24,25) or in the classical R-matrix formalism, 26) and constructions of multi-soliton solutions. [13][14][15]18,27) As is well known, Hirota bilinear relations and -function of the Sato theory play an important role in the analysis of solutions and symmetries of the KP 28) and the mKP hierarchies. They also provide a consistent and coherent treatment for various properties of the KP and mKP hierarchies.…”