Classical Lie symmetries and reductions of a nonisospectral Lax pair

The classical Lie method is applied to a nonisospectral problem associated with a system of partial differential equations in 2+1 dimensions (Maccari A, J. Math. Phys. 39, (1998), 6547-6551). Identification of the classical Lie symmetries provides a set of reductions that give rise to different nontrivial spectral problems in 1+1 dimensions. The form in which the spectral parameter of the 1+1 Lax pair is introduced is carefully described.Comment: 11 pages (v2: A typo corrected in the authors' names

About the self-dual Chern-Simons system and Toda field theories on the noncommutative plane

The relation of the noncommutative self-dual Chern-Simons (NCSDCS) system to the noncommutative generalizations of Toda and of affine Toda field theories is investigated more deeply. This paper continues the programme initiated in $JHEP {\bf 10} (2005) 071$, where it was presented how it is possible to define Toda field theories through second order differential equation systems starting from the NCSDCS system. Here we show that using the connection of the NCSDCS to the noncommutative chiral model, exact solutions of the Toda field theories can be also constructed by means of the noncommutative extension of the uniton method proposed in $JHEP {\bf 0408} (2004) 054$ by Ki-Myeong Lee. Particularly some specific solutions of the nc Liouville model are explicit constructed.Comment: 24 page

How to find discrete contact symmetries

This paper describes a new algorithm for determining all discrete contact symmetries of any differential equation whose Lie contact symmetries are known. The method is constructive and is easy to use. It is based upon the observation that the adjoint action of any contact symmetry is an automorphism of the Lie algebra of generators of Lie contact symmetries. Consequently, all contact symmetries satisfy various compatibility conditions. These conditions enable the discrete symmetries to be found systematically, with little effort

Scattering of Noncommutative Waves and Solitons in a Supersymmetric Chiral Model in 2+1 Dimensions

Interactions of noncommutative waves and solitons in 2+1 dimensions can be analyzed exactly for a supersymmetric and integrable U(n) chiral model extending the Ward model. Using the Moyal-deformed dressing method in an antichiral superspace, we construct explicit time-dependent solutions of its noncommutative field equations by iteratively solving linear equations. The approach is illustrated by presenting scattering configurations for two noncommutative U(2) plane waves and for two noncommutative U(2) solitons as well as by producing a noncommutative U(1) two-soliton bound state.Comment: 1+13 pages; v2: reference added, version published in JHE

Darboux Transformations for a Lax Integrable System in 2n2n-Dimensions

A $2n$-dimensional Lax integrable system is proposed by a set of specific spectral problems. It contains Takasaki equations, the self-dual Yang-Mills equations and its integrable hierarchy as examples. An explicit formulation of Darboux transformations is established for this Lax integrable system. The Vandermonde and generalized Cauchy determinant formulas lead to a description for deriving explicit solutions and thus some rational and analytic solutions are obtained.Comment: Latex, 14 pages, to be published in Lett. Math. Phy

Chern-Simons Solitons, Chiral Model, and (affine) Toda Model on Noncommutative Space

We consider the Dunne-Jackiw-Pi-Trugenberger model of a U(N) Chern-Simons gauge theory coupled to a nonrelativistic complex adjoint matter on noncommutative space. Soliton configurations of this model are related the solutions of the chiral model on noncommutative plane. A generalized Uhlenbeck's uniton method for the chiral model on noncommutative space provides explicit Chern-Simons solitons. Fundamental solitons in the U(1) gauge theory are shaped as rings of charge `n' and spin `n' where the Chern-Simons level `n' should be an integer upon quantization. Toda and Liouville models are generalized to noncommutative plane and the solutions are provided by the uniton method. We also define affine Toda and sine-Gordon models on noncommutative plane. Finally the first order moduli space dynamics of Chern-Simons solitons is shown to be trivial.Comment: latex, JHEP style, 23 pages, no figur

Symmetries, Currents and Conservation Laws of Self-Dual Gravity

We describe an infinite-dimensional algebra of hidden symmetries for the self-dual gravity equations. Besides the known diffeomorphism-type symmetries (affine extension of w(infinity) algebra), this algebra contains new hidden symmetries, which are an affine extension of the Lorentz rotations. The full symmetry algebra has both Kac-Moody and Virasoro-like generators, whose exponentiation maps solutions of the field equations to other solutions. Relations to problems of string theories are briefly discussed.Comment: 14 pages, LaTeX, the paper was reformatte

Interaction of Noncommutative Plane Waves in 2+1 Dimensions

In this paper the interaction of extended waves in a noncommutative modified 2+1 dimensional U(2) sigma model are studied. Using the dressing method, we construct an explicit two-wave solution of the noncommutative field equation. The scattering of these waves and large time factorization are discussed.Comment: 12 pages, 4 figures, latex; V.2: two references adde