On Noncommutative Sinh–Gordon Equation

In this talk, we present a group-theoretical symmetry analysis for the supersymmetric versions of specific nonlinear equations. Specifically, we consider supersymmetric extensions of the (1 + 1)-dimensional sine-Gordon equation, the (1 + 1)-dimensional sinh-Gordon equation and the following generalized polynomial form of the Klein-Gordon equationIn each case, the supersymmetric version of the equation is constructed on the 4dimensional Grassmannian superspace {(x, t, θ 1 , θ 2 )}. Here, the variables x and t represent the bosonic coordinates of 2-dimensional Minkowski space, while the quantities θ 1 and θ 2 are anticommuting fermionic variables. The bosonic function u(x, t) is replaced by the scalar bosonic superfieldwhere φ and ψ are fermionic-valued fields and F is a bosonic field. The supersymmetric extension is constructed in such a way that it is invariant under a set of supersymmetry transformations (generated by vector fields Q x and Q t ) which link the bosonic independent variables x and t to the fermionic independent variables θ 1 and θ 2 respectively. This is ensured by writing the supersymmetric equation in terms of covariant derivative operators D x and D t which anticommute with the supersymmetry generators Q x and Q t respectively. For each of the supersymmetric equations under consideration, we use a generalization of the method of prolongation in order to determine the Lie superalgebra of symmetries of the equation, and we present a systematic classification of all one-dimensional subalgebras of this resulting Lie superalgebra. The method of symmetry reduction then allows us to derive invariant solutions of the supersymmetric model. Some interpretation of the obtained results is given. References: 1. A. M. Grundland, A J. Hariton and L. Snobl, J. Phys. A: Math. Theor. 42, 335203 (2009). 2. A. M. Grundland, A J. Hariton and L. Snobl, "Invariant solutions of supersymmetric nonlinear wave equations", in preparation (2010).

Section: Symmetries Of the Supersymmetric Polynomial Klein-gordon Equ...mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Invariant solutions of supersymmetric nonlinear wave equations

Grundland

Hariton

Šnobl

2011

“…The other aspect that we shall be interested in investigating is the generalization of our results to the case of the noncommutative principal chiral model. In recent times, there has been an increasing interest in the study of noncommutative integrable field models (see, e.g., [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30]). The interest in these studies is partly due to the fact that the noncommutative field theories play an important role in string theory, D-brane dynamics, quantum Hall effect, etc (see, e.g., [33][34][35]).…”

Section: Introductionmentioning

confidence: 99%

TheU(N) chiral model and exact multi-solitons

Haider¹,

Hassan²

2008

Self Cite

“…1751-8113/07/195205+13$30.00 © 2007 IOP Publishing Ltd Printed in the UK noncommutativity of time variables leads to non-unitarity and affects the causality of the theory [16,17]. The noncommutative extension of integrable equations has also attracted a great deal of interest during the last decade and these noncommutative versions of integrable equations have been shown to admit integrability structures such as the Lax pair, the Bäcklund transformation, the existence of an infinite number of conserved quantities and zero-curvature representation [18][19][20][21][22][23][24][25][26][27][28].…”

Section: Introductionmentioning

confidence: 99%

Non-local continuity equations and binary Darboux transformation of noncommutative (anti) self-dual Yang–Mills equations

Saleem

Hassan

Siddiq

2007

Self Cite

We present an infinite number of non-local continuity equations of noncommutative (anti) self-dual Yang–Mills (nc-(A)SDYM) equations using the induction method of Brézin et al (1979 Phys. Lett. B 82 442) and relate it to the Lax pair and the parametric Bäcklund transformation of the system. From the Lax pair, we derive a binary Darboux transformation to generate solutions of the nc-(A)SDYM equations.