Invariant solutions of supersymmetric nonlinear wave equations

Abstract

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).

Supersymmetric versions of the equations of conformally parametrized surfaces

“…{P + + η 1 η 2 P − }, since there is no limit to the number of odd parameters η k that can be used to construct even coefficients. While such subalgebras may allow for more freedom in the choice of invariants, we then encounter the problem of non-standard invariants [24], [58]. Such non-standard invariants, which do not lead to standard reductions or invariant solutions, are found for several other SUSY hydrodynamic-type systems, e.g.…”

Supersymmetric Version of the Euler System and Its Invariant Solutions

“…If these arbitrary functions depend on x, the postulated form will change as the solution evolves in time. Such a large number of arbitrary function degrees of freedom were not found in the previous analyses by the authors of supersymmetric hydrodynamic systems (i.e., polytropic gas [4], integrable models (sine-Gordon, sinh-Gordon, polynomial Klein-Gordon [23,24]). Some of the obtained solutions involved damping and growth.…”

“…For instance, the subalgebra, L 2 = {νQ 2 }, has the nonstandard invariant, νf (x, t, θ 1 , θ 2 , U, R, P), where f is an arbitrary function of its arguments. Such non-standard invariants were found by the authors for several other supersymmetric hydrodynamic-type systems, including the supersymmetric sinh-Gordon Equation [23], the supersymmetric Klein-Gordon polynomial Equation [23] and supersymmetric polytropic gas dynamics [4].…”

Interaction solutions for supersymmetric mKdV-B equation

“…The soliton solution of BSmKdV-B system reads in the following form by using the line solution (13) and the nonauto-BT theorem…”

“…The B-supersymmetric of the Korteweg-de Vries (KdV) [5], dispersionless two boson hierarchy [6], Sawada-Kotera [7], modified KdV and Camassa-Holme quations [8] have been constructed. In the meanwhile, the methodologies involved in the study of integrable systems have been expanded to the supersymmetric framework [9][10][11][12][13]. The soliton solutions of supersymmetric systems have been constructed via many methodologies.…”