Supersymmetric formulation of polytropic gas dynamics and its invariant solutions

Abstract

In this paper, a supersymmetric extension of the polytropic gas dynamics equations is constructed through the use of a superspace involving two independent fermionic variables and two bosonic superfields. A superalgebra of symmetries of the proposed extended model is determined and a systematic classification of the one-dimensional subalgebras of this superalgebra is performed. Through the use of the symmetry reduction method, a number of invariant solutions of the supersymmetric polytropic gas dynamics equations are found. Several types of solutions are obtained, including algebraic-type solutions and propagation waves (simple and double waves). Many of the obtained solutions involve arbitrary functions of one or two bosonic or fermionic variables. In the case where the arbitrary functions involve only the independent fermionic variables, the solutions are expressed in terms of Taylor expansions.Byline: Supersymmetric polytropic gas and its solutions PACS: 02.20.Sv, 12.60.Jv, 02.30.Jr

Supersymmetric versions of the equations of conformally parametrized surfaces

“…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. in [7], [59].…”

Algebraic Aspects of the Supersymmetric Minimal Surface Equation

“…Each reduced system can be solved in order to construct an invariant solution of the SUSY MS Equation (15). It should be noted that, as has been observed for other similar supersymmetric extensions [6], some of the subalgebras listed in the Appendix A have a non-standard invariant structure in the sense that they do not reduce the system to ordinary differential equations in the usual sense. These are the nine subalgebras: 24 , and L 33 .…”

“…A number of supersymmetric extensions have been formulated for both classical and quantum mechanical systems. In particular, such supersymmetric generalizations have been constructed for hydrodynamic-type systems (e.g., the Korteweg-de Vries equation [2,3], the Sawada-Kotera equation [4], polytropic gas dynamics [5,6] and a Gaussian irrotational compressible fluid [7]) as well as other nonlinear wave equations, e.g., the Schrödinger equation [8] and the sine/sinh-Gordon equation [9][10][11]. Parameterizations of strings and Nambu-Goto membranes have been used to supersymmetrize the Chaplygin gas in (1 + 1) and (2 + 1) dimensions respectively [12].…”

Supersymmetric Version of the Euler System and Its Invariant Solutions

“…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].…”

“…Such extensions have been constructed recently for certain hydrodynamic-type models (see, e.g., [1][2][3][4]). A study of polytropic supersymmetric models has also been performed by Das and Popowicz [5] in the specific case when p = kρ k .…”