N = 2 supersymmetric extension of a hydrodynamic system in Riemann invariants

Abstract

In this paper, we formulate an N = 2 supersymmetric extension of a hydrodynamic-type system involving Riemann invariants. The supersymmetric version is constructed by means of a superspace and superfield formalism, using bosonic superfields, and consists of a system of partial differential equations involving both bosonic and fermionic variables. We make use of group-theoretical methods in order to analyze the extended model algebraically. Specifically, we calculate a Lie superalgebra of symmetries of our supersymmetric model and make use of a general classification method to classify the one-dimensional subalgebras into conjugacy classes. As a result we obtain a set of 401 one-dimensional nonequivalent subalgebras. For selected subalgebras, we use the symmetry reduction method applied to Grassmann-valued equations in order to determine analytic exact solutions of our supersymmetric model. These solutions include travelling waves, bumps, kinks, double-periodic solutions and solutions involving exponentials and radicals. *

Supersymmetric formulation of polytropic gas dynamics and its invariant solutions

“…5 When γ = 3, the equations represent the dispersionless limit of two non-interacting Korteweg-de Vries equations (known as Riemann invariants equations). 1,6,7 Finally, the γ = 4 case is the dispersionless limit of a Boussinesq equation. 8,9 In the context of hydrodynamic-type equations, other models have been supersymmetrized and analyzed from the group-theoretical point of view.…”

“…Examples of such models include the Korteweg-de Vries equation, 10,11 the Kadomtsev-Petviashvili equation, 12 the scalar Born-Infeld model, 13 a Gaussian fluid flow 14 and a hydrodynamic system expressed in terms of Riemann invariants. 6,7,15 The results are of interest since we can construct certain classes of solutions with the freedom of arbitrary functions of one or two arguments involving bosonic and fermionic variables.…”