Invariant solutions of a supersymmetric fluid model

In this paper, we present a method for constructing invariant solutions of the supersymmetric Chaplygin gas in (1 + 1) dimensions. This approach is based on the use of a generalized Legendre transformation, through which we transform the original field equations into a new set of equations involving the velocity and sound speed of the fluid as independent variables. We describe the Lie symmetry properties of the equations in both coordinate systems, and make use of a systematic subgroup classification to determine certain classes of group-invariant solutions of the transformed field equations. Where it is possible, the Legendre transformation is applied in reverse in order to obtain equivalent solutions of the standard field equations. A number of analytic solutions of the supersymmetric Chaplygin gas in one spatial dimension are found. In addition, certain basic elements of a possible extension of our method to the supersymmetric Chaplygin gas in two spatial dimensions have been formulated. In particular, the (2 + 1)-dimensional analogue of the transformed equation has been determined, and some of its Lie point symmetries have been identified. For a certain specific case, it has been demonstrated how an invariant solution can be inverted and then extended to a solution of the planar supersymmetric model