N=2 supersymmetric extension of a hydrodynamic system in Riemann   invariants
  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. 
A. M. Grundland, A. J. Hariton