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

TL;DR

This paper develops an N=2 supersymmetric extension of a hydrodynamic system with Riemann invariants, analyzing its symmetries and deriving exact solutions including waves and kinks using group-theoretical methods.

Contribution

It introduces a novel supersymmetric extension of a hydrodynamic model and classifies its symmetry subalgebras, providing explicit solutions for the extended system.

Findings

Identified 401 symmetry subalgebras of the supersymmetric model.

Derived exact solutions such as travelling waves, kinks, and periodic solutions.

Demonstrated the use of symmetry reduction on Grassmann-valued equations.

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.