Invariant solutions of the supersymmetric version of a two-phase fluid flow system

Abstract

A supersymmetric extension of a two-phase fluid flow system is formulated. A superalgebra of Lie symmetries of the supersymmetric extension of this system is computed. The classification of the one-dimensional subalgebras of this superalgebra into 63 equivalence classes is performed. For some of the subalgebras, it is found that the invariants have a non-standard structure. For six selected subalgebras, the symmetry reduction method is used to find invariants, orbits of the subgroups and reduced systems. Through the solutions of the reduced systems, the most general solutions are expressed in terms of arbitrary functions of one or two fermionic and one bosonic variables.

Appendix

The following list constitutes the classification of the one-dimensional subalgebras of the Lie symmetry superalgebra associated with the supersymmetric system (21), (22) and (23) into conjugacy classes under the action of the associated Lie group. Here $\varepsilon =\pm 1$, the parameters a and b are non-zero bosonic constants, and ${\underline{\mu }}$ and ${\underline{\nu }}$ are non-zero fermionic constants.

$$\begin{aligned} \begin{aligned}&{{\mathcal {L}}}_1=\{P_1\}, \quad {{\mathcal {L}}}_2=\{P_2\}, \quad {{\mathcal {L}}}_3=\{P_1+aP_2\}, \quad {\mathcal L}_4=\{{\underline{\mu }}Q_1\}, \\&{\mathcal L}_5=\{P_1+{\underline{\mu }}Q_1\}, \quad {{\mathcal {L}}}_6=\{P_2+{\underline{\mu }}Q_1\}, \quad {\mathcal L}_7=\{P_1+aP_2+{\underline{\mu }}Q_1\}, \\&{{\mathcal {L}}}_8=\{{\underline{\mu }}Q_2\}, \quad {\mathcal L}_9=\{P_1+{\underline{\mu }}Q_2\}, \quad {\mathcal L}_{10}=\{P_2+{\underline{\mu }}Q_2\}, \\&{{\mathcal {L}}}_{11}=\{P_1+aP_2+{\underline{\mu }}Q_2\}, \quad {\mathcal L}_{12}=\{{\underline{\mu }}Q_1+{\underline{\nu }}Q_2\}, \quad {{\mathcal {L}}}_{13}=\{P_1+{\underline{\mu }}Q_1+{\underline{\nu }}Q_2\}, \\&{{\mathcal {L}}}_{14}=\{P_2+{\underline{\mu }}Q_1+{\underline{\nu }}Q_2\}, \quad {\mathcal L}_{15}=\{P_1+aP_2+{\underline{\mu }}Q_1+{\underline{\nu }}Q_2\}, \quad {{\mathcal {L}}}_{16}=\{M_1\}, \\&{{\mathcal {L}}}_{17}=\{M_1+\varepsilon P_1\}, \quad {\mathcal L}_{18}=\{M_1+\varepsilon P_2\}, \quad {{\mathcal {L}}}_{19}=\{M_1+\varepsilon P_1+aP_2\}, \\&{{\mathcal {L}}}_{20}=\{M_1+{\underline{\mu }}Q_1\}, \quad {{\mathcal {L}}}_{21}=\{M_1+\varepsilon P_1+{\underline{\mu }}Q_1\}, \quad {{\mathcal {L}}}_{22}=\{M_1+\varepsilon P_2+{\underline{\mu }}Q_1\}, \\&{{\mathcal {L}}}_{23}=\{M_1+\varepsilon P_1+aP_2+{\underline{\mu }}Q_1\}, \quad {{\mathcal {L}}}_{24}=\{M_1+{\underline{\mu }}Q_2\}, \\&{{\mathcal {L}}}_{25}=\{M_1+\varepsilon P_1+{\underline{\mu }}Q_2\}, \quad {{\mathcal {L}}}_{26}=\{M_1+\varepsilon P_2+{\underline{\mu }}Q_2\},\\&{{\mathcal {L}}}_{27}=\{M_1+\varepsilon P_1+aP_2+{\underline{\mu }}Q_2\}, \quad {{\mathcal {L}}}_{28}=\{M_1+{\underline{\mu }}Q_1+{\underline{\nu }}Q_2\}, \\&{{\mathcal {L}}}_{29}=\{M_1+\varepsilon P_1+{\underline{\mu }}Q_1+{\underline{\nu }}Q_2\}, \quad {\mathcal L}_{30}=\{M_1+\varepsilon P_2+{\underline{\mu }}Q_1+{\underline{\nu }}Q_2\}, \\&{{\mathcal {L}}}_{31}=\{M_1+\varepsilon P_1+aP_2+{\underline{\mu }}Q_1+{\underline{\nu }}Q_2\}, \quad {\mathcal L}_{32}=\{M_2\}, \quad {{\mathcal {L}}}_{33}=\{M_2+\varepsilon P_1\}, \\&{{\mathcal {L}}}_{34}=\{M_2+\varepsilon P_2\}, \quad {\mathcal L}_{35}=\{M_2+\varepsilon P_1+aP_2\}, \quad {\mathcal L}_{36}=\{M_2+{\underline{\mu }}Q_1\}, \\&{{\mathcal {L}}}_{37}=\{M_2+\varepsilon P_1+{\underline{\mu }}Q_1\}, \quad {{\mathcal {L}}}_{38}=\{M_2+\varepsilon P_2+{\underline{\mu }}Q_1\},\\&{{\mathcal {L}}}_{39}=\{M_2+\varepsilon P_1+aP_2+{\underline{\mu }}Q_1\}, \quad {{\mathcal {L}}}_{40}=\{M_2+{\underline{\mu }}Q_2\}, \\&{{\mathcal {L}}}_{41}=\{M_2+\varepsilon P_1+{\underline{\mu }}Q_2\}, \quad {{\mathcal {L}}}_{42}=\{M_2+\varepsilon P_2+{\underline{\mu }}Q_2\},\\&{{\mathcal {L}}}_{43}=\{M_2+\varepsilon P_1+aP_2+{\underline{\mu }}Q_2\}, \quad {{\mathcal {L}}}_{44}=\{M_2+{\underline{\mu }}Q_1+{\underline{\nu }}Q_2\}, \\&{{\mathcal {L}}}_{45}=\{M_2+\varepsilon P_1+{\underline{\mu }}Q_1+{\underline{\nu }}Q_2\}, \quad {\mathcal L}_{46}=\{M_2+\varepsilon P_2+{\underline{\mu }}Q_1+{\underline{\nu }}Q_2\}, \\&{{\mathcal {L}}}_{47}=\{M_2+\varepsilon P_1+aP_2+{\underline{\mu }}Q_1+{\underline{\nu }}Q_2\}, \quad {\mathcal L}_{48}=\{M_2+aM_1\}, \\&{{\mathcal {L}}}_{49}=\{M_2+aM_1+\varepsilon P_1\}, \quad {{\mathcal {L}}}_{50}=\{M_2+aM_1+\varepsilon P_2\}, \\&{{\mathcal {L}}}_{51}=\{M_2+aM_1+\varepsilon P_1+bP_2\}, \quad {{\mathcal {L}}}_{52}=\{M_2+aM_1+{\underline{\mu }}Q_1\}, \\&{{\mathcal {L}}}_{53}=\{M_2+aM_1+\varepsilon P_1+{\underline{\mu }}Q_1\}, \quad {{\mathcal {L}}}_{54}=\{M_2+aM_1+\varepsilon P_2+{\underline{\mu }}Q_1\},\\&{{\mathcal {L}}}_{55}=\{M_2+aM_1+\varepsilon P_1+bP_2+{\underline{\mu }}Q_1\}, \quad {\mathcal L}_{56}=\{M_2+aM_1+{\underline{\mu }}Q_2\}, \\&{{\mathcal {L}}}_{57}=\{M_2+aM_1+\varepsilon P_1+{\underline{\mu }}Q_2\}, \quad {{\mathcal {L}}}_{58}=\{M_2+aM_1+\varepsilon P_2+{\underline{\mu }}Q_2\},\\&{{\mathcal {L}}}_{59}=\{M_2+aM_1+\varepsilon P_1+bP_2+{\underline{\mu }}Q_2\}, \quad {\mathcal L}_{60}=\{M_2+aM_1+{\underline{\mu }}Q_1+{\underline{\nu }}Q_2\}, \\&{{\mathcal {L}}}_{61}=\{M_2+aM_1+\varepsilon P_1+{\underline{\mu }}Q_1+{\underline{\nu }}Q_2\}, \\&{{\mathcal {L}}}_{62}=\{M_2+aM_1+\varepsilon P_2+{\underline{\mu }}Q_1+{\underline{\nu }}Q_2\}, \\&{{\mathcal {L}}}_{63}=\{M_2+aM_1+\varepsilon P_1+bP_2+{\underline{\mu }}Q_1+{\underline{\nu }}Q_2\} \end{aligned} \end{aligned}$$

(65)