The Even and Odd Supersymmetric Hunter - Saxton and Liouville Equations

It is shown that two different supersymmetric extensions of the Harry Dym equation lead to two different negative hierarchies of the supersymmetric integrable equations. While the first one yields the known even supersymmetric Hunter - Saxton equation, the second one is a new odd supersymmetric Hunter - Saxton equation. It is further proved that these two supersymmetric extensions of the Hunter - Saxton equation are reciprocally transformed to two different supersymmetric extensions of the Liouville equation.

💡 Research Summary

The paper investigates two distinct supersymmetric extensions of the classical Harry Dym (HD) equation—one based on even supersymmetry and the other on odd supersymmetry—and shows how each leads to a separate negative hierarchy of supersymmetric integrable equations. In the even case the authors introduce a fermionic super‑field (W=\chi+\theta u) and two even supersymmetric Hamiltonian operators (\hat K_1=D\partial_x^2) and (\hat K_2=\frac12