We give a systematic study about supersymmetric Euler equations on the smooth dual NSreg*(N) of the Neveu-Schwarz algebra NS(N) for N ≤ 3. Let A be the inertia operator and c1,c2R, we will show that the N = 2 supersymmetric Euler equation with A=c1+c2D2 is local bi-super-Hamiltonian with the freezing point (c1,c2)NSreg*(2), which is similar to that of the Virasoro algebra NS(0), and the N = 3 supersymmetric Euler equation with A=c2D31 is local bi-super-Hamiltonian with the freezing point (0,c2)NSreg*(3), which is similar to that of the N = 1 Neveu-Schwarz algebra NS(1).

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39.

It is well-known that for n = 1, 2, 3, there each exists a unique such extension. For n = 4, there are three kinds of extensions. However, for n > 4, the Lie algebra K0(1|n) has no nontrivial central extension.6,7 We remark that the Virasoro algebra can be viewed as NS(0).

40.
 The adjoint of A is the unique operator, denoted with A*, such that
for each pair of superfields f and g. For the product of operators, it is found that (AB)* = (−1)|A||B|B*A* implying that (A1)*=(1)|A|(A*)1 if A is invertible. Moreover, Di*=Di,*=; if f is a superfield of any parity, for the corresponding multiplication operator, we have f* = f.
41.
For a functional f[U], the variational derivatives δfδU are defined by
42.

In Ref. 35, Popowicz has obtained two super-Hamiltonian operators J1 and J2. In order to check the compatibility of two Hamiltonian operators, he used computer algebra Reduce and special computer package SUSY2. However, from the above construction, using the central extension and the “frozen-point” method, it is natural to obtain the compatibility.

43.
The reduction N = 3 ↦ N = 2 goes as follows. The N = 3 supercurrent U(x, t; θ1, θ2, θ3) contains the N = 2 supercurrent W(x, t; θ1, θ2) as a coefficient before θ3 in its θ3 expansion, while all the additional currents are contained in the θ3 independent part of U. In other words, we should put the ansatz U = θ3W in and get
which is exactly the N = 2 sKdV0 equation (1.4) [or see (3.15) in the Remark 3.7].
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