| Abstract: |
| A supersymmetric nonlinear partial differential equation (NPDE) with fermionic fields combines supersymmetric field theory with the integrable system theory. The fermionic fields describe the spin$-1/2$ particles (fermions, particles with half-integer spin) in supersymmetric theories and are crucial for understanding particle interactions and symmetries in field theory. In this talk, we consider the NPDE
$\begin{equation}
\frac{{\rm d} \Phi}{{\rm d} t} =
{\Phi} \left(\frac{{\rm d} {\Phi}}{{\rm d} x}\right) \left(\frac{{\rm d}^{2}{\Phi}}{{\rm d} x^{2}}\right) F \! \left({\mathcal{D}}{\Phi}, \frac{{\rm d} {\mathcal{D}}{\Phi}}{{\rm d} x}, \frac{{\rm d}^{2}{\mathcal{D}}{\Phi}}{{\rm d} x^{2}}, \frac{{\rm d}^{3}{\mathcal{D}}{\Phi}}{{\rm d} x^{3}}\right)
\end{equation}$
with $F$ being an arbitrary function of the bosonic fields using the dot product first to combine the fermions appearing in it and then unearth its higher order supersymmetries with arbitrary functions that only related to the bosonic fields and their derivatives. It is interesting that the dot product of two fermions can be related to symmetry properties, and the study of supersymmetry sets up a bridge between the fermionic and bosonic fields. |
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