N=2 and N=4 Supersymmetric mKdV and sinh-Gordon Hierarchies

Integrable models with higher N=2 and N=4 supersymmetries are formulated on reductions of twisted loop superalgebras $\widehat{sl}(2\mathrm{\mid }2)$ and $\widehat{sl}(4\mathrm{\mid }4)$ endowed with principal gradation. In case of the $\widehat{sl}(4\mathrm{\mid }4)$ loop algebra a sequence of progressing reductions leads both to the N=4 and N=2 supersymmetric mKdV and sinh-Gordon equations. The reduction scheme is induced by twisted automorphism and allows via dressing approach to associate to each symmetry flow of half-integer degree a supersymmetry transformation involving only local expressions in terms of the underlying fields.