With the aid of the spectral gradient method of Fuchssteiner, the compatible pair of Hamiltonian operators for the coupled NLS hierarchy is rediscovered. This result enables us to construct a hierarchy, which contains a vector generalization of Fokas-Lenells system. The vector Fokas-Lenells system is shown to be bi-Hamiltonian and to possess a Lax pair.
In this paper, Painlevé analysis of supersymmetric extensions of the Sawada-Kotera (SK) equation is performed. It is shown that only two simple supersymmetric extensions of the Sawada-Kotera equation pass the Painlevé test. One of them was proposed by Tian and Liu, the other one is a Bextension of the SK equation.
A system proposed by Ito is reconsidered. The corresponding Darboux transformation is presented explicitly. The resulted Bäcklund transformation is shown to be equivalent to the one found by Hirota. Also, a nonlinear superposition formula, which is of differential-algebraic, is obtained.
The study of new integrable defects leads to new type of Bäcklund transformations named as the type-II Bäcklund transformations. In this article, we show, for the MKdV hierarchy, that the type-II Bäcklund transformation is the compound type-I Bäcklund transformation.
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