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It is shown that there exist two inner automorphisms and a discrete transformation besides the usual Bäcklund transformation for the integrable focusing nonlinear Schrödinger equation. By virtue of the inner automorphisms, the discrete transformation can generate multi-soliton solutions. The expressions for multi-soliton solutions generated by the discrete and the Bäcklund transformations turn out to be the same.

Inner automorphisms, discrete and Bäcklund transformations of integrable systems