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The plane wave stability properties of the conservative schemes of Besse andFei et al. for the cubic Schr\"{o}dinger equation are analysed. Although thetwo methods possess many of the same conservation properties, we show thattheir stability behaviour is very different. An energy preservinggeneralisation of the Fei method with improved stability is presented.

Plane wave stability of some conservative schemes for the cubic Schrödinger equation