A supertrace identity on Lie superalgebras is established. It provides a tool for constructing super-Hamiltonian structures of zero curvature equations associated with Lie superalgebras. Applications in the case of the Lie superalgebra B͑0,1͒ present super-Hamiltonian structures of a super-AKNS soliton hierarchy and a super-Dirac soliton hierarchy.

General higher-order rogue waves of a vector nonlinear Schrödinger equation (Manakov system) are derived using a Darboux-dressing transformation with an asymptotic expansion method. The N th-order semirational solutions containing 3N free parameters are expressed in separation-of-variables form. These solutions exhibit rogue waves on a multisoliton background. They demonstrate that the structure of rogue waves in this two-component system is richer than that in a one-component system. Our results would be of much importance in understanding and predicting rogue wave phenomena arising in nonlinear and complex systems, including optics, fluid dynamics, Bose-Einstein condensates, and finance. Introduction.It is well known that many nonlinear wave equations of physical interest support solitons, which are localized waves arising from a balance between dispersion and nonlinearity, and which can propagate steadily for a long time. Recently, over the last two decades, it has been recognized that another class of solutions, namely breathers, are also of fundamental importance. Breathers propagate steadily and are localized in either time or space, while being periodic in either space or time. Further, due to their localization properties, breathers have been invoked as models of rogue waves, also called freak waves, which are large amplitude waves which apparently appear without warning and then disappear without trace. While they have been most often found in the context of water waves [1, 2, 3], they have also been found in other physical contexts such as optical fibres [4,5,6].The breather solutions of the focusing nonlinear Schrödinger equation (NLSE) have been widely invoked as models of rogue waves; see the references above and Akhmediev, and Ankiewicz [7], for instance. The NLSE is integrable [8], and many kinds of exact solutions have been found. In particular, the Peregrine breather [9], the Akhmediev breather (AB) [7], and the Kuznetsov-Ma breathers (KM) [10,11] have been associated with rogue waves as the potential outcome of the modulational instability of a plane wave. AB is periodic in space and localized in

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We report new types of matter rogue waves of a spinor (three-component) model of the Bose-Einstein condensate governed by a system of three nonlinearly coupled Gross-Pitaevskii equations. The exact first-order rational solutions containing one free parameter are obtained by means of a Darboux transformation for the integrable system where the mean-field interaction is attractive and the spin-exchange interaction is ferromagnetic. For different choices of the parameter, there exists a variety of different shaped solutions including two peaks in bright rogue waves and four dips in dark rogue waves. Furthermore, by utilizing the relation between the three-component and the one-component versions of the nonlinear Schrödinger equation, we can devise higher-order rational solutions, in which three components have different shapes. In addition, it is noteworthy that dark rogue wave features disappear in the third-order rational solution.