A construction is proposed for a supersymmetric generalization of the cubic Schrödinger equation, resulting in two supersymmetric systems, one of which contains a free parameter. Both systems are proven to admit an infinite set of (higher-order) local and nonlocal symmetries and a seemingly infinite set of conservation laws, the lowest-order terms of which are given explicitly. Moreover, the theory of coverings (equivalent to the prolongation method of Wahlquist and Estabrook) is applied to both systems. Both are seen to admit an infinite-dimensional covering algebra, the structure of which is determined explicitly, resulting in a related supersymmetric system of differential equations, as well as an auto-Bäcklund transformation for each equation. This indicates the complete integrability of both systems.
An efficient method to construct Hamiltonian structures for nonlinear evolution equations is described. It is based on the notions of variational Schouten bracket and ℓ * -covering. The latter serves the role of the cotangent bundle in the category of nonlinear evolution PDEs. We first consider two illustrative examples (the KdV equation and the Boussinesq system) and reconstruct for them the known Hamiltonian structures by our methods. For the coupled KdV-mKdV system, a new Hamiltonian structure is found and its uniqueness (in the class of polynomial (x, t)-independent structures) is proved. We also construct a nonlocal Hamiltonian structure for this system and prove its compatibility with the local one.
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