B\"acklund-Darboux Transformations and Discretizations of Super KdV Equation

For a generalized super KdV equation, three Darboux transformations and the corresponding B\"acklund transformations are constructed. The compatibility of these Darboux transformations leads to three discrete systems and their Lax representations. The reduction of one of the B\"acklund-Darboux transformations and the corresponding discrete system are considered for Kupershmidt's super KdV equation. When all the odd variables vanish, a nonlinear superposition formula is obtained for Levi's B\"acklund transformation for the KdV equation

Achieving Secrecy Capacity of the Gaussian Wiretap Channel with Polar Lattices

In this work, an explicit wiretap coding scheme based on polar lattices is proposed to achieve the secrecy capacity of the additive white Gaussian noise (AWGN) wiretap channel. Firstly, polar lattices are used to construct secrecy-good lattices for the mod-$\Lambda_s$ Gaussian wiretap channel. Then we propose an explicit shaping scheme to remove this mod-$\Lambda_s$ front end and extend polar lattices to the genuine Gaussian wiretap channel. The shaping technique is based on the lattice Gaussian distribution, which leads to a binary asymmetric channel at each level for the multilevel lattice codes. By employing the asymmetric polar coding technique, we construct an AWGN-good lattice and a secrecy-good lattice with optimal shaping simultaneously. As a result, the encoding complexity for the sender and the decoding complexity for the legitimate receiver are both O(N logN log(logN)). The proposed scheme is proven to be semantically secure.Comment: Submitted to IEEE Trans. Information Theory, revised. This is the authors' own version of the pape

Supersymmetric KdV equation: Darboux transformation and discrete systems

For the supersymmetric KdV equation, a proper Darboux transformation is presented. This Darboux transformation leads to the B\"{a}cklund transformation found early by Liu and Xie \cite{liu2}. The Darboux transformation and the related B\"{a}cklund transformation are used to construct integrable super differential-difference and difference-difference systems. The continuum limits of these discrete systems and of their Lax pairs are also considered.Comment: 13pages, submitted to Journal of Physics