It is shown that the generalized N=4 supersymmetric Toda lattice hierarchy contains the N=4 super-KdV hierarchy with the first flow time in the role of space coordinate. Two different N=2 superfield form of the generalized N=4 TL equation which are useful when solving the N=4 KdV and (1,1)-GNLS hierarchies are discussed.

We discuss a differential integrable hierarchy, which we call the (N,M)–th KdV hierarchy, whose Lax operator is obtained by properly adding M pseudo–differential terms to the Lax operator of the N–th KdV hierarchy. This new hierarchy contains both the higher KdV hierarchy and multi– field representation of KP hierarchy as sub–systems and naturally appears in multi–matrix models. The N + 2M − 1 ...

We construct N = 4 supersymmetric KdV equation as a hamiltonian flow on the N = 4 SU(2) super Virasoro algebra. The N = 4 KdV superfield, the hamiltonian and the related Poisson structure are concisely formulated in 1D N = 4 harmonic superspace. The most general hamiltonian is shown to necessarily involve SU(2) breaking parameters which are combined in a traceless rank 2 SU(2) tensor. First non...

On the exact solutions of integrable models, there is a new classification way recently based on the property of associated spectral parameters [1]. Negatons, related to the negative spectral parameter, are usually expressed by hyperbolic functions, and positons are expressed by means of trigonometric functions related to the positive spectral parameters. The so-called complexiton, which is exp...

A new Darboux transformation is presented for the Hirota-Satsuma coupled KdV system. It is shown that this Darboux transformation can be constructed by means of two methods: Painlevé analysis and reduction of a binary Darboux transformation. By iteration of the Darboux transformation, the Grammian type solutions are found for the coupled KdV system.

We degenerate the finite gap solutions of the KdV equation from the general formulation in terms of abelian functions when the gaps tends to points, to recover solutions of KdV equations given a few years ago in terms of wronskians called solitons or positons. For this we establish a link between Fredholm determinants and Wronskians.

We propose a hamiltonian formulation of the N = 2 supersymmetric KP type hierarchy recently studied by Krivonos and Sorin. We obtain a quadratic hamiltonian structure which allows for several reductions of the KP type hierarchy. In particular , the third family of N = 2 KdV hierarchies is recovered. We also give an easy construction of Wronskian solutions of the KP and KdV type equations.

We provide an overview of elliptic algebro-geometric solutions of the KdV and AKNS hierarchies, with special emphasis on Floquet theoretic and spectral theoretic methods. Our treatment includes an effective characterization of all stationary elliptic KdV and AKNS solutions based on a theory developed by Hermite and Picard.