Assuming that there exist at least two fermionic parameters, the classical N= 1 supersymmetric Korteweg-de Vries (SKdV) system can be transformed to some coupled bosonic systems. The boson fields in the bosonized SKdV (BSKdV) systems are defined on even Grassmann algebra. Due to the intrusion of other Grassmann parameters, the BSKdV systems are different from the usual non-supersymmetric integrable systems, and many more abundant solution structures can be unearthed. With the help of the singularity analysis, the Painlev\'e property of the BSKdV system is proved and a B\"acklund transformation (BT) is found. The BT related nonlocal symmetry, we call it as residual symmetry, is used to find symmetry reduction solutions of the BSKdV system. Hinted from the symmetry reduction solutions, a more generalized but much simpler method is established to find exact solutions of the BSKdV and then the SKdV systems, which actually can be applied to any fermionic systems.

Bosonization, Singularity Analysis, Nonlocal Symmetry Reductions and Exact Solutions of Supersymmetric KdV Equation

Assuming that there exist at least two fermionic parameters, the classical $ \mathcal{N}=1 $
 supersymmetric Korteweg-de Vries (SKdV) system can be transformed to some coupled bosonic systems. The boson fields in the bosonized SKdV (BSKdV) systems are defined on even Grassmann algebra. Due to the intrusion of other Grassmann parameters, the BSKdV systems are different from the usual non-supersymmetric integrable systems, and many more abundant solution structures can be unearthed. With the help of the singularity analysis, the Painleve property of the BSKdV system is proved and a Backlund transformation (BT) is found. The BT related nonlocal symmetry, we call it as residual symmetry, is used to find symmetry reduction solutions of the BSKdV system. Hinted from the symmetry reduction solutions, a more generalized but much simpler method is established to find exact solutions of the BSKdV and then the SKdV systems, which actually can be applied to any fermionic systems.

Bosonization, singularity analysis, nonlocal symmetry reductions and exact solutions of supersymmetric KdV equation

The Lax formulation of the “new” N=2 SUSY KdV equation

N = 4 super KdV equation

Towards the construction of N=2 supersymmetric integrable hierarchies

There are two known integrable N=2 space supersymmetric extensions of the KdV equation. Both can be written as Hamiltonian systems with a common Poisson structure which corresponds to the N=2 supersymmetric form of the second KdV structure. By analyzing the conservation laws of the one parameter family of equations that share this Hamiltonian structure, one finds that a third system is singled out by the existence of nontrivial conservation laws. It is conjectured to be integrable.

A new N=2 supersymmetric Korteweg–de Vries equation

We give closed form expressions for the numbers of multi-rooted plane trees with specified degrees of root vertices. This results in an infinite number of integer sequences, some of which are known to have an alternative interpretation. We also propose recursion relations for numbers of such trees as well as for the corresponding generating functions. Explicit expressions for the generating functions corresponding to plane trees having two and three roots are derived. As a by-product, we obtain a new binomial identity and a conjecture relating hypergeometric functions. 

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P. Labelle

Papers

A new N=2 supersymmetric Korteweg–de Vries equation

Enumeration of Multi-rooted Plane Trees