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

In this paper, we generalize classical Bell polynomials into super version, which are found to be effective in systematically constructing super bilinear representation, bilinear Backlund transformation, Lax pair, and infinite conservation laws of supersymmetric equations. We take N=1 supersymmetric KdV equation and N=2 supersymmetric sine-Gordon equation to illustrate this procedure.

Super extension of Bell polynomials with applications to supersymmetric equations

(2 < p < 4) [200]. (Uq(∫u(1, 1)), oq1/2(2n)) [92]. 1 [273, 79, 304, 119]. 1 + 1 [252]. 2 [352, 318, 226, 40, 233, 157, 299, 60]. 2× 2 [185]. 3 [456, 363, 58, 18, 351]. ∗ [238]. 2 [277]. 3 [350]. p [282]. B−L [427]. α [216, 483]. α− z [322]. N = 2 [507]. D [222]. ẍ+ f(x)ẋ + g(x) = 0 [112, 111, 8, 5, 6]. Eτ,ηgl3 [148]. g [300]. κ [244]. L [205, 117]. L [164]. L∞ [368]. M [539]. P [27]. R [147]. Z2 [565]. Z n 2 [131]. Z2 × Z2 [25]. D(X) [166]. S(N) [110]. ∫l2 [154]. SU(2) [210]. N [196, 242]. O [386]. osp(1|2) [565]. p [113, 468]. p(x) [17]. q [437, 220, 92, 183]. R, d = 1, 2, 3 [279]. SDiff(S) [32]. σ [526]. SLq(2) [185]. SU(N) [490]. τ [440]. U(1) N [507]. Uq(sl 2) [185]. φ 2k [283]. φ [553]. φ4 [365]. ∨ [466]. VOA[M4] [33]. Z [550].

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A Complete Bibliography of Publications in the Journal of Mathematical Physics: 2005{2009

Bosonization of supersymmetric KdV equation

The nonlocal conserved quantities of the N=1 Super KdV are obtained using a Gardner map. A fermionic substitution semigroup and the resulting Gardner category are defined and several propositions concerning their algebraic structure are obtained. This algebraic framework makes it possible to define general transformations between different nonlinear SUSY differential equations. A SUSY ring extension is then introduced to deal with the nonlocal conserved quantities of SKdV. The algebraic version of the nonlocal conserved quantities is solved in terms of the exponential function applied to the D−1 of the local conserved quantities of SKdV. Finally the same formulas are shown to work for rapidly decreasing superfields.

The Gardner category and nonlocal conservation laws for N=1 Super KdV

An extension of the super Korteweg–de Vries (KdV) integrable system in terms of operator valued functions is obtained. Following the ideas of Gardner, a general algebraic approach for finding the infinitely many conserved quantities of integrable systems is presented. The approach is applied to the above described system and infinitely many conserved quantities are constructed. In a particular case they reduce to the corresponding conserved quantities of super KdV.

An operator valued extension of the super Korteweg–de Vries equations

An extension of the Super KdV integrable system in terms of operator valued functions is obtained. Following the ideas of Gardner, a general algebraic approach for finding the infinitely many conserved quantities of integrable systems is presented. The approach is applied to the above described system and infinitely many conserved quantities are constructed. In a particular case they reduce to the corresponding conserved quantities of Super KdV.

An Operator Valued Extension of the Super KdV Equations

The non-local conserved quantities of N=1 Super KdV are obtained using a complete algebraic framework where the Gardner category is introduced. A fermionic substitution semigroup and the resulting Gardner category are defined and several propositions concerning their algebraic structure are proven. This algebraic framework allows to define general transformations between different nonlinear SUSY differential equations. We then introduce a SUSY ring extension to deal with the non-local conserved quantities of SKdV. The algebraic version of the non-local conserved quantities is solved in terms of the exponential function applied to the $D^{-1}$ of the local conserved quantities of SKdV. Finally the same formulas are shown to work for rapidly decreasing superfields.

S. Andrea

Papers

The Gardner category and nonlocal conservation laws for N=1 Super KdV

An operator valued extension of the super Korteweg–de Vries equations

An Operator Valued Extension of the Super KdV Equations

The Gardner Category and Non-local Conservation Laws for N=1 Super KdV

On the non-abelian superalgebra spanned by the conserved quantities of N=1 supersymmetric Korteweg–de Vries equation