The solution of the N = 2 supersymmetric f-Toda chain with fixed ends

Abstract

The integrability of the recently introduced N = 2 supersymmetric f-Toda chain, under appropriate boundary conditions, is proven. The recurrent formulae for its general solutions are derived. As an example, the solution for the simplest case of boundary conditions is presented in explicit form.

The N = 2 supersymmetric Toda lattice hierarchy

“…Ancestors of the present paper can be considered refs. [1,2,3] as well as [5]. As one might suspect, there is a one-to-one correspondence between the lattice hierarchy defined above and the differential N = 2 supersymmetric NLS hierarchy.…”

“…and reproduce the set of the restricted f-Toda chain equations [3], which form a subset of the f-Toda chain equations (see section 6). The canonical basis (70) is not important only for the Lagrangian formulation, there is also another reason to introduce it.…”

Periodic Supersymmetric Toda Lattice Hierarchy

“…where r-matrix r(λ − µ) = P/(µ − λ) is defined by the permutation matrix P ij;kl = (−1) p(i)p(j) δ i,l δ j,k and for the supermatrix L j (λ) (12) we have the Grassmann parity of rows and columns to be p(1) = p(2) = 0, p(3) = p(4) = 1. As a consequence of (13), the monodromy matrix T m (λ) = …”

“…One-dimensional reductions of these hierarchies -the N = 4 and N = 2 supersymmetric Toda lattice hierarchies -were studied in [7,8], while their finite reductions corresponding to different boundary conditions (e.g., fixed ends, periodic boundary conditions, etc.) were investigated in [9][10][11][12][13]. Quite recently, a dispersionless limit of the N = (1|1) supersymmetric Toda lattice hierarchy was constructed in [14,15].…”

Supersymmetric Toda Lattice Hierarchies

“…(132), completely disappears. The equations (134) reproduce the component form of the N = (0|2) 2DTL equation [4] which can be reduced to the onedimensional N = 2 supersymmetric Toda chain equations [22] by the reduction constraint ∂ + 2 = ∂ − 2 . Let us also point out that bosonic and fermionic symmetries of this reduction were analyzed in detail in [23].…”