Effect of scalings and translations on the supersymmetric quantum mechanical structure of soliton systems

Abstract

We investigate a peculiar supersymmetry of the pairs of reflectionless quantum mechanical systems described by $n$-soliton potentials of a general form that depends on $n$ scaling and $n$ translation parameters. We show that if all the discrete energy levels of the subsystems are different, the superalgebra, being insensitive to translation parameters, is generated by two supercharges of differential order $2n$, two supercharges of order $2n+1$, and two bosonic integrals of order $2n+1$ composed from Lax integrals of the partners. The exotic supersymmetry undergoes a reduction when $r$ discrete energy levels of one subsystem coincide with any $r$ discrete levels of the partner; the total order of the two independent intertwining generators reduces then to $4n\ensuremath{-}2r+1$, and the nonlinear superalgebraic structure acquires a dependence on $r$ relative translations. For a complete pairwise coincidence of the scaling parameters which control the energies of the bound states and the transmission scattering amplitudes, the emerging isospectrality is detected by a transmutation of one of the Lax integrals into a bosonic central charge. Within the isospectral class, we reveal a special case giving a new family of finite-gap first order Bogoliubov-de Gennes systems related to the Ablowitz-Kaup-Newell-Segur integrable hierarchy.