Supersymmetric Structure of two Families of Solitons

Solitons have generated considerable interest in the cold atoms and condensed matter communities. We demonstrate that two families of $n$-soliton solutions (with $n$ an integer) -- one for the attractive nonlinear Schr\"{o}dinger (NLS) equation, and one for the sine-Gordon (sG) equation -- originate from a quantum-mechanical supersymmetric (QM-SUSY) chain connecting a set of reflectionless operators $\hat{H}_n$. The families consist of breather-type solitons for NLS\footnote{D. Schrader, IEEE J. Quantum Electron. {\bf 31}, 2221 (1995).} and multi-(anti)kink solitons with specific velocities for sG. The operators $\hat{H}_n$, which we refer to as Akulin`s Hamiltonians\footnote{V. M. Akulin, \underline{Coherent Dynamics of Complex Quantum Systems} (Springer, Heidelberg, 2006).}, form reflectionless direct-scattering initial conditions for the inverse scattering method. Such a QM-SUSY chain is analogous to the known connection between QM-SUSY chains of P\"{o}schl-Teller potentials and solitons of the Korteweg-de Vries (KdV) equation\footnote{Sukumar, J. Phys. A {\bf 19}, 2297 (1986)}. The existence of QM-SUSY chains connecting soliton solutions, now for three different integrable nonlinear equations, sheds light on the underlying mechanisms responsible for soliton generation.