Solitons of q-deformed quantum lattices and the quantum soliton

We use the classical N-soliton solution of a q-deformed lattice, the Maxwell-Bloch (MB) lattice, which we reported recently (Rybin A V, Varzugin G G, Timonen J and Bullough R K Year 2001 J. Phys. A: Math. Gen. 34 157) in order, ultimately, to fully comprehend the 'quantum soliton'. This object may be the source of a new information technology (Abram I 1999 Quantum solitons Phys. World 21-4). We suggested in Rybin et al 2001 that a natural quantum mechanical matrix element of the q-deformed quantum MB lattice becomes in a suitable limit the classical 1-soliton solution of the classical q-deformed MB lattice explicitly derived by a variant of the Darboux-Baecklund method. The classical q-deformed MB lattice was introduced in Bogoliubov N M, Rybin A V, Bullough R K and Timonen J 1995 Phys. Rev. A 52 1487. In this short paper the quantum inverse method is viewed as a Darboux-Baecklund transformation at quantum level, two q-deformed quantum lattices are introduced and solved, and relevant matrix elements are formally derived. Further investigation of the classical limits of these matrix elements must however be deferred until future work. (author)