Solitons of \(q\)-deformed quantum lattices and the quantum soliton (Q2773076)

The authors of this short paper discuss the quantum inverse method viewed as a Darboux-Bäcklund transformation at quantum level. Two \(q\)-deformed quantum lattices are introduced and solved and at the same time relevant matrix elements are formally derived. The \(N\)-string bound state solutions of the quantum \(q\)-deformed lattice Maxwell-Bloch (LMB) equations are investigated. It is shown that there are at least two such \(q\)-deformed LMB equations with two different Hamiltonians which have \(N\)-string bound state solutions and that this is a generic property which is important to a class of lattices and their various continuum limits. Their eigenstates are taken in the strict Bethe ansatz (or quantum inverse method) form rather than in the coordinate Bethe ansatz. These Bethe states are then needed to evaluate the matrix elements taken in an explicit form.