Novel multi-band quantum soliton states for a derivative nonlinear Schroedinger model

We show that localized N-body soliton states exist for a quantum integrable derivative nonlinear Schroedinger model for several non overlapping ranges (called bands) of the coupling constant η. The number of such distinct bands is given by Euler's phi-function which appears in the context of number theory. The ranges of η within each band can also be determined completely using concepts from number theory such as Farey sequences and continued fractions. We observe that N-body soliton states appearing within each band can have both positive and negative momentum. Moreover, for all bands lying in the region η>0, soliton states with positive momentum have positive binding energy (called bound states), while the states with negative momentum have negative binding energy (anti-bound states)