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

We show that localized N-body soliton states exist for a quantum integrable derivative nonlinear Schrodinger model for several non-overlapping ranges (called bands) of the coupling constant η. The number of such distinct bands is given by Euler's ϕ-function which appears in the context of number theory. The ranges of ηwithin 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).