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In this paper we consider soliton and breather gases for one dimensionalintegrable focusing Nonlinear Schr\"{o}dinger Equation (fNLS). We deriveaverage densities and fluxes for such gases by studying the thermodynamic limitof the fNLS finite gap solutions. Thermodynamic limits of quasimomentum,quasienergy and their connections with the corresponding $g$-functions werealso established.  We then introduce the notion of periodic fNLS gases and calculate for themthe average densities, fluxes and thermodynamic limits of meromorphicdifferentials. Certain accuracy estimates of the obtained results are alsoincluded.  Our results constitute another step towards the mathematical foundation forthe spectral theory of fNLS soliton and breather gases that appeared in work ofG. El and A. Tovbis, Phys. Rev. E, 2020.

journal of physics. a, mathematical and theoretical

Recent developments in spectral theory of the focusing NLS soliton and  breather gases: the thermodynamic limit of average densities, fluxes and  certain meromorphic differentials; periodic gases