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

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In this paper we consider soliton and breather gases for one dimensional integrable focusing Nonlinear Schr\"{o}dinger Equation (fNLS).

• We derive average densities and fluxes for such gases by studying the thermodynamic limit of the fNLS finite gap solutions.
• Thermodynamic limits of quasimomentum, quasienergy and their connections with the corresponding $g$-functions were also established.
• We then introduce the notion of periodic fNLS gases and calculate for them the average densities, fluxes and thermodynamic limits of meromorphic differentials.