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We prove existence, uniqueness and non-negativity of solutions of certainintegral equations describing the density of states $u(z)$ in the spectraltheory of soliton gases for the one dimensional integrable focusing NonlinearSchr\"{o}dinger Equation (fNLS) and for the Korteweg de Vries (KdV) equation.Our proofs are based on ideas and methods of potential theory. In particular,we show that the minimizing (positive) measure for certain energy functional isabsolutely continuous and its density $u(z)\geq 0$ solves the required integralequation. In a similar fashion we show that $v(z)$, the temporal analog of$u(z)$, is the difference of densities of two absolutely continuous measures.Together, integral equations for $u,v$ represent nonlinear dispersion relationfor the fNLS soliton gas. We also discuss smoothness and other properties ofthe obtained solutions. Finally, we obtain exact solutions of the aboveintegral equations in the case of a KdV condensate and a bound state fNLScondensate. Our results is a first step towards a mathematical foundation forthe spectral theory of soliton and breather gases, which appeared in work of Eland Tovbis, Phys. Rev. E, 2020. It is expected that the presented ideas andmethods will be useful for studying similar classes of integral equationdescribing, for example, breather gases for the fNLS, as well as soliton gasesof various integrable systems.

On minimal energy solutions to certain classes of integral equations related to soliton gases for integrable systems