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

Abstract

Abstract We prove existence, uniqueness and non-negativity of solutions of certain integral equations describing the density of states u(z) in the spectral theory of soliton gases for the one dimensional integrable focusing nonlinear Schrödinger 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 minimising (positive) measure for a certain energy functional is absolutely continuous and its density u(z) ⩾ 0 solves the required integral equation. 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, the integral equations for u, v represent nonlinear dispersion relation for the fNLS soliton gas. We also discuss smoothness and other properties of the obtained solutions. Finally, we obtain exact solutions of the above integral equations in the case of a KdV condensate and a bound state fNLS condensate. Our results is a step towards a mathematical foundation for the spectral theory of soliton and breather gases, which appeared in work of El and Tovbis (2020 Phys. Rev. E 101 052207). It is expected that the presented ideas and methods will be useful for studying similar classes of integral equation describing, for example, breather gases for the fNLS, as well as soliton gases of various integrable systems.