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

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 minimizing (positive) measure for certain energy functional is absolutely continuous and its density $u(z)\ge 0$ 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, 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 first step towards a mathematical foundation for the spectral theory of soliton and breather gases, which appeared in work of El and Tovbis, Phys. Rev. E, 2020. 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.

arXiv comment32 pages, 1 figure

References in corpus (9)

• Soliton gases and generalized hydrodynamics
• Kinetic equation for a dense soliton gas
• A geometric viewpoint on generalized hydrodynamics
• The thermodynamic limit of the Whitham equations
• Spectral theory of soliton and breather gases for the focusing nonlinear Schrödinger equation
• Dam break problem for the focusing nonlinear Schrödinger equation and the generation of rogue waves
• Formation of rogue waves from the locally perturbed condensate
• Equations of state in generalized hydrodynamics
• Rigorous asymptotics of a KdV soliton gas

Cited by in corpus (1)

• Soliton gas in integrable dispersive hydrodynamics