Kinetic equation for a soliton gas - SOLITONS, COLLAPSES AND ...

Thermodynamic limitThe total density of statesρ N = 1 ∫0π Re−1dp N (λ ′ ) = 12πIn the thermodynamic limit limN→∞ ρ N = O(1).This is achieved by the following ( thermodynamic) spectral scalingN∑j=1k j|gap j | ∼1ϕ(η j )N|band j | ∼ exp {−γ(η j )N}, j = 1, . . .,Nwhere φ(η), γ(η) are some continuous positive functions on [0, 1].Venakides (1989) The continuum limit of theta functions;El, Krylov, Molchanov & Venakides (1999) Soliton turbulence as thethermodynamic limit od soliton latticesNote: |band j |/|gap j | → 0 as N → ∞ ∀j i.e. “infinite-soliton” limit.Gennady ElKinetic equation for a soliton gas