Kinetic equation for a soliton gas - SOLITONS, COLLAPSES AND ...
Kinetic equation for a soliton gas - SOLITONS, COLLAPSES AND ...
Kinetic equation for a soliton gas - SOLITONS, COLLAPSES AND ...
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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 <strong>soliton</strong> latticesNote: |band j |/|gap j | → 0 as N → ∞ ∀j i.e. “infinite-<strong>soliton</strong>” limit.Gennady El<strong>Kinetic</strong> <strong>equation</strong> <strong>for</strong> a <strong>soliton</strong> <strong>gas</strong>