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

The thermodynamic limit of the Whitham equationsThe modulation system ∂ t N N (−η 2 ) = ∂ x V N (−η 2 )In the thermodynamic limit as N → ∞:dN N → πf (η)dη > 0 , dV N → −πf (η)s(η)dηs(η) and f (η) are related via:s(η) = −4η 2 + 1 η∫ 10lnη − µ∣η + µ ∣ f (µ)[s(η) − s(µ)]dµ, (1)Now, we postulate that on a larger scale, ∆x, ∆t ≫ 1:f (η) = f (η, x, t), s(η) = s(η, x, t)Then the modulation system transforms intof t = (fs) x , (2)Equations (2), (1) form a closed system : the kinetic equation for theKdV soliton gas of finite density (El 2003)Gennady ElKinetic equation for a soliton gas