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

Hydrodynamic reductions of the kinetic equation.f t +(sf ) x = 0 ,s(η, x, t) = S(η)+∫ ∞0G(η, µ)[s(η, x, t)−s(µ, x, t)]f (µ, x, t)dµWe introduce u(η, x, t) = ηf (η, x, t), v(η, x, t) = −s(η, x, t) andconsider N-component ‘cold-gas’ ansatzN∑u = u i (x, t)δ(η − η i ),i=1which reduces (1) to a system of N hydrodynamic conservation laws,∂ t u i = ∂ x (u i v i ), i = 1, . . .,N ,where the velocities v i = v(η i , x, t) and the ‘densities’ u i are related viaN∑v i = ξ i + ǫ ik u k (v k − v i ), ǫ ik = ǫ ki ,k=1ξ i = S(η i ), ǫ ik = 1η i η kG(η i , η k ) > 0 , v i = s(η i ).Gennady ElKinetic equation for a soliton gas(1)