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

Wave field moments in a soliton gas: the KdV caseFor the two first moments we haveu(x, t) = 4∫ ∞Important restriction:0ηf (η, x, t)dη , u 2 (x, t) = 163∫ ∞0η 3 f (η, x, t)dησ = u 2 − u 2 ≥ 0 (1)For instance, for a one-component soliton gas f = f 0 (x, t)δ(η − η 0 ), sou = 4η 0 f 0 (x, t), u 2 = 163 η3 0 f 0(x, t).i.e. the varianceσ = 16f 0 η0( 2 η 03 − f 0).Now one can see that (1) imposes a restriction on the possible densityvalues f 0 for a one-component soliton gas with a given η 0 :Critical density for a soliton gas.f 0 < f cr = η 03 .Gennady ElKinetic equation for a soliton gas