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

N=3: explicit formulae (El, Kamchatnov, Pavlov & Zykov, 2008).The three-component ‘cold-gas’ hydrodynamic reduction of the nonlocalkinetic equation∂ t u i = ∂ x (u i v i ), i = 1, 2, 3 ,v i = ξ i +3∑ǫ ik u k (v k − v i ) ǫ ik = ǫ ki .k=1has the Riemann invariant representationwhere∂ t r j = V j (r)∂ x r j , j = 1, 2, 3,V 1 = ζ 2r 2 − ζ 3 r 3r 2 − r 3 , V 2 = ζ 3r 3 − ζ 1 r 1r 3 − r 1 , V 3 = ζ 1r 1 − ζ 2 r 2r 1 − r 2ζ 1 = ξ 3ǫ 12 − ξ 2 ǫ 13ǫ 12 − ǫ 13, ζ 2 = ξ 1ǫ 23 − ξ 3 ǫ 12ǫ 23 − ǫ 12, ζ 3 = ξ 1ǫ 23 − ξ 2 ǫ 13ǫ 23 − ǫ 13Gennady ElKinetic equation for a soliton gas