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

Wave field moments in a soliton gas: the KdV caseWe have obtained that in the thermodynamic limit dN N → πf (η, x, t)dη.On the other hand, it is well known that the integrated density of statesN N (λ) is the generating function for the averaged Kruskal integrals I (N)k:N N (λ) = 2 √ −λ +I (N) 10 = limL→∞ L∫ L0∞∑k=0I (N)k(−2λ) k , −λ ≫ 1 ,u N (x)dx , I (N) 11 = limL→∞ L∫ L0u 2 N (x)dx , . . .Then the thermodynamic limit of Kruskal integrals is found as∫∞I (N)k→ 22k+22k + 1 (−1)k+1 η 2k+1 f (η)dη , k = 0, 1, 2, . . .0Gennady ElKinetic equation for a soliton gas