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

Linearly degenerate conservation lawsIt follows from Pavlov (1987) and Ferapontov (1991) that the system ofconservation lawsu i t = (ui v i ) x , v i = v i (u(r)) i = 1, . . .,Nis a semi-Hamiltonian linearly degenerate hydrodynamic type system iffthe densities u i and velocities v i (u) admit the representationsu i = det∆(1) idet ∆ (−1)i+1 P i (r i ),v i = det∆(2) idet∆ (1)iin terms of the Stäkel matrix ∆ via N functions r k ; here P i (r i ) arearbitrary functions.For the N - component hydrodynamic reductions of the kineticequation it was proved in (El, Kamchatnov, Pavlov & Zykov 2008)that such a parametrization exists for any N.Hence: integrability of the ‘cold-gas’ hydrodynamic reductions for any N.Gennady ElKinetic equation for a soliton gas