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

Linearly degenerate hydrodynamic type systemsDefinition. A diagonal hydrodynamic type system rt i = V i (r)rx i is calledlinearly degenerate if ∂ i V i = 0 ∀i. (∂ k ≡ ∂/∂r k )A semi-Hamiltonian (i.e. integrable) linearly degenerate hydrodynamictype system rt i = V i (r)rx i is characterised by the so-called Stäkel matrix⎛φ 1 1 (r1 ) ... φ 1 ⎞N (rN )... ... ...∆ =⎜ φ N−21 (r 1 ) φ N−2N(r N )⎟⎝ φ N−11 (r 1 ) φ N−1N(r N ) ⎠1 ... 1where φ i k (rk ) are certain functions, so that (Ferapontov 1991)where ∆ (k)iV i (r) = det∆(2) i,det∆ (1)iis the matrix ∆ without k-th row and i-th column.Gennady ElKinetic equation for a soliton gas