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

Whitham equations as the equations for the spectralmeasureAssume that the finite-band part of the spectrum λ lies in [−1, 0].We integrate the Whitham system ∂ t dp N (λ) = ∂ x dq N (λ) on the real linefrom −1 to −η 2 ∈ [−1, 0] and take the real part:Here∂ t N N (−η 2 ) = ∂ x V N (−η 2 ).N N (λ) = 1 ∫λπ Re−1dp N (λ ′ )is the integrated density of states (Johnson & Moser 1982), andV N (λ) = 1 π Re ∫ λ dq N (λ ′ ) – its temporal analog.−1Importantly, dN N (−η 2 ) is a measure.Gennady ElKinetic equation for a soliton gas