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

A2. Kinetic equation for the focusing NLS soliton gasN=2: collision of two cold soliton gases.Jump conditions:−c − [ρ 1c − ρ − 1 ] + [ρ 1cs 1c − ρ − 1 s− 1 ] = 0,−c − [ρ 2c − ρ − 2 ] + [ρ 2cs 2c − ρ − 2 s− 2 ] = 0,−c + [ρ 1c − ρ + 1 ] + [ρ 1cs 1c − ρ + 1 s+ 1 ] = 0,−c + [ρ 2c − ρ + 2 ] + [ρ 2cs 2c − ρ + 2 s+ 2 ] = 0.As a result, the densities and velocities of the components in theinteraction region areρ 1c = ρ 10(1 − κρ 20 )1 − κ 2 ρ 10 ρ 20, ρ 2c = ρ 20(1 − κρ 10 )1 − κ 2 ρ 10 ρ 20.s 1c = 4α 1 − κ(ρ 1c − ρ 2c )1 − κ(ρ 1c + ρ 2c ) , s 2c = −4α 1 + κ(ρ 1c − ρ 2c )1 − κ(ρ 1c + ρ 2c ) .The speeds of the ‘shocks’ :c − = −4α 1 + κρ 101 − κρ 10, c + = 4α 1 + κρ 201 − κρ 20.Gennady ElKinetic equation for a soliton gas