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

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Appendix 2. Kinetic equation for the focusing NLS solitongasN=2: collision of two cold soliton gases.Let ρ 1 (x, t) = ρ 10 = constant 1 , ρ 2 (x, t) = ρ 10 = constant 2 .No continuous solution.Two hyperbolic conservation laws available:∂ρ 1∂t + ∂(s 1ρ 1 )= 0,∂x∂ρ 2∂t + ∂(s 2ρ 2 )= 0,∂xs 1 = s 1 (ρ 1 , ρ 2 ) s 2 = s 2 (ρ 1 , ρ 2 )Weak solution: three constant states separated by two strongdiscontibuities.ρ10ρ 1c+ρ2cρ1cρ2cρ20c - tc + txGennady ElKinetic equation for a soliton gas
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
Page 1 and 2: Kinetic equation for a soliton gasG
Page 3 and 4: From N-solitons/N-gap potentials to
Page 5 and 6: OutlineKinetic equation for soliton
Page 7 and 8: The Whitham equationsThe differenti
Page 9 and 10: Thermodynamic limitThe total densit
Page 11 and 12: Kinetic equation for solitons: smal
Page 13 and 14: Hydrodynamic reductions of the kine
Page 15 and 16: Hydrodynamic reductions: N ≥ 3.Th
Page 17 and 18: Linearly degenerate conservation la
Page 19 and 20: N=3: explicit formulae (El, Kamchat
Page 21 and 22: References[1] El, G.A., Krylov A.L.
Page 23 and 24: Wave field moments in a soliton gas
Page 25 and 26: Appendix 1. N=3: Exact solutions.1.
Page 27: Appendix 2. Kinetic equation for th

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

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