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

Kinetic equation for a soliton gasGennady ElChernogolovka, July 2009Gennady ElKinetic equation for a soliton gas

MotivationDescription of “integrable turbulence”.V.E. Zakharov, Turbulence in integrable systems, Stud. Appl. Math.(2009)A particular important aspect: theory of soliton gases.V.E. Zakharov, Kinetic equation for solitons, JETP (1971)Here we consider only strongly integrable systems (like KdV, NLS etc.)Gennady ElKinetic equation for a soliton gas

From N-solitons/N-gap potentials to a soliton gasN-solitons: two approachesIST: reflectionless potentials (N-soliton solutions);Finite-gap theory: closing all spectral bands in the N-gap potentialleads to the N-soliton.Soliton gas: N → ∞; a generalised reflectionless potential (Marchenko)with shift invariant probability measure on it.IST:Gurevich, Zybin et. al. (2000, 2002): stochastic version of theLax-Levermore approach (KdV and defocusing NLS)Kotani (2008): KdV flow on generalized reflectionless potentials(Sato ’s Grassmanian approach)Finite-gap theory: El, Krylov, Molchanov and Venakides (1999):soliton gas as the thermodynamic limit of finite-gap potentials:N → ∞ , L → ∞ N/L ∼N∑k j = O(1)j=1Gennady ElKinetic equation for a soliton gas

From the Whitham equations to kinetic equation forsolitonsAssuming the existence of the invariant spectral measurecharacterising spatially homogeneous soliton gas, we want todescribe slow evolution of this measure for an inhomogeneousgas, i.e. to derive kinetic equation for solitons.Similarity with the formulation of the Whitham modulation theory(but with the averaging over ensemble rather than period!).P.D. Lax (1991): The zero dispersion limit, a deterministic analogueof turbulence.Hence our interest in the thermodynamic limit of the Whithamequations.Gennady ElKinetic equation for a soliton gas

OutlineKinetic equation for solitons as the thermodynamic limit ofthe Whitham equationsGeneralised kinetic equationHydrodynamic reductions and integrabilityConclusionsAlso, if time permits;Moments of the wave fieldFurther research directionsGennady ElKinetic equation for a soliton gas

The Whitham equations (KdV case)The generating equation for the KdV-Whitham system is (Flaschka,Forest, McLaughlin 1979)(dp N ) t = (dq N ) x ,where dp N and dq N are the quasimomentum and quasienergydifferentials on the hyperelliptic Riemann surface of genus N:R 2 (λ) =∏(λ − λ j ), λ ∈ C, λ j ∈ R .2N+1j=1Asymptotic behaviour near λ = −∞:−λ ≫ 1 : dp N ∼ − dλ(−λ) 1/2 ,dq N ∼ (−λ) 1/2 dλGennady ElKinetic equation for a soliton gas

The Whitham equationsThe differentials dp N and dq N are uniquely defined by the normalisation:∮dp N (λ) = 0 ,β i∮dq N (λ) = 0 ,β ii = 1, . . .,N . j 1 2 j3 42 N 1gapband2 j 1The fundamental wavenumbers k j and frequencies ω j are found as∮∮k j (λ 1 , . . .,λ 2N+1 ) = dp N (λ), ω j (λ 1 , . . . , λ 2N+1 ) = dq N (λ),α j 2 jα jGennady ElKinetic equation for a soliton gas

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

Thermodynamic limitThe total density of statesρ N = 1 ∫0π Re−1dp N (λ ′ ) = 12πIn the thermodynamic limit limN→∞ ρ N = O(1).This is achieved by the following ( thermodynamic) spectral scalingN∑j=1k j|gap j | ∼1ϕ(η j )N|band j | ∼ exp {−γ(η j )N}, j = 1, . . .,Nwhere φ(η), γ(η) are some continuous positive functions on [0, 1].Venakides (1989) The continuum limit of theta functions;El, Krylov, Molchanov & Venakides (1999) Soliton turbulence as thethermodynamic limit od soliton latticesNote: |band j |/|gap j | → 0 as N → ∞ ∀j i.e. “infinite-soliton” limit.Gennady ElKinetic equation for a soliton gas

The thermodynamic limit of the Whitham equationsThe modulation system ∂ t N N (−η 2 ) = ∂ x V N (−η 2 )In the thermodynamic limit as N → ∞:dN N → πf (η)dη > 0 , dV N → −πf (η)s(η)dηs(η) and f (η) are related via:s(η) = −4η 2 + 1 η∫ 10lnη − µ∣η + µ ∣ f (µ)[s(η) − s(µ)]dµ, (1)Now, we postulate that on a larger scale, ∆x, ∆t ≫ 1:f (η) = f (η, x, t), s(η) = s(η, x, t)Then the modulation system transforms intof t = (fs) x , (2)Equations (2), (1) form a closed system : the kinetic equation for theKdV soliton gas of finite density (El 2003)Gennady ElKinetic equation for a soliton gas

Kinetic equation for solitons: small-density expansionWe replace s → −s. Now s is the velocity of soliton gas.s(η) = 4η 2 + 1 ηf T + (fs) X = 0 , (1)∫ ∞0lnη + µ∣η − µ ∣f (µ)[s(η) − s(µ)]dµ, (2)The small-density, ρ = ∫ ∞0fdη ≪ 1, expansion of (2), yieldss(η) = 4η 2 + 1 η∫ ∞0lnη + µ∣η − µ ∣ f (µ)[4η2 − 4µ 2 ]dµ + O(ρ 2 ), (3)– the velocity of a ‘trial’ soliton in a rarefied soliton gas (Zakharov 1971).So Eqs. (1), (2) represent a generalisation of Zakharov’s kinetic equationfor a rarefied soliton gas to the case of the gas of finite density.Gennady ElKinetic equation for a soliton gas

Generalised kinetic equations for soliton gases with elasticcollisions El & Kamchatnov (PRL 2005)Main ingredients: (i) the speed of a free soliton S(η) and (ii) thephase shift ∆x η,µ = G(η, µ) due to the soliton-soliton collision.Introduce the spectral distribution function f (η) ≡ f (η, x, t) and themean speed of a ‘trial’ η- soliton s(η) ≡ s(η, x, t)Then the self-consistent definition of the soliton velocity s(η) in adense soliton gas with the spectral distribution f (η) is given by theintegral equations(η) = S(η) +∫ ∞0G(η, µ)[s(η) − s(µ)]f (µ)dµ ,Isospectrality implies the conservation equation for the spectraldistribution function f (η, x, t):f t + (sf ) x = 0 .Gennady ElKinetic equation for a soliton gas

Hydrodynamic reductions of the kinetic equation.f t +(sf ) x = 0 ,s(η, x, t) = S(η)+∫ ∞0G(η, µ)[s(η, x, t)−s(µ, x, t)]f (µ, x, t)dµWe introduce u(η, x, t) = ηf (η, x, t), v(η, x, t) = −s(η, x, t) andconsider N-component ‘cold-gas’ ansatzN∑u = u i (x, t)δ(η − η i ),i=1which reduces (1) to a system of N hydrodynamic conservation laws,∂ t u i = ∂ x (u i v i ), i = 1, . . .,N ,where the velocities v i = v(η i , x, t) and the ‘densities’ u i are related viaN∑v i = ξ i + ǫ ik u k (v k − v i ), ǫ ik = ǫ ki ,k=1ξ i = S(η i ), ǫ ik = 1η i η kG(η i , η k ) > 0 , v i = s(η i ).Gennady ElKinetic equation for a soliton gas(1)

Hydrodynamic reductions: N = 2For N = 2 the system of hydrodynamic laws assumes the form∂ t u 1 = ∂ x (u 1 v 1 ), ∂ t u 2 = ∂ x (u 2 v 2 )u 1 = 1ǫ 12v 2 − ξ 2v 1 − v 2 , u2 = 1ǫ 12v 1 − ξ 1v 2 − v 1 .Passing to the Riemann invariants we obtainv 1 t = v 2 v 1 x , v 2 t = v 1 v 2 x . (1)The system (1) is linearly degenerate, i.e. its characteristic velocities donot depend on the corresponding Riemann invariants.What about N > 2?Gennady ElKinetic equation for a soliton gas

Hydrodynamic reductions: N ≥ 3.Theorem (El, Kamchatnov, Pavlov & Zykov 2008)N-component hydrodynamic type system∂ t u i = ∂ x (u i v i ), i = 1, . . .,N ,v i = ξ i +N∑ǫ ik u k (v k − v i ), ǫ ik = ǫ ki ,k=1where ξ 1 , ξ 2 , . . . , ξ N are constants and ǫ is a constant symmetric matrix,ǫ ik = ǫ ki , is:diagonalizablelinearly degenerate,semi-Hamiltonian (i.e. integrable - Tsarev 1985, 1991),for any N.The proof is based on the theory of integrable linearly degeneratehydrodynamic type systems developed by Pavlov (1987) and Ferapontov(1991).Gennady ElKinetic equation for a soliton gas

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

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

N=3: explicit formulae (El, Kamchatnov, Pavlov & Zykov, 2008).The three-component ‘cold-gas’ hydrodynamic reduction of the nonlocalkinetic equation∂ t u i = ∂ x (u i v i ), i = 1, 2, 3 ,v i = ξ i +3∑ǫ ik u k (v k − v i ) ǫ ik = ǫ ki .k=1has the Riemann invariant representationwhere∂ t r j = V j (r)∂ x r j , j = 1, 2, 3,V 1 = ζ 2r 2 − ζ 3 r 3r 2 − r 3 , V 2 = ζ 3r 3 − ζ 1 r 1r 3 − r 1 , V 3 = ζ 1r 1 − ζ 2 r 2r 1 − r 2ζ 1 = ξ 3ǫ 12 − ξ 2 ǫ 13ǫ 12 − ǫ 13, ζ 2 = ξ 1ǫ 23 − ξ 3 ǫ 12ǫ 23 − ǫ 12, ζ 3 = ξ 1ǫ 23 − ξ 2 ǫ 13ǫ 23 − ǫ 13Gennady ElKinetic equation for a soliton gas

N=3: explicit formulae (El, Kamchatnov, Pavlov & Zykov, 2008).The Riemann invariants r 1 , r 2 , r 3 are expressed in terms of the densitiesu 1 , u 2 , u 3 asr 1 =r 2 =r 3 =(ǫ 12 − ǫ 13 )(ǫ 12 ǫ 13 u 1 + ǫ 12 ǫ 23 u 2 + ǫ 13 ǫ 23 u 3 + ǫ 23 )[(ξ 3 − ξ 1 )ǫ 12 + (ξ 1 − ξ 2 )ǫ 13 ]u 1 − (ξ 2 − ξ 3 )(ǫ 12 u 2 + ǫ 13 u 3 + 1) ,(ǫ 23 − ǫ 12 )(ǫ 12 ǫ 13 u 1 + ǫ 12 ǫ 23 u 2 + ǫ 13 ǫ 23 u 3 + ǫ 13 )[(ξ 1 − ξ 2 )ǫ 23 + (ξ 2 − ξ 3 )ǫ 12 ]u 2 − (ξ 3 − ξ 1 )(ǫ 12 u 1 + ǫ 23 u 3 + 1) ,(ǫ 13 − ǫ 23 )(ǫ 12 ǫ 13 u 1 + ǫ 12 ǫ 23 u 2 + ǫ 13 ǫ 23 u 3 + ǫ 12 )[(ξ 2 − ξ 3 )ǫ 13 + (ξ 3 − ξ 1 )ǫ 23 ]u 3 − (ξ 1 − ξ 2 )(ǫ 13 u 1 + ǫ 23 u 2 + 1) .Gennady ElKinetic equation for a soliton gas

ConclusionsThe thermodynamic limit of the Whitham equations associated withhyperelliptic Riemann surfaces leads to the kinetic equations for thecorresponding soliton gases.The original Zakharov (1971) prescription for the determination ofthe velocity of a trial soliton in a rarefied soliton can be directlyextended to the case of the soliton gas of finite density.N-component cold gas hydrodynamic reductions of the generalizedkinetic equation for solitons represent linearly degeneratesemi-Hamiltonian (integrable) systems of hydrodynamic type for anyN.Gennady ElKinetic equation for a soliton gas

References[1] El, G.A., Krylov A.L., Molchanov, S.A. and Venakides, S., ”Solitonturbulence as a thermodynamic limit of stochastic soliton lattices”,Advances in Nonlinear Mathematics and Science, PHYSICA D, 152/153,2001, pp 653-664.[2] El, G.A., ”The thermodynamic limit of the Whitham equations”,PHYS LETT A, 311, 2003, pp 374-383.[3] El, G.A. and Kamchatnov, A.M., ”Kinetic equation for a densesoliton gas”, PHYS REV LETT 95, 2005, Art No 204101, 4 pp.[4] El, G.A., Kamchatnov, A.M., Pavlov, M.V., and Zykov, S.A.,”Nonlocal kinetic equation: integrable hydrodynamic reductions,symmetries and exact solutions” nlin. arXive 2008.Gennady ElKinetic equation for a soliton gas

Wave field moments in a soliton gas: the KdV caseWe have obtained that in the thermodynamic limit dN N → πf (η, x, t)dη.On the other hand, it is well known that the integrated density of statesN N (λ) is the generating function for the averaged Kruskal integrals I (N)k:N N (λ) = 2 √ −λ +I (N) 10 = limL→∞ L∫ L0∞∑k=0I (N)k(−2λ) k , −λ ≫ 1 ,u N (x)dx , I (N) 11 = limL→∞ L∫ L0u 2 N (x)dx , . . .Then the thermodynamic limit of Kruskal integrals is found as∫∞I (N)k→ 22k+22k + 1 (−1)k+1 η 2k+1 f (η)dη , k = 0, 1, 2, . . .0Gennady ElKinetic equation for a soliton gas

Wave field moments in a soliton gas: the KdV caseFor the two first moments we haveu(x, t) = 4∫ ∞Important restriction:0ηf (η, x, t)dη , u 2 (x, t) = 163∫ ∞0η 3 f (η, x, t)dησ = u 2 − u 2 ≥ 0 (1)For instance, for a one-component soliton gas f = f 0 (x, t)δ(η − η 0 ), sou = 4η 0 f 0 (x, t), u 2 = 163 η3 0 f 0(x, t).i.e. the varianceσ = 16f 0 η0( 2 η 03 − f 0).Now one can see that (1) imposes a restriction on the possible densityvalues f 0 for a one-component soliton gas with a given η 0 :Critical density for a soliton gas.f 0 < f cr = η 03 .Gennady ElKinetic equation for a soliton gas

Future researchConnection with hydrodynamic chains and 2+1 dispersionlesssystems.Motivation: The collisionless Boltzmann kinetic equation for thedistribution function f (p, x, t)∂f∂t + p ∂f (∫ ) ∂f ∂f∂x − ∂x dp ∂p = 0is equivalent to the Benney moment chain (Zakharov 1981, Gibbons1981)∂A n+ ∂A ∫n+1 ∂A 0+ nA n−1∂t ∂x ∂x = 0 , A n = p n fdpand to the dispersionless KP-II equation (Kupershmidt & Manin ??,Gibbons 1984, Kodama 1988)(u t + 6uu x ) x + u yy = 0 .Construction of physically relevant exact solutions.PDF, correlation function.Gennady ElKinetic equation for a soliton gas

Appendix 1. N=3: Exact solutions.1. Similarity solutionsThe family of the similarity solutions.r i = 1 ( x)t αli , i = 1, 2, 3,tis implicitly specified by the algebraic systemxt= c 1 ζ 1 (l 1 ) γ + c 2 ζ 2 (l 2 ) γ + c 3 ζ 3 (l 3 ) γ ,−1 = c 1 (l 1 ) γ + c 2 (l 2 ) γ + c 3 (l 3 ) γ ,0 = c 1 (l 1 ) γ−1 + c 2 (l 2 ) γ−1 + c 3 (l 3 ) γ−1 ,where γ = −1/α and c 1 , c 2 , c 3 are arbitrary constants.Gennady ElKinetic equation for a soliton gas

Appendix 1. N=3: Exact solutions.2. Quasi-periodic solutionsThe family of the quasi-periodic (3 periods) solution is implicitly specifiedby the system∫r 1x = ζ 1ξdξ√R7 (ξ) + ζ 2∫r 2ξdξ√R7 (ξ) + ζ 3∫r 3ξdξ√R7 (ξ) ,−t =0 =∫r 1∫r 1∫r 2ξdξ√R7 (ξ) +∫r 2dξ√R7 (ξ) +∫r 3ξdξ√R7 (ξ) +∫r 3dξ√R7 (ξ) +ξdξ√R7 (ξ) ,dξ√R7 (ξ) ,where7∏R 7 (ξ) = (ξ − E m ),m=1E 1 < E 2 < · · · < E 7 are arbitrary real constants.Gennady El Kinetic equation for a soliton gas

Appendix 2. Kinetic equation for the focusing NLS solitongasThe focusing nonlinear Schrödinger (NLS) equation has the formiu t + u xx + 2|u| 2 u = 0Then the kinetic equation for a dense gas of bright NLS solitons iss(α, γ) = −4α + 1 ∫ ∞2γ −∞Here∫ ∞0f T + (sf ) X = 0 ,lnλ − µ2∣λ − µ ∣ f (ξ, η)[s(α, γ) − s(ξ, η)]dξdη .f ≡ f (α, γ; X, T), s ≡ s(α, γ; X, T)λ = α + iγ ,µ = ξ + iηGennady ElKinetic equation for a soliton gas

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