Kinetic equation for a soliton gas and its hydrodynamic reductions

Kinetic equation for a soliton gas and its h ydro dynamic reductions G.A. El 1 , A.M. Kamc hatnov 2 , M.V. P avlo v 3 and S.A. Zyko v 4 1 Department of Mathemati cal Sciences, Loughbor ough Universit y , UK 2 Institut e of Sp ectrosco p y , Russian Academy of S ci ences, T roitsk, Moscow Regio n, Russia 3 Leb edev Physical Instit ute, Russia n Aca dem y o f Sciences, Moscow 4 SISSA, T rieste, Ital y , and Institut e of Metal Ph ysi cs, Ura ls Divisi on o f Russia n Academy of Sciences, Ek at erinburg, Russia Abstract W e in tr o duce and study a new class of kin etic equations, which arise in the descrip - tion of nonequilibrium macroscopic dyn amics of solito n gases with elastic collisions b et w een solitons. These equ ations represent n onlinear integro- diﬀeren tial sys tems and ha ve a nov el structure, whic h w e in vestiga te b y studying in detail the class of N - comp onen t ‘cold-gas’ h ydro dyn amic redu ctions. W e pr o ve that these redu ctions repr e- sen t integrable linearly degenerate h ydro dyn amic t yp e s ystems for arbitrary N whic h is a strong evidence in fa vour of in tegrabilit y of the full kinetic equation. W e derive compact explicit representa tions for th e Riemann inv ariants and charac teristic v elo c- ities of the h ydro dyn amic reductions in terms of th e ‘cold-gas’ comp onen t d ensities and construct a n um b er of exact solutions having sp ecial p rop erties (quasi-p erio d ic, self-similar). Hydro dyn amic symmetries are then deriv ed and in v estigated. T he ob- tained results shed th e ligh t on the stru cture of a con tinuum limit for a large class of in tegrable systems of h ydro dyn amic t yp e and are also relev an t to the d escription of turbulent motion in conserv ativ e compressible ﬂows. 1 In tro ductio n and summary of r e sults The p ossibilit y of mo delling certain ty p es of turbulen t motion with the aid o f the equations for weak limits of highly o scillatory disp ersiv e compressible ﬂo ws (the so-called Whitham mo dulation equations [42], [15], [31]) was p ointed out b y P .D. Lax in [30]. While this “de- terministic analogue of turbulence” has ob vious limitatio ns to its p o ssible applications to the description of hydrodynamic (incompressible) turbulent ﬂows , it op ens a whole new p er- sp ectiv e for constructing the statistical description of purely conserv a tiv e w a v e regimes in in tegrable disp ersiv e system s b y a ssigning appro pr ia te probabilistic measures to the wa v e 1 sequence s, so that their we ak limits could then b e rega r ded as ensemble aver ages . Suc h an uncon v en tional union of in tegrability and sto c hasticity has a natural phys ical motiv ation: nonlinear disp ersiv e w av es, while often b eing successfully mo delled by in tegrable systems, could demonstrate a v ery complex behaviour calling for a statistical description c haracteristic of the classical turbulence theories. Recen tly , a closely related prog ramme for t he construc- tion of the theory of wav e turbulenc e in the framew orks of integrable systems has b een put forw a r d b y V.E. Zakharo v [49]. One of the imp ortan t problems arising in this connection is the description of “soliton gases” — ra ndom distributions o f solitons whic h can b e ma t hematically deﬁned in terms of generalised reﬂectionless p otentials with shift inv ariant probabilit y measure on them (see e.g. [29]). D ue to isosp ectrality of the “pr imitive” microscopic ev olution, the macroscopic dynam- ics of a homogeneous soliton ga s is trivial (for the so-called ‘strongly integrable’ systems, suc h as the Kortew eg – de V ries (KdV), nonlinear Schr¨ odinger (NLS) or Kadomtsev -P etviash vili (KP-I I) equations — see [49]), namely , all the statistical c haracteristics can b e sp eciﬁed arbitrarily at the initial momen t and remain unch anged in time. Ho w ev er, if the soliton gas is spatially inhomogeneous, i.e. if t he probability distribution function dep ends on the space co ordinate, then nontrivial macroscopic dynamics can o ccur due to phase shifts of individ- ual solitons in their collisions with eac h o ther. An approximate kinetic equation describing spatial ev olution of the soliton distribution function in a rareﬁed gas of the KdV solitons, when these phase shifts can b e tak en in to accoun t explicitly , w a s deriv ed b y Zakharov back in 197 1 [46]. Generalization of Za kharo v’s kinetic equation to the case of a solito n gas of ﬁnite den- sit y has b een made p ossible rather recen tly [6] and required consideration of the contin uum thermo dynamic limit of the Whitham mo dulation equations asso ciat ed with ﬁnite-gap p o- ten tials. In t he thermo dynamic limit, the nonlinear interacting wa v e mo des tr a nsform in t o randomly distributed lo calised states (solito ns) and the mo dulation system assumes the form of a nonlinear kinetic equation. This new kinetic equation w as extended, using ph ysical r ea- soning, in [7] to o ther integrable sys tems with t w o-particle elastic in teractions of solitons (i.e. when m ulti-part icle eﬀects are absen t). The kinetic equation for solitons in general form represen ts a nonlinear integro-diﬀeren tial system f t + ( sf ) x = 0 , s ( η ) = S ( η ) + 1 η ∞ Z 0 G ( η , µ ) f ( µ )[ s ( µ ) − s ( η )] dµ . (1) Here f ( η ) ≡ f ( η , x, t ) is the distribution function and s ( η ) ≡ s ( η , x, t ) is the a sso ciated transp ort velocity . The (given) functions S ( η ) and G ( η , µ ) do not dep end on x and t . The function G ( η , µ ) is assumed to b e symmetric, i.e. G ( η , µ ) = G ( µ , η ). The c ho ice S ( η ) = 4 η 2 , G ( η , µ ) = log     η − µ η + µ     (2) corresp onds to the K dV soliton gas [6 ], where the KdV equation is take n in the canonical form φ t − 6 φφ x + φ xxx = 0 . (3) 2 In the KdV con text, η ≥ 0 is a real-v alued spectral parameter (to b e precise, b efore the passage to the contin uum limit one has λ k = − η 2 k , where λ k , k = 1 , . . . , N a r e the dis- crete eigen v alues of the Schr¨ odinger op erator) , thus the f unction f ( η , x, t ) is the distribution function of solitons o v er spectrum so tha t κ = R ∞ 0 f ( η ) dη = O (1) is the spatial densit y of solitons. If κ ≪ 1, the ﬁrst order approximation of (1), (2) yields Zakharov ’s kinetic equation for a dilute gas of KdV solitons [46] (see equations (28), (29) b elow). The quan tit y S ( η ) in (1) has a natura l meaning of the v elo cit y of an isolated (free) soliton with the sp ectral parameter η and the function 1 η G ( η , µ ) is the expression for a phase shift of this soliton o ccurring af ter its collision with a no ther soliton hav ing the sp ectral par a meter µ < η . Then s ( η , x, t ) acquires the meaning of the self-consisten tly deﬁned mean lo cal v elo city of solitons with the sp ectral parameter close to η (see [7]). Theory of nonlo cal kinetic equations of the form (1) is not dev elop ed y et. P ossible approac hes to their treatmen t w ere discuss ed in [2] in connection with sp ecial classes of exact solutions fo r the Boltzmann kinetic equation for Maxw ellian pa rticles. The deriv ation of (1), (2) as a certain (a lb eit singular) limit of the in tegrable KdV-Whitham system suggests tha t this new kinetic equation is also an in tegra ble system, a t least for sp ecial c hoices of functions G ( η , µ ). A natural question arising in this connection is: what is the exact meaning of in tegrability f or the equations of the type (1)? In tegrability of kinetic equations has b een the ob ject of intens iv e studies in recen t decades. F o r instance, integrabilit y of the collisionless Boltzmann equation (whic h is sometimes called the Vlasov equation) can b e deﬁned in terms of tw o o ther closely connected (ev en equiv alen t in some sense) ob jects: the Benney h ydro dynamic c hain [3 ], [47 ], [17 ] and the disp ersionless limit of the Kado m tsev–P etviash vili equation ([2 7 , 2 8], [21]. It turns out that all these t hree diﬀeren t nonlinear pa rtial diﬀeren tial equations p ossess t he same inﬁnite set of N -comp onen t h ydro dynamic reductions parameterised by N arbitrary functions of a single v ariable [18 , 19 ] (w e note tha t the solutions to these N -component reductions a re parameterised, in their turn, b y another N arbitrary functions of a single v ariable). This prop erty w as used in [12, 13](see also [48], [14], [20], [34]) when in tro ducing the in t egr a bilit y criterion for a wide class of kinetic equations, corresp onding h ydro dynamic c hains and 2+1 quasilinear equations. Moreo ve r, it w as prov ed in [36] that the existence of at least one N -comp onent hydrodynamic reduction written in the so-called symmetric form is suﬃcien t for in tegrabilit y in the sense of [12]. Another p ossible approac h to analyse an in tegrable kinetic equation is to use the fact tha t it po ssess es inﬁnitely many particular solutions determined by the corresp onding h ydro dynamic reductions (see [3 4] for details). The distinctiv e feature of the kinetic equation (1) is its nonlo cal structure, which repre- sen ts an obstacle to the direct application to it of t he approac hes deve lop ed in [36] and [34]. F o r instance, the p ossibilit y of a n explicit construction of symmetric hy dro dynamic reduc- tions, and ev en the existence of such reductions f or (1), are op en questions at the moment. In this pap er, w e study a particular, y et probably the most imp o r t an t fro m the viewpoint of capturing the essen tia l prop erties of the full equation, f a mily of the ‘cold-g a s’ N -comp onen t h ydro dynamic reductions of (1) obtained via the delta-function ansatz for the distribution function f ( η , x, t ) = P N i =1 f i ( x, t ) δ ( η − η i ), where η N > η N − 1 > · · · > η 1 > 0 ar e a rbitrary n umbers. Then the v elo city distribution s ( η , x, t ) ov er the ‘sp ectrum’ b ecomes a discrete set of functions { s i ( x, t ) : s i = s ( η i , x, t ) , i = 1 , . . . , N } and t he sough t reductions family 3 assumes the form of a system of h ydro dynamic conserv ation laws u i t = ( u i v i ) x , i = 1 , . . . , N , (4) where the the ‘densities ’ u i = η i f i ( x, t ) and the v elo cities v i = − s i ( x, t ) ar e related a lge- braically: v i = ξ i + X m 6 = i ǫ im u m ( v m − v i ) , ǫ ik = ǫ k i . (5) Here ξ i = − S ( η i ) , ǫ ik = 1 η i η k G ( η i , η k ) , i 6 = k . Despite the deceptiv ely simple for m o f system (4), (5), a n attempt of the analysis of its inte grabilit y prop erties b y employing standard metho ds of the theory of hydrodynamic t yp e systems (v eriﬁcation of the Haantjes tensor v anishing, computation of the Riemann in v ariants in terms of the densities of conserv at io n la ws, establishing the semi-Hamiltonian prop ert y etc. — see, e.g. [37]) rev eals serious tec hnical problems already fo r a mo dest N = 4. The reason f or suc h unexp ected diﬃculties in the apparen tly straigh tfo rw ard pro ce- dure lies in the fact that t he existing theory heavily relies on the knowled ge of the exp l i c it dep endence of the co eﬃcien t matrix of the h ydro dynamic t yp e system on ﬁeld v ariables while the dep endence v i ( u ) in (4) is given im p licitly b y alg ebraic system (5). It turns out that the resolution of t his system for v i using standard computer algebra pac k ages b ecomes notoriously resource consuming with the gro wth of N and do es not hold an y promise of get- ting structurally transparen t r esults for the Riemann inv ariants and c haracteristic v elo cities. This makes the standard direct r oute completely prosp ectless from the viewp o in t o f pro ving in tegrability of (4), (5) a nd obtaining explicit analytic results for an arbitr ary N . T o deal with the sp eciﬁc structure of system (4), (5) w e dev elop in this pap er a new approach, whic h has enabled us to p erform the complete analysis of its integrabilit y for an arbitrary N and, in part icular, to deriv e compact and elegan t represen tat ions for the Riemann in v a rian ts a nd c haracteristic ve lo cities. The main results of the pap er can b e summarized as follows: • W e prov e that reductions ( 4), (5) represen t line arly de gener ate inte gr able systems of h ydro dynamic type for arbitr ary N . This is done b y pro ving the existence of a certain represen tation of the densities u i and v elo cities v i in terms of the so-called St¨ ac ke l matrix whic h depends on N functions r i ( x, t ), whic h are t he R iemann inva riants of equations (4), (5). W e a lso prov e that system (4) , (5) b elongs to the Egorov class (see Def. 7.1 in Section 7). Moreo ve r, as a by -pro duct of our analysis, w e conclude that the system under study is the only (up to unessen tial transformations) system of h ydro dynamic t yp e whic h is sim ultaneously Egoro v and linearly degenerate. The c haracteristic v elo cities, conserv atio n law densities and symmetries (commuting ﬂo ws) for such systems are ﬁxed b y N ( N − 1) / 2 symmetric constan ts ǫ ik , i 6 = k , and N constan ts ξ i (i.e. b y N ( N + 1) / 2 constan ts in to tal). • W e deriv e an explicit Riemann in v arian t represen tat ion of system (4), (5), r i t = v i ( r ) r i x , i = 1 , . . . , N , (6) 4 where the Riemann inv arian ts r i are expressed in terms o f the comp onen t densities u 1 , u 2 , . . . , u N as r i = − 1 u i 1 + X m 6 = i ǫ im u m ! , i = 1 , . . . , N (7) and for the c ha racteristic ve lo cities v i ( r ) w e o bta in v i = 1 u i N X m =1 ξ m β im , where u i = N X m =1 β im . (8) Here the matrix β = − ǫ − 1 where the oﬀ-diagona l elemen t s of the symmetric mat r ix ǫ are ﬁxed b y system (5) while the diagonal elemen ts are deﬁned as ǫ ii = r i . Remark ably , the o ﬀ -diagonal symmetric elemen ts of the matrix β are nothing than the r otation c o eﬃcients of the curvilinear conjugate co ordinate net asso ciated with system (6). W e also note t hat the second formula in (8) is in fact the in v ersion of the explicit represen tation (7). Imp ortantly , the characteristic v elo cities in (6) coincide with the transp ort ve lo cities in the conserv ation laws (4) — this is the consequence o f linear degeneracy of system (4), (5). • W e construct the full set of comm uting ﬂo ws to (4), (5), of whic h N − 2 a re linearly degenerate. This has allow ed us, in particular, to obtain the family of quasi-p erio dic solutions x + ξ i t = r i Z ξ dξ p R K ( ξ ) + X m 6 = i ǫ im r m Z dξ p R K ( ξ ) , i = 1 , 2 , ..., N , (9) where R K ( ξ ) = K Y n =1 ( ξ − E n ) , and E 1 < E 2 < · · · < E K are real constan ts ( K = 2 N + 1 if N is o dd and K = 2 N + 2 if N is ev en) • W e show t ha t f or the sp ecial case N = 3 there exists a f amily of similarit y solutions to (6), (8) ha ving the form ˜ r i = t − α l i ( x/t ), i = 1 , 2 , 3, α 6 = 0 , where eac h ˜ r i is a certain rational function of the corr esp onding Riemann inv ariant r i (7) (and hence, is also a Riemann in v ariant). These solutions are found in an implicit (ho do graph) form. F o r N > 3 suc h solutions generally do no t exist. In tegrability , for arbitrary N , of the class of the h ydro dynamic reductions studied in this pap er is a strong evidence in fa vour of integrabilit y of the full nonlo cal kinetic equation (1), a t least for certain c hoices of the functions S ( η ) and G ( eta, µ ) in the inte gral closure equation. Of course, suc h an outcome do es not lo ok surprising for t he particular c ho ice (2) of S ( η ) and G ( η , µ ) corresp onding to the t hermo dynamic limit of the integrable KdV-Whitham equations but our ana lysis suggests that the g eneral in tegro- diﬀeren tial kinetic equation (1) is a represen t ativ e of a whole new unexplored class o f integrable equations with p oten tially imp ortant physic al applications. 5 The structure o f the pap er is as follows. In Section 2 we outline the deriv ation of the kinetic equations for the g a s o f the KdV solitons follo wing the thermo dynamic limit pro ce- dure of [6] a nd extending it to the entire KdV-Whitham hierarc hy . W e then in tro duce the generalised kinetic equation (1), and in Section 3 consider its N -comp onen t ‘cold-gas’ hy dro- dynamic reductions (4), (5) having the fo rm of hy dro dynamic conserv ation laws. W e then form ula te our main Theorem 3.1 stating that the hydrodynamic reductions under study are linearly degenerate and integrable (in Tsarev’s g eneralised ho dogra ph sense ) h ydro dynamic t yp e systems for any N . Section 4 is dev oted to the accoun t of the main r esults of the the- ory of linearly degenerate hy dro dynamic type systems. In Section 5 w e prov e the statemen t of the main Theorem 3.1 for the case N = 3 by explicitly constructing the corresp onding St¨ ack el matr ix a nd presen ting expressions f or t he Riemann in v ariants and ch aracteristic v e- lo cities in terms of the conserv ed comp onen t densities. W e also construct t w o distinguished families of exact solutions ( self-similar and quasi-p erio dic) to t he 3-comp onen t reduction. In Section 6, the existence of the Riemann inv ariant parametrization of the cold-gas hy dro- dynamic reduction, via a single St¨ ack el matrix, is pro ved for arbitrary N , whic h enables us to complete the pro of of the main Theorem 3.1 for a general case. In Section 7, w e derive explicit expressions (7) and (8) for the Riemann in v ariants and c ha racteristic v elo cities in terms of the comp o nen t densities. And at last, in Section 8 we deriv e h ydro dynamic sym- metries (comm uting ﬂows ) of the N -comp onen t hy dro dynamic reductions under study and then extract the family of linearly degenerate comm uting ﬂo ws. W e conclude in Section 9 with a general outlo ok a nd p ersp ectiv es arising from our study . 2 Kinetic e quation for a sol iton gas as the thermo dy- namic limit of th e Whitham mo d u lation syste m W e start with an outline of the deriv ation of the kinetic equation for t he gas of t he KdV solitons as the thermodynamic limit of the KdV-Whitham system fo llo wing [6 ]. W e then naturally extend this deriv ation to the en tire Whitham-KdV hierarc h y . Let us consider the Whitham mo dulation system asso ciat ed with the N -gap p oten tials φ N ( x, t ) of the KdV equation (3). This system is most con v enien tly represen ted as a single generating equation in the form [15]: ( dp N ) t = ( dq N ) x , (10) where dp N and dq N are the quasimomen tum and quasienergy diﬀeren tials deﬁned on the t wo-shee ted h yp erelliptic Riemann surface o f gen us N : Γ : µ 2 ( λ ) = 2 N +1 Y j =1 ( λ − λ j ) , λ ∈ C , λ j ∈ R . (11) λ 1 < λ 2 < · · · < λ 2 N < λ 2 N +1 , with cuts along spectral bands [ λ 1 , λ 2 ], . . . [ λ 2 j − 1 , λ 2 j ], . . . , [ λ 2 N +1 , ∞ ]. W e in tro duce the canonical system of cycles on Γ as follo ws (see Fig. 1): the α j -cycle surrounds the j -th cut clo c kwise on the upp er sheet, and the β j - cycle is canonically conjugated to α j ’s suc h that 6 band 2 O 1 O 2 1 j O  2 j O 3 O 2 1 N O  4 O gap j D j E Figure 1: The canonical system of cycles on the hyperelliptic Riemann surface of gen us N . the closed contour β j starts at λ 2 j , go es to + ∞ on the upp er sheet and r eturns to λ 2 j on the low er sheet. The meromorphic diﬀ erentials dp N and dq N are uniquely deﬁned b y their asymptotic b eha viour near λ = −∞ : − λ ≫ 1 : dp N ∼ − dλ ( − λ ) 1 / 2 , dq N ∼ ( − λ ) 1 / 2 dλ (12) and the normalization I β i dp N = 0 , I β i dq N = 0 , i = 1 , . . . , N ; c N = − 1 2 2 N +1 X j =1 λ j . (13) The integrals of dp N and d q N o ver the α - cycles give the comp onen ts of the wa v e n um b er and the frequency ve ctors respectiv ely I α j dp N ( λ ) = k j ( λ 1 , . . . , λ 2 N +1 ) , I α j dq N ( λ ) = ω j ( λ 1 , . . . , λ 2 N +1 ) , j = 1 , . . . , N . (14) Let λ 1 = − 1, λ 2 N +1 = 0. F ollowing V enakides [44] w e intro duce a lattice of p o in ts 1 ≈ η 1 > η 2 > . . . > η N ≈ 0 , (15) where − η 2 j = 1 2 ( λ 2 j − 1 + λ 2 j ) (16) are the cen tres of bands. W e no w assume that the sp ectral bands are distributed suc h that one can in tro duce t w o p ositiv e con tin uous f unctions on [0 , 1]: 1. The no r malized densit y of bands ϕ ( η ): ϕ ( η ) d η ≈ n umber of lattice p oin ts in ( η , η + dη ) N . That is, ϕ ( η j ) = 1 N ( η j − η j +1 ) + O ( 1 N ) , 1 Z 0 ϕ ( η ) d η = 1 , η 2 = − λ ∈ [0 , 1] . (17) 2. The no r malized logarithmic band width γ ( η ): γ ( η j ) = − 1 N log δ j + O ( 1 N ) , δ j = λ 2 j − λ 2 j − 1 . (18) 7 The f unctions ϕ ( η ) and γ ( η ) asymptotically deﬁne the lo c al structure of the Riemann surface Γ (11) for N ≫ 1. In other w ords, instead of 2 N + 1 discrete parameters λ j w e ha ve tw o con tinuous functions of η on [0 , 1] whic h do not depend on x, t on the scale o f the t ypical c hange of λ j ’s in (10), sa y ∆ x ∼ ∆ t ∼ l . The existence o f the con tin uous distributions ϕ ( η ) and γ ( η ) implies the follo wing band- gap scaling for N ≫ 1: | gap j | ∼ 1 ϕ ( η j ) N , | band j | ∼ exp {− γ ( η j ) N } , j = 1 , . . . , N (19) In tro duction of the distribution (19) is motiv ated b y the structure of t he sp ectrum o f Hill’s op erator in the semi-classical limit [41], [43] although the scaling (19) alone, of course, do es not imply exact p erio dicit y of the ( ﬁnite- g ap) p otential. The scaling (19) has an imp ortant pro p ert y: it preserv es the ﬁniteness of the in tegrated densit y of states as N → ∞ . The in tegrated densit y of states is deﬁned in terms of the r eal part o f the quasimomen tum integral (see [2 4]): N N ( λ ) = 1 π Re λ Z − 1 dp N ( λ ′ ) , λ ∈ [ − 1 , 0 ] . (20) No w, using ( 1 4) one can readily see that N N ( λ ) = 1 2 π M X j =1 k j if λ ∈ [ λ 2 M , λ 2 M +1 ] , M = 1 , . . . , N , (21) whic h is a particular (ﬁnite-gap) case of the g eneral g ap-lab eling theorem for quasi-p erio dic p oten tials [24]. It is not diﬃcult to sho w that the scaling (19) implies that k j ∼ 1 / N so the total densit y of states N N (0) = 1 2 π N X j =1 k j (22) remains ﬁnite in the limit as N → ∞ . F or this reason w e shall call the con tinuum limit as N → ∞ , deﬁned on the sp ectral scaling (19), the thermo dynamic limit . W e shall not b e concerned here with t he existence and the exact meaning of the thermo- dynamic limit for the ﬁnite-gap p oten tials u N ( x, t ) (whic h is a separate interesting problem closely connected with V enakides’ con tin uum limit of t heta- functions [44]) but shall rather directly consider this limit for the asso ciated Whitham system (1 0). It is how ev er, instruc- tiv e to note that it f ollo ws from (19) that in the thermo dynamic limit the band/gap ratio v anishes for eac h oscillating mo de (i.e. k j → 0 ∀ j = 1 , 2 . . . . , N ), so the thermo dynamic limit of the sequence of ﬁnite-gap p oten tials asso ciated with the spectral scaling (19) is es- sen tially an inﬁnite-soliton limit. It w as prop osed in [8] tha t this limiting p oten tial should b e described in terms of ergo dic random pro cesses and can b e view ed a s a homo gen e ous soliton gas (or homogeneous soliton turbulence – dep ending on whic h of t he t w o “iden tities” of a soliton is emphasized: the particle or the w av e one). Then it is na tural to supp ose that the same thermo dynamic limit for t he asso ciated Whitham system should describ e macroscopic ev olution of the spatially inhomo gene ous soliton gas. Indeed, as we shall see, the thermo dy- namic limit of the Whitham equations turns out to b e consisten t (in the small-density limit) 8 with the kinetic equation for solitons deriv ed by Z akharo v [46] using the inv erse scattering problem f ormalism. W e ﬁrst note that N N ( λ ) deﬁned by (20) is a monotone increasing p ositiv e function so d N N ( λ ) is a measure supp orted on the sp ectrum of the ﬁnite-gap p otential u N ( x ) [24]. Next w e in t r o duce a ‘t emp o ral’ ana logue of the densit y of states (20) b y the form ula V N ( λ ) = 1 π Re λ Z − 1 dq N ( λ ′ ) , λ ∈ [ − 1 , 0 ] . (23) Then in tegration of the generating mo dulation equation (10) on the real a xis o f λ from − 1 to − η 2 ∈ [ − 1 , 0] yields ∂ t d N N ( − η 2 ) = ∂ x d V N ( − η 2 ) , η ∈ [0 , 1] . (24) Th us t he ﬁnite-gap Whitham- KdV system can b e regarded as the system go verning the ev olution of the sp ectral measure. No w w e consider the t hermo dynamic limits of d N N and d V N whic h w e denote as d N N → π f ( η ) dη , d V N → − π f ( η ) s ( η ) dη as N → ∞ , (25) where the limit is t ak en on t he thermo dynamic sp ectral scaling (19). Since π f ( η ) dη is the limiting spectral measure, the function f ( η ) has the na t ur a l meaning of the distribution function of the solitons o ver the sp ectrum (the meaning of the function s ( η ) will b ecome clear so o n). The functions f ( η ) and s ( η ) we re show n in [8], [6] to b e express ed in terms of the rat io σ ( η ) = ϕ ( η ) /γ ( η ) of the lattice distribution functions (17), (18) by certain integral equations, whic h are then com bined in to a single equation directly connecting f ( η ) a nd s ( η ) [6]: s ( η ) = 4 η 2 + 1 η 1 Z 0 log     η + µ η − µ     f ( µ )[ s ( η ) − s ( µ )] dµ . (26) W e stress tha t in the con tin uum (thermo dynamic) limit give n b y equations (25), (2 6) the ex- plicit dep endence of the densit y o f states on the discrete sp ectral branch p oints λ j disapp ears. The o nly ‘reminder’ of the h yp erelliptic Riemann surface Γ (11) is the ke rnel ln | η + µ | / | η − µ | whic h a rises as the con t in uum limit of the o ﬀ-diagonal elemen ts of the p erio d matrix B of the Riemann theta-function Θ N ( x, t | B ) deﬁning, via the Its-Matv eev f o rm ula, the ﬁnite-gap p oten tial (see [44] and [6]). Th us, in tegral equation (2 6) can b e view ed as a lo c al (in the x, t - plane) relationship b et wee n the functions f ( η ) and s ( η ) characterizing the soliton gas. Let l ≫ 1 b e the c haracteristic length at whic h the c hange o f functions f ( η ), s ( η ) is small (o f order 1 /l ≪ 1). Next, in the spirit of the mo dulation theory (see [42], [1 5]) w e assume that on a larger spatiotemp oral scale, ∆ x ≫ l , ∆ t ≫ l , w e hav e f ( η ) ≡ f ( η , x, t ), s ( η ) ≡ s ( η , x, t ) and p ostulate, using (25), that ∂ t d N N → π ∂ t f ( η , x, t ) dη , ∂ x d V N → − π ∂ x [ f ( η , x, t ) s ( η , x, t )] dη . (27) Then mo dulation equation ( 2 4) assumes the form of a conserv at io n equation for f , f t + ( sf ) x = 0 , (28) 9 whic h is clearly the express ion of the isospectrality of the KdV ev olution. Since ρ ( x, t ) = R 1 0 f dη is the densit y of solitons the quantit y s ( η , x, t ) can naturally b e interpre ted as the v elo city of the soliton gas (or, more precisely , the v elo city of a ‘trial’ soliton with the spectral parameter λ = − η 2 – see [23]). One can see from (26) that this velocity diﬀers from the v elo city 4 η 2 of the f ree soliton with the same sp ectral parameter. This diﬀerence is ob viously due to the collisions of the ‘trial’ η - soliton with other ‘ µ ’ - solito ns in the soliton gas. Indeed, for small densities ρ = R f dη ≪ 1 one can consider the second term in (26) as a small correction to the free-soliton v elo cit y and obtain t ha t to the ﬁrst order in ρ s ( η ) ≈ 4 η 2 + 1 η 1 Z 0 ln     η + µ η − µ     f ( µ )[4 η 2 − 4 µ 2 ] dµ , (29) whic h is Z akharo v’s expression for the av erage v elo city of a trial soliton in a rareﬁed soliton gas, obtained in [46] by taking in to accoun t the c hange in the soliton p osition due to phase shifts in its pairwise collisions with o t her solitons. W e w o uld lik e to emphasize crucial diﬀerence b etw een the ma t hematical structure of Zakharo v’s asymptotic for mula (29), whic h represen ts an exp l i c it expr ession for the trial soliton velocity s ( η ) in terms of the sp ectral distribution function f ( η ), and that of form ula (26) whic h is a non- p erturbativ e inte g r al e quation for s ( η ). Equations (28) and (26) th us prov ide a self-consisten t kinetic description of the KdV soliton gas of ﬁnite density . W e note that the upp er limit in the integrals in (26), (29) can b e replaced by + ∞ to mak e the kinetic equation indep enden t on the original sp ectral lattice normalization ( 15). The outlined pro cedure o f the thermo dynamic limit can b e readily extended to the entire Whitham-KdV hierarch y , ( dp N ) t n = ( dq ( n ) N ) x , n ∈ N , (30) where n is the n umber of the “higher” Whitham-KdV equation in the hierarc h y (the original mo dulation equation (10) corresp onding to the KdV equation itself has the num b er n = 1) and t n is the corresp onding “higher” time, so that ( d p N ) t n t m = ( dp N ) t m t n for all n 6 = m . The meromorphic diﬀeren tial dq ( n ) N is uniquely deﬁned by its asymptotic b eha viour near λ = −∞ , dq ( n ) ∼ ( − λ ) n − 1 / 2 dλ , (31) and the normalization I β j dq ( n ) ( λ ) = 0 , j = 1 , . . . , N (32) analogous to (13). No w, applying the ab ov e pro cedure of the thermo dynamic limiting transition to equation (30) w e obtain the same transp ort equation (28) for the distribution function f ( η , x, t ) f t n + ( s n f ) x = 0 , (33) while the in tegral closure equation for s n assumes the fo rm s n ( η ) = C n η 2 n + 1 η 1 Z 0 log     η + µ η − µ     f ( µ )[ s n ( η ) − s n ( µ )] dµ , (34) 10 where C n are certain constan ts whose sp eciﬁc v alues w on’t b e required b elo w. Moreo v er, since the c haracteristic sp eeds of the comm uting K dV- Whitham ﬂows, and, therefore, the corresp onding transp ort v elo cities s n in the thermo dynamic limit equation (34), are deﬁned up to a constan t factor , hereafter o ne can assume C n to b e arbitrary constan ts. W e note that equation (34) diﬀers from ( 26) only in the ﬁrst term corresp onding t o the free-soliton v elo city . Also note that the ‘phase-shift’ lo g arithmic k ernel in the inte gral equation (34) is the same f or all n , whic h is not surprising as the entire ﬁnite- g ap Whitham- KdV hierarc hy (30) is asso ciat ed with the same Riemann surface, i.e. with the same p erio d matrix B resp onsible f o r the for m of the in tegral k ernel in the limit. No w it is only natural to consider a g eneralization of t he deriv ed kinetic equations (33), (34) b y in tro ducing in (34) an arbitrary function S ( η ) instead of the free-soliton v elo cit y term and an arbitrary symmetric f unction G ( η , µ ) instead of the logarithmic ‘phase-shift k ernel’ in the in t egr a l term. Also, as w a s already men tioned, w e replace the upp er limit of in tegrat io n in the closure equation (26) by + ∞ . As a result, we arrive at the g eneralised kinetic equation (1), whic h will b e our main concern in the remainder of the pap er. 3 ‘Cold-g as’ h ydro dynamic reduct i ons W e in tro duce an N -comp onen t ‘cold-gas’ ansa tz for the distribution function f ( η , x, t ): f = N X i =1 f i ( x, t ) δ ( η − η i ) , (35) where η N > η N − 1 > · · · > η 1 > 0 are arbitrary num b ers and f i ( x, t ), n = 1 , . . . , N are unkno wn functions. Before w e pro ceed with t he analysis of mathematical consequences of this ‘cold-gas’ ansatz it is instructiv e t o sa y a couple of words ab o ut it s ph ysical meaning (see [7]). T o b e deﬁnite, w e shall refer to the KdV case here. The distribution (35 ) represen ts a n idealized description of the distribution function in a soliton ga s with the solitons hav ing their sp ectral parameters η distributed in nar r o w vicinities of N discrete v alues η i . As a matter of fact, o wing to non- degeneracy of discrete sp ectrum of the linear Sc hr¨ odinger op erator, all individual sp ectral parameters within the i -th comp onent of the soliton gas m ust b e diﬀeren t. The soliton p ositions in suc h a ‘quasi-mono c hromatic’ comp onen t of the soliton gas a re statistically indep enden t whic h results in t he P oisson distribution with the mean densit y f i for the n umber of solitons in a unit space interv al (the Poiss on distribution na t urally arises in the thermo dynamic limit of ﬁnite-gap p oten tials [8]). It is also clear that one can neglect the eﬀect of the interactions b etw een the solitons b elonging to the same gas comp onen t compared with the cross-comp onen t in teractions (the t ypical time o f the interactions b et we en solitons with close v alues of t he sp ectral parameter is m uc h larger than when these parameters are m utually spaced within the sp ectral in t erv a l — see, e.g., [33]). This will b e show n b elo w to ha ve imp ortant mathematical consequences. Substitution (35) r educes (1) to a syste m of h ydro dynamic conserv ation la ws, u i t = ( u i v i ) x , i = 1 , . . . , N , (36) where the comp onen t ‘densities’ u i and the v elo cities v i deﬁned as u i ( x, t ) = η i f i ( x, t ) , v i ( x, t ) = − s ( η i , x, t ) , (37) 11 are related algebraically v i = ξ i + X m 6 = i ǫ im u m ( v m − v i ), ǫ ik = ǫ k i . (38) Here ξ i = − S ( η i ) , ǫ ik = 1 η i η k G ( η i , η k ) i 6 = k . (39) Note that the quantities ǫ ii are not deﬁned. In a t w o- comp onen t case, the a b o ve algebraic system (38) can b e easily resolv ed for u 1 , 2 in terms of v 1 , 2 : u 1 = 1 ǫ 12 v 2 − ξ 2 v 1 − v 2 , u 2 = 1 ǫ 12 v 1 − ξ 1 v 2 − v 1 . (40) Substituting (40) in to (36) w e arriv e at the Lemma 3.1 (El & Kamc hatnov 2005 [7]): Hydr o dynamic typ e system (36 ), (38) for N = 2 r e duc es to a diagona l form in the ﬁeld varia bles v 1 and v 2 : v 1 t = v 2 v 1 x , v 2 t = v 1 v 2 x . (41) Remark ably , the h ydro dynamic t yp e system (4 1) is line arly de gener ate b ecause its c hara c- teristic v elo cities do not dep end on the corresp onding R iemann in v ariants. Ph ysically this linear degeneracy reﬂects the already men tioned do mination of the ‘cross-comp onent’ soliton in teractions o ver t he interactions within a given comp onen t consisting of solitons with close amplitudes. It is worth noting t ha t system (41) is equiv alen t to the 1D Bo r n- Infeld equation (Born & Infeld 1934) arising in nonlinear electromagnetic ﬁeld theory (see [42], [1]) (1 + ϕ 2 x ) ϕ y y − 2 ϕ x ϕ y ϕ xy + (1 − ϕ 2 y ) ϕ xx = 0 . As any tw o-comp onen t h ydro dynamic ty p e system, (4 1) is in tegrable (linearizable) via the classical ho dogra ph transform. How ev er, for an y N ≥ 3 in tegra bility o f the o r ig inal system (36), (38) is no longer o b vious. As a matter of fact, most N -comp onent hy dro dynamic type systems are no t inte gr able f o r N ≥ 3. Also, it is ev en no t clear whether N -comp onen t system (36), (38) is linearly degenerate. It migh t seem that this sys tem is simple enough for one to b e able to v erify these prop erties b y a direct computation, using general deﬁnitions of linear degeneracy and in tegrabilit y for h ydro dynamic type systems [3 5], [39, 40 ] (also see t he next Section). T o our surprise, ev en the simplest non-trivial case N = 3 turned out to b e complicated enough to r equire computer alg ebra to get the conﬁrmation o f our hypothesis. The iden tiﬁcation of the system (36), (38) for N = 3 as an in tegrable linearly degenerate h ydro dynamic system can b e considered as a strong indication that b oth prop erties (linear degeneracy and in tegrabilit y) could hold true for this system for arbitra r y N . Th us w e form ula te our main Theorem 3.1 N -c omp onent r e ductions (36), (3 8 ) of the gene r alise d kinetic e quation (1) ar e line arly de gener ate inte gr able hydr o dynamic typ e systems for any N . T o pro v e this t heorem, w e tak e adv an tag e of the w ell-dev elop ed theory of integrable (semi- Hamiltonian) linearly degenerate hydrodynamic t yp e systems [35 ], [11 ]. F o r conv enience, in the next section w e presen t a brief review o f the main r esults of this t heory whic h will b e extensiv ely used in Sections 5 – 8 of the pap er. 12 4 Linearly de generate in t e grable h ydro dynamic t yp e systems: accoun t of prop erties A hy dro dynamic t yp e system U i t = v i j ( U ) U j x , i, j = 1 , 2 , ..., N (42) is called semi-Hamiltonian (see [39, 40]) if it (i) has N m utually distinct eigen v alues λ = λ i ( U ) deﬁned b y the equation det   λδ i j − v i j ( U )   = 0; (43) (ii) admits inv ertible p oin t transformations U k ( r ), suc h that this h ydro dynamic type system can b e written in diagonal form r i t = V i ( r ) r i x , i = 1 , . . . , N . (44) The v ar ia bles r k ( U ) are called Riemann in v ariants and V k ( r ) = λ k ( U ( r )) – c ha racteristic v elo cities. Eac h Riemann inv ariant r i is determined up t o a n arbitrary function of a single v ariable R i ( r i ). (iii) satisﬁes t he iden tity ∂ j ∂ k V i V k − V i = ∂ k ∂ j V i V j − V i , i 6 = j 6 = k (45) for eac h three distinct characteristic v elo cities ( ∂ k ≡ ∂ /∂ r k ). A semi-Hamiltonian hydrodynamic type syste m p o ssess es inﬁnitely man y conserv ation la ws parameterised by N arbitrary functions of a single v ariable. Its general lo cal solution for ∂ x r i 6 = 0, i = 1 , . . . , N is giv en b y the generalised ho dograph formula [39, 40] x + V i ( r ) t = W i ( r ) , (46) where functions W i ( r ) are found from the linear system of PDEs: ∂ i W j W i − W j = ∂ i V j V i − V j , i, j = 1 , . . . , N , i 6 = j. (47) Th us, the semi-Hamiltonia n prop erty (45) implies in tegra bilit y of diagonal hy dro dynamic t yp e system in the ab o v e g eneralised ho dog raph sense. It is know n [40] that solutions W j of (47) sp ecify commu ting hydrodynamic ﬂows to (44): r j τ = W j ( r ) r j x , j = 1 , . . . , N , (48) where τ is a new time (group parameter). Indeed, one can readily sho w that equations (4 4), (48), (47) imply ( r j τ ) t = ( r j t ) τ . A sub-class of linearly degenerate h ydro dynamic type systems is distinguished b y the prop ert y ∂ i V i = 0 (49) 13 for each index i . It means that each characteristic v elo cit y do es not dep end on the corre- sp onding Riemann inv arian t r i . Theorem 4.1 (P avlo v 1987 [35]): If s e mi-Hamiltonian hydr o dynamic typ e system (44) p ossesses c onserva tion laws (36) with u i = U i ( r ) and v i ( U ( r )) = V i ( r ) then this system is line arly de gener ate. These c onservation laws (36) a r e p ar ameterise d b y N arbi tr ary functions of a sin gle variable. Pro of : The semi-Hamiltonian prop erty (i.e. inte gr ability ) is giv en b y the condition (45). W e in tro duce, following Tsarev [40], the so-called Lame co eﬃcien ts ¯ H i b y ∂ k ln ¯ H i = ∂ k V i V k − V i , i 6 = k . (50) Supp ose that some semi-Hamiltonian h ydro dynamic t yp e system (44) can b e written in the conserv ative for m (36) with v i ( U ( r )) = V i ( r ). In such a case ∂ k U i · r k t = ∂ k ( U i V i ) · r k x . Since r ( x, t ) is an arbitrary solution of (44 ) w e obtain N equations V k · ∂ k U i = ∂ k ( U i V i ) . (51) If k 6 = i , then ∂ k ln U i = ∂ k V i V k − V i , (52) i.e. eac h of the conserv ation law densities U i is determined up to an arbitrary function of a single v ariable P i ( r i ) (cf. (5 0) and (52)), U i = ¯ H i · P i ( r i ) . (53) If k = i , then it f ollo ws fro m (51) that ∂ i V i = 0 i.e. t he system is linearly degenerate. The theorem is pro v ed. Remark 1 : A subset { u k } of t he conserv ation law densities { U k } satisfying a g iven system of conserv ation la ws (e.g. (36), (3 8)) is selected b y ﬁxing the functions P k (e.g. P k ( r k ) ≡ 1 — see (9 9) in Section 7). While con v erse of Theorem 4.1 is a lso true, one should note that not every conserv ation la w of a semi-Hamiltonian linearly degenerate system satisﬁes the k ey prop erty v i = V i . Indeed, let us consider the tw o-comp onen t system of conserv a tion laws, U 1 t = ( U 1 v 1 ( U 1 , U 2 )) x , U 2 t = ( U 2 v 2 ( U 1 , U 2 )) x . (54) Supp ose this h ydro dynamic type system is linearly degenerate, then it can b e written in Riemann in v arian ts r 1 ( U 1 , U 2 ), r 2 ( U 1 , U 2 ) as fo llo ws: r 1 t = V 1 ( r 1 , r 2 ) r 1 x , r 2 t = V 2 ( r 1 , r 2 ) r 2 x , where V 1 , 2 ( r ) = v 1 , 2 ( U ( r )). Let us in tro duce new conserv a tion law densities ˜ U 1 = U 1 + U 2 and ˜ U 2 = U 1 − U 2 . Then the system of conserv ation laws (54 ) assumes a n equiv alent f o rm ˜ U 1 t = ( ˜ U 1 ˜ v 1 ( ˜ U 1 , ˜ U 2 )) x , ˜ U 2 t = ( ˜ U 2 ˜ v 2 ( ˜ U 1 , ˜ U 2 )) x , 14 where the c haracteristic v elo cities ˜ v 1 = U 1 v 1 + U 2 v 2 U 1 + U 2 , ˜ v 2 = U 1 v 1 − U 2 v 2 U 1 − U 2 no longer coincide with V 1 ( r 1 , r 2 ) and V 2 ( r 1 , r 2 ). The full t heory of linearly degenerate semi-Hamiltonian hydrodynamic t yp e systems w as constructed b y F era p on tov in [11] using the St¨ ac k el matrices ∆ =           φ 1 1 ( r 1 ) ... φ 1 N ( r N ) ... ... ... φ N − 2 1 ( r 1 ) φ N − 2 N ( r N ) φ N − 1 1 ( r 1 ) φ N − 1 N ( r N ) 1 ... 1           (55) where φ i k ( r k ) are N ( N − 1) arbitrary functions ( it is clear that without loss of generalit y one can put φ N − 1 k ( z ) = z and the n umber of arbitra r y function reduces to N ( N − 2)). Then the c haracteristic v elo cities of suc h linearly degenerate hydrodynamic type systems are giv en b y the formula V i = det ∆ (2) i det ∆ (1) i , (56) where ∆ ( k ) i is the matrix ∆ without k th row and i th column. The family of the conserv a t ion la w densities U i corresp onding to the semi-Hamiltonian system (44), (56) is determined b y (cf. (53)) U i = det ∆ (1) i det ∆ ( − 1) i +1 P i ( r i ) , (57) where P i ( r i ), i = 1 , . . . , N are arbitrary functions. Corollary 4.1 The system of c onservation la w s (36) is a semi - Ham iltonian li n e arly de - gener ate hydr o dynamic typ e system if an d only if the densities u i and velo cities v i ( u ) admit r epr esentations u i = U i ( r ) a n d v i ( U ( r )) = V i ( r ) , s p e ciﬁ e d by (57), (56), via N functions r k ( x, t ) s a tisfying diagonal system (44). Prop osition 4.1 (F erap on tov 1991 [11]): Semi-Hamil toni an line arly d e ge n er ate hydr o- dynamic typ e system ( 44), (56) has N − 2 nontrivial line arly de ge n er ate c ommuting ﬂows r j t k = V j ( k ) ( r ) r j x , j = 1 , . . . , N , k = 3 , 4 , ..., N , (58) whose ch aracteristic v elo cities are determined as (cf. (5 6)) V i ( k ) = det ∆ ( k ) i det ∆ (1) i . (59) An y c haracteristic v elo cit y v ector W ( r ) = ( W 1 ( r ) , W 2 ( r ) , . . . , W N ( r )) sp ecifying linearly degenerate hy dro dynamic ﬂow r j τ = W j ( r ) r j x , j = 1 , . . . , N , commu ting with (4 4), (56), can 15 b e represen ted as a linear com bination of the “basis” c haracteristic v elo cit y v ectors V ( k ) (59) (including “trivial” ones V (2) ≡ V (see (5 6 )) and V (1) ≡ 1 ) with some constan t co eﬃcien ts b k . Th us, for any comp onen t W i there exists a decomp osition W i = N X k =1 b k V i ( k ) . (60) Theorem 4.2 (F erap onto v 199 1 [11]): Gener al solution r ( x, t ) of the semi-Hamiltonian line arly de gener ate system (44) is p ar ameterise d by N arbi tr ary functions of one variable f k ( r k ) and is gi v en in an implicit form by the algebr aic system x = N X k =1 r k Z φ 1 k ( ξ ) dξ f k ( ξ ) , − t = N X k =1 r k Z φ 2 k ( ξ ) dξ f k ( ξ ) (61) 0 = N X k =1 r k Z φ M k ( ξ ) dξ f k ( ξ ) , M = 3 , 4 , ..., N . (note the c hange of sign for t compared to [11] due to a slightly diﬀeren t represen tation of the diagona l system (44) in this pap er). W e also note that formulae (6 1) represen t an equiv alen t of the symmetric generalised ho dog r aph solution (46) for semi-Hamiltonian linearly degenerate h ydro dynamic t yp e systems. It is instructiv e to introduce, follo wing Darb oux [1 0], the so-called rotatio n co eﬃcien ts β ik = ∂ i ¯ H k ¯ H i , i 6 = k , (62) where the Lam ´ e co eﬃcien ts ¯ H i are deﬁned by (50) . Then expression (45) for the semi- Hamiltonian pro p ert y assumes the f orm of a Darb oux system ∂ i β j k = β j i β ik , i 6 = j 6 = k . (63) Using ( 6 2) linear system (47) can b e related to another linear system ∂ i H k = β ik H i , i 6 = k , (64) via the so-called Com b escure transfor ma t io n (see [10 ]) W i = H i ¯ H i . (65) In other w ords, one can sho w ( see [40]) that the ratio of any t w o solutio ns to (6 4) satisﬁes system (47) for the c haracteristic v elo cities of the commuting ﬂo ws (48 ). W e note that t he particular solution ˜ H i of (64) corr espo nding to the characteristic v elo cities V i of the o r ig inal system (4 4) is expressed in terms of the Lam´ e co eﬃcien t ¯ H i as ˜ H i = V i ¯ H i . (66) 16 Of course, general solution H i of system (64), as w ell as general solution W i of t he generalised ho dograph equations (47), is para meterised b y N arbitrary functions of a single v ariable. Theorem 4.3 (Pa vlo v 1987 [35]): The class of the semi-Hamiltoni a n line arly de gener ate systems of hydr o dynamic typ e is sele cte d, in addition to (63), by the set of r estrictions on the r otation and L ame c o eﬃcients ∂ i ln ¯ H i = ∂ i ln β j i (67) for any inde x j 6 = i . Pro of : Let us consider the Lam ´ e co eﬃcien ts for the linearly degenerate systems. Using (50), (49) w e ha ve ∂ j V i = ∂ j ln ¯ H i · ( V j − V i ) , i 6 = j, ∂ i V i = 0 . The compatibility condition ∂ i ( ∂ j V i ) = ∂ j ( ∂ i V i ) implies that ∂ i ∂ j ln ¯ H i = ∂ j ln ¯ H i · ∂ i ln ¯ H j , i 6 = j. (68) No w one can see that the l.h.s. of (68) can b e written in the fo r m ∂ i ∂ j ln ¯ H i = ∂ i  ¯ H j ¯ H i β j i  = β ij β j i + ¯ H j ¯ H i ∂ i β j i − ¯ H j ¯ H 2 i β j i ∂ i ¯ H i . (69) On the other hand, the r.h.s. o f (68) is nothing but the pro duct β ij β j i . Now (67) immediately follo ws fro m (68) and (69). The Theorem is prov ed. No w, supp ose that the rotation co eﬃcien ts (62) for some linearly degenerate h ydro dy- namic type system are g iven. Then restrictions (67) determine not only the Lam´ e co eﬃcien ts (50) but also all o t her solutions of (64 ) asso ciated, via (65), with the ch aracteristic v elo cities (56), (59) of the complete set o f linearly degenerate comm uting ﬂows. Indeed, one can see that equations (62), (67) actually represen t N system s of or dinary diﬀer ential e quations so that eac h system con tains diﬀeren tia t ion with resp ect to only o ne Riemann in v ariant. Thus , the general solution ¯ H i of system (62), (67) is parameterised b y N arbitrary constan ts (see Prop osition 4.1). Let us in tro duce N particular solutions ¯ H ( k ) i of system (62), (67) suc h that (see (56), (59)) V i ( k ) = ¯ H ( k ) i ¯ H i , k = 1 , 2 , ..., N , where ¯ H i = ¯ H (1) i , ˜ H i = ¯ H (2) i (see (59)). As a matter o f fact, V i (2) ≡ V i , V i (1) ≡ 1. Then (6 7) can b e written in a sligh tly more general form, ∂ i ln β j i = ∂ i ln ¯ H ( k ) i , – fo r an y k and j 6 = i . Th us, the f ull class o f linearly degenerate semi-Hamiltonian hy dro dynamic type systems is determined by conditions (6 7), (62) and (63). W e note that system (67), (62) and (63) is an o v erdetermined system in in volution. Its inte gration leads to the afor emen tioned set of particular solutions of (64) t ha t can b e parameterised via a St¨ ac k el mat r ix ( see (55) , (56), (57) and (59)). 17 5 N = 3 : explicit form ulae W e no w consider the ﬁrst non trivial (from the viewp oint of integrabilit y) case N = 3 of the h ydro dynamic reduction (36 ), (38). T o pro ve our main Theorem 3.1 for N = 3 w e shall mak e use o f Corollary 4.1. Let us supp ose that hy dro dynamic system of conserv ation la ws (36), (38) is linearly degenerate and can b e written in a diagonal f orm (44 ), i.e. w e supp ose that there exists an in ve rtible c hange of v a riables r j ( u ) , j = 1 , 2 , 3 , suc h tha t system (36) assume s a diagonal form r j t = V j ( r ) r j x , j = 1 , 2 , 3 , (70) where V j ( r ) = v j ( u ( r )). W e in tro duce the St¨ ac k el matrix (55), whic h for N = 3 can b e written in the f orm ∆ =   B 1 ( r 1 ) B 2 ( r 2 ) B 3 ( r 3 ) A 1 ( r 1 ) A 2 ( r 2 ) A 3 ( r 3 ) 1 1 1   , (71) where A k ( z ) and B k ( z ) are arbitrary functions. No w, by Corollary 4.1, if system (36), (38) is linearly degenerate and semi-Hamiltonian then its diagonal represen tation (70) m ust ha v e characteristic velocities in the form (56), i.e. for N = 3 w e hav e V 1 = B 2 ( r 2 ) − B 3 ( r 3 ) A 2 ( r 2 ) − A 3 ( r 3 ) , V 2 = B 3 ( r 3 ) − B 1 ( r 1 ) A 3 ( r 3 ) − A 1 ( r 1 ) , V 3 = B 1 ( r 1 ) − B 2 ( r 2 ) A 1 ( r 1 ) − A 2 ( r 2 ) . (72) Then, using (57) the corresp onding conserv at io n la w densities u k are found in terms of Riemann in v arian ts as u 1 = P 1 ( r 1 ) det ∆ [ A 2 ( r 2 ) − A 3 ( r 3 )], u 2 = P 2 ( r 2 ) det ∆ [ A 3 ( r 3 ) − A 1 ( r 1 )], u 3 = P 3 ( r 3 ) det ∆ [ A 1 ( r 1 ) − A 2 ( r 2 )] , (73) where P j ( r j ) are a rbitrary functions and the determinan t of the St¨ ack el matrix is giv en by det ∆ = A 1 ( r 1 )[ B 2 ( r 2 ) − B 3 ( r 3 )] + A 2 ( r 2 )[ B 3 ( r 3 ) − B 1 ( r 1 )] + A 3 ( r 3 )[ B 1 ( r 1 ) − B 2 ( r 2 )] . (74 ) Substitution of (72)–(74) into (38) yields expressions for the functions A k ( z ), B k ( z ), P k ( z ), k = 1 , 2 , 3 . Before w e presen t these expressions, w e note that it fo llo ws f rom (7 2), (74) that functions B k ( z ) are determined up to a constant shift whic h is then translated in to a certain shift for the functions P k ( z ). It turns out that this shift can b e remo v ed by the simplest c hange of the R iemann in v arian ts, ( r k + constant) 7→ r k (although the relationships b et we en the shift constan ts for B k , P k and r k are rather cum b ersome) so tha t w e ev entually obta in A i ( r i ) = r i , B i ( r i ) = ζ i r i , i = 1 , 2 , 3 , (75) where ζ 1 = ξ 3 ǫ 12 − ξ 2 ǫ 13 ǫ 12 − ǫ 13 , ζ 2 = ξ 1 ǫ 23 − ξ 3 ǫ 12 ǫ 23 − ǫ 12 , ζ 3 = ξ 1 ǫ 23 − ξ 2 ǫ 13 ǫ 23 − ǫ 13 , (76) 18 P 1 = ξ 2 − ξ 3 ǫ 12 − ǫ 13 r 1 + ǫ 23 , P 2 = ξ 1 − ξ 3 ǫ 12 − ǫ 23 r 2 + ǫ 13 , P 3 = ξ 1 − ξ 2 ǫ 13 − ǫ 23 r 3 + ǫ 12 . (77) Direct v eriﬁcation sho ws tha t t he diago nal system (70), (7 2), (75), (76) is indeed equiv a len t, via (73), (74), (77), to the original set of conserv at ion la ws (36), (38 ) , where v k ( u ( r )) = V k ( r ). Th us, system (36), (38) is consisten t with form ulae (72), (73) deﬁned b y the St ¨ ac kel matrix (71 ) . Therefore, b y Corollary 4.1 , the three-comp onen t h ydro dynamic reduction (36), (38) is a linearly degenerate semi-Hamiltonian (i.e. in tegrable) h ydro dynamic t yp e system. Remark. As w e ha ve seen, the outlined construction has some additional inheren t “de- grees of freedom”, namely , thr ee ar bitr ary constan ts due to non-uniquenes s of the St¨ ac k el matrix sp ecifying a giv en linearly degenerate semi-Hamiltonian system. The full set of a rbi- trary constan ts remov able by an appropriate c hange of the Riemann in v ariants will app ear later in Section 5 where we shall consider N -comp onen t h ydro dynamic reductions with a r- bitrary N ≥ 3. Using ( 7 2)–(77) w e obtain explicit expressions fo r the c hara cteristic v elo cities V k and densities u k in terms o f Riemann inv ariants, V 1 = ζ 2 r 2 − ζ 3 r 3 r 2 − r 3 , V 2 = ζ 3 r 3 − ζ 1 r 1 r 3 − r 1 , V 3 = ζ 1 r 1 − ζ 2 r 2 r 1 − r 2 , (78) u 1 = P 1 r 2 − r 3 det ∆ , u 2 = P 2 r 3 − r 1 det ∆ , u 3 = P 3 r 1 − r 2 det ∆ , (79) where det ∆ = ( ζ 1 − ζ 2 ) r 1 r 2 + ( ζ 2 − ζ 3 ) r 2 r 3 + ( ζ 3 − ζ 1 ) r 3 r 1 . (80) W e note that, unlik e in the case N = 2, algebraic system ( 3 8) cannot b e resolv ed for u k in terms o f v n for any o dd N (cf. corresp onding formulae in Section 2), b ecause determinan t of the matrix ˆ A of linear system (38) ˆ Au = b , where A ik = ǫ ik ( v k − v i ) and b i = v i − ξ i , equals zero due to its sk ewsymmetry . F or instance, for N = 3, the consistency condition of this linear system (i.e. the condition that the rank of the augmen ted matrix equals 2) is giv en b y the relation ǫ 23 ( v 3 − v 2 )( ξ 1 − v 1 ) + ǫ 12 ( v 2 − v 1 )( ξ 3 − v 3 ) + ǫ 13 ( v 1 − v 3 )( ξ 2 − v 2 ) = 0 . (81) Direct substitution of v j = V j ( r ) (78) in to (81) show s that it satisﬁes iden tically . Using (79), (8 0), (7 6), (7 7) one can express the Riemann in v arian ts in terms of the densities u k explicitly , r 1 = ( 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) , r 2 = ( 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) , (82) r 3 = ( 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) . 19 Direct substitution sho ws that express ions (8 2) and ( 78) are consisten t with or iginal a lgebraic system (3 8) where v j = V j ( r ( u )). It is instructiv e to lo ok at what happ ens to the diag onal system (70 ) when the densit y of one of the comp onents in conserv ation law s (36), sa y u 3 , v anishes. O ne can see from (82) that if u 3 = 0 (this corresp onds to v anishing of P 3 in (79)) then the Riemann in v ariant r 3 b ecomes a constan t, u 3 = 0 : r 3 = − ( ǫ 23 − ǫ 13 ) ǫ 12 ξ 1 − ξ 2 ≡ r 3 0 , so that the equation for r 3 satisﬁes iden tically and system (70) reduces to its 2- comp onen t coun terpart (41) for v 1 ( u 1 , u 2 ) = V 1 ( r 2 ( u 1 , u 2 , 0) ) , v 2 ( u 1 , u 2 ) = V 2 ( r 1 ( u 1 , u 2 , 0) ) , as one should expect. Similar reductions o ccur for u 1 = 0 and u 2 = 0, whic h lead to r 1 = r 1 0 = constant and r 2 = r 2 0 = constant resp ectiv ely . As a matter of fact, an y function R j ( r j ) is also a Riemann inv ariant so one can choose a new set of Riemann inv ariants say R j = r j − r j 0 so that R j = 0 when u j = 0. This normalisation could b e useful for applications. No w we consider some sp ecial families of solutions to linearly degenerate system (70), (78). a) Similari ty s o lutions One can see that, owing to ho mogeneit y of the characteristic v elo cities (78) as functions of R iemann in v ariants, system (70) admits similarit y solutio ns of t he form r i = 1 t α l i  x t  , i = 1 , 2 , 3 , (83) where α is a n arbitrary p ositiv e real n um b er and the functions l i ( τ ), where τ = x/t , satisfy the system of ordinary diﬀeren tial equations ( V i ( l ) + τ ) dl i dτ + αl i = 0 , i = 1 , 2 , 3 . (84) Here the functions V i ( l ) are o btained from (7 8) b y replacing r i with l i . It is not diﬃcult to see that, due to the structure of the characteristic v elo cities, the case α = 0 implies o nly a constan t solution l i = l i 0 , where l 1 0 , l 2 0 , l 3 0 are arbitrary constan ts. If α 6 = 0, the general solution of (84) can b e found in an implicit form using the generalised ho dograph form ulae (61), where for N = 3 w e substitute, a ccording to (71) , (75), φ 1 k ( ξ ) ≡ B k ( ξ ) = ζ k ξ , φ 2 k ( ξ ) ≡ A k ( ξ ) = ξ . T o obtain similarit y solutions (83) one should use in (6 1) f i ( ξ ) = ξ β /c i , where β = 2 + 1 /α and c i , i = 1 , 2 , 3, are arbitrar y nonzero constan ts. Then the requiremen t that the functions l i m ust dep end on τ = x/t alo ne leads to the algebraic system τ = 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 , (85) 0 = c 1 ( l 1 ) γ − 1 + c 2 ( l 2 ) γ − 1 + c 3 ( l 3 ) γ − 1 , 20 where γ = − 1 /α and w e ha v e also replaced c i /γ 7→ c i . Direct substitution sho ws that solution l i deﬁned b y (85) indeed satisﬁes system (84 ) . W e note that this family o f solutions is unique to the case N = 3 and generally do es not exist f o r N > 3. b) Quasip erio dic solutions Another intere sting t yp e of solutions arises when one in tro duces in (61) (for N = 3) f 1 ( ξ ) = f 2 ( ξ ) = f 3 ( ξ ) = p R 7 ( ξ ) , R 7 ( ξ ) = 7 Y n =1 ( ξ − E n ) , where E 1 < E 2 < · · · < E 7 are real constants. Then, according to (75), solution (6 1) assumes the form x = ζ 1 r 1 Z ξ dξ p R 7 ( ξ ) + ζ 2 r 2 Z ξ dξ p R 7 ( ξ ) + ζ 3 r 3 Z ξ dξ p R 7 ( ξ ) , (86) − t = r 1 Z ξ dξ p R 7 ( ξ ) + r 2 Z ξ dξ p R 7 ( ξ ) + r 3 Z ξ dξ p R 7 ( ξ ) , (87) 0 = r 1 Z dξ p R 7 ( ξ ) + r 2 Z dξ p R 7 ( ξ ) + r 3 Z dξ p R 7 ( ξ ) , (88) whic h resem bles the celebrated system for the mu lti-gap (here – three-gap) solutions of t he KdV equation. Unlike (86) - (88), ho w ev er, t he three-gap KdV solutions corresp ond to the St¨ ack el matrix (71) with the rows A k ( ξ ) = ξ , B k ( ξ ) = ξ 2 , k = 1 , 2 , 3 [11]. Prop osition 5.1. F or any c onstants ζ 1 6 = ζ 2 6 = ζ 3 6 = 0 ther e exis ts at le ast one set { E 1 , . . . , E 6 } such that the solution r i ( x, t ) , i = 1 , 2 , 3 describ e d by (86) - (8 8) is quasi- p erio dic in x and p ossibly in t . W e presen t here a sk etch of the pro of. Av aila bilit y of the solution in the for m (86) - (88) implies the existence of separate dynamics of r j -s with resp ect to x and t . Indee d, diﬀeren tiating (86 ) - (88) with resp ect to x for ﬁxed t one readily obtains ∂ r i ∂ x = ( r j − r k ) p R 7 ( r i ) Π , i, j, k = 1 , 2 , 3 , i 6 = j 6 = k , (89) where Π( r 1 , r 2 , r 3 ) = ( ζ 1 − ζ 2 ) r 1 r 2 + ( ζ 2 − ζ 3 ) r 2 r 3 + ( ζ 3 − ζ 1 ) r 3 r 1 = det ∆ (90) – see (8 0). Analogously , diﬀeren tiating (86) - (88) with resp ect to t for ﬁxed x one obtains ∂ r i ∂ t = ( ζ j r j − ζ k r k ) p R 7 ( r i ) Π , i, j, k = 1 , 2 , 3 , i 6 = j 6 = k . (91) One can see tha t t he ﬂows (89) a nd (91) a re consisten t with the spatio- temp oral dynamics (44), (78). W e also note that equations ( 8 9), (91) resem ble Dubrovin’s equations for the 21 auxiliary spectrum dynamics in the KdV ﬁnite-gap in tegratio n problem (see, for instance, [33]). Let us no w suppose that r 1 ∈ [ E 1 , E 2 ] , r 2 ∈ [ E 3 , E 4 ] , r 3 ∈ [ E 5 , E 6 ] , (92) so that all p R 7 ( r i ) are real. The ab o ve condition ( 9 2) means that the p oint p = ( r 1 , r 2 , r 3 ) ∈ R 3 lies within the rectangular b ox K ij k ∈ R 3 with the v ertices at ( E i , E j , E k ), i, j, k = 1 , . . . , 6, i 6 = j 6 = k . No w, for an y set of the constants ζ 1 , ζ 2 , ζ 3 there exists at least one b o x K i,j,k = K ∗ ∈ R 3 , whic h is not in tersected by the cone Π( r 1 , r 2 , r 3 ) = 0. That is, inside K ∗ the denominator Π( r 1 , r 2 , r 3 ) in (8 9) nev er v anishes. Assume no w that the ‘initial’ v alues of r 1 , r 2 , r 3 for some x = x 0 b elong to K ∗ . Then it follow s from (89) t ha t , under the x -ﬂow ( t = const), the p oin t p remains inside K ∗ and undergo es “elastic” reﬂections at the faces of K ∗ as x v aries (note that, since r j 6 = r k for j 6 = k , t he f actor ( r j − r k ) in (89) nev er v anishes so t he reﬂections o ccur only at the faces of K ∗ ). Therefore, the motion is quasi-p erio dic with respect to x a s long as conditio ns (92) are satisﬁed. Indeed, the system (89) p ossesses t w o in tegra ls (87) and (88) outside the “resonan t” p oin ts, where Π = 0, so it sp eciﬁes a quasi-p erio dic motion o n a 3-torus provided conditions (92) are satisﬁed. Of course, if conditions (92) are no t satisﬁed at x = x 0 the solutions r i ( x ) may blow up and not b e quasi-p erio dic. The pro of of quasi-p erio dicity of t he t - ﬂow is similar, ho wev er, there is an additional requiremen t that the factor ( ζ j r j − ζ k r k ) in (90 ) should not v anish for all r ∈ K ∗ whic h migh t imp ose additional restrictions on the c hoice of E i (that is for some { E j } the motion can b e quasi-p erio dic in x but not in t ). W e note that t he quasi-p erio dicity of the x - and t -ﬂows can b e prov ed directly from the solution (8 6 ) – (88 ), how ev er t he outlined pro of using the dynamical systems argumen t s is qualitativ ely more transparent and more readily yields the “ r esonant” restrictions for x - and t -ﬂo ws. W e also note that the quasip erio dic solutions could b e constructed for N > 3 as w ell (see Section 8.2). 6 In tegr ab i l it y o f N -comp onen t h ydro dynamic re duc- tions W e no w prov e our main Theorem 3.1 stating that t he N -comp onen t ‘cold-gas’ h ydro dynamic reduction (36), (38) represen ts a semi-Hamiltonian (i.e. integrable) linearly degenerate hy - dro dynamic t yp e system. F or that, according to Corollary 3.1, it is suﬃcien t to show tha t the conserv ation la w densities u i and the tra nsp o rt ve lo cities v i admit para metric represen- tations (57) and (56), u i = U i ( r ) and v i ( U ( r )) = V i ( r ), via N functions r k in terms of the St¨ ack el matrix (55). W e supp ose that h ydro dynamic t yp e system (36), (3 8 ) can b e rewritten in a diag o nal form (44), a nd, moreo v er, the characteristic v elo cities V i ( r ) coincide with the expressions v i ( U ( r )). 22 No w, substitution of (56), (57) in to (38) leads to the algebraic system N X k =1 ǫ ik ( − 1) k P k det ∆ (12) ik = det ∆ (2) i − ξ i det ∆ (1) i , i = 1 , . . . , N , (93) for P k ( r k ) and φ i k ( r k ). Here the matr ix ∆ (12) ik is the matrix ∆ with ﬁrst two rows and i th and k th columns deleted. In the deriv atio n of (93) w e ha ve used the d e term i n ant Sylvester identity ( see, for instance, Gantmac her 1959 ) det ∆ (12) ik = det ∆ (1) k det ∆ (2) i − det ∆ (1) i det ∆ (2) k det ∆ . Expanding the determinan ts, det ∆ (1) i = N X k =1 h ( − 1) k +1 φ 2 k det ∆ (12) ik i , det ∆ (2) i = N X k =1 h ( − 1) k +1 φ 1 k det ∆ (12) ik i , w e rewrite equations (93) as N nonline ar systems for φ n k and P k , where k , n = 1 , . . . , N , N X k =1 ( − 1) k ( φ 1 k − ξ i φ 2 k + ǫ ik P k ) det ∆ (12) ik = 0 , i = 1 , . . . , N . (94) W e recall tha t φ N − 1 k = r k , φ N k = 1. One can now in tro duce N mat r ices δ i obtained fro m the matr ix ∆ b y deleting the ﬁrst t wo rows and the i -t h column, and adding the ﬁrst r ow with the elemen ts φ 1 k − ξ i φ 2 k + ǫ ik P k . Th us, eac h matrix δ i has dimension ( N − 1) × ( N − 1). Then the a b o v e set of equations (94) can b e rewritten as det δ i = 0 , i = 1 , . . . , N , (95 ) whic h implies that the rows o f each o f the matrices δ i m ust b e line arly d e p en dent : C i, 1  ǫ ik P k + φ 1 k − ξ i φ 2 k  + N − 2 X n =3 C i,n − 1 φ n k = C i,N − 2 r k + C i,N − 1 , k = 1 , . . . N , i = 1 , . . . , N − 1 , k 6 = i, (96) where C i,k are arbitrary constan ts. These conditio ns can b e considered as N linear systems , for ﬁxed k each. Since a ll these systems are consisten t the f unctions φ i k and P k can b e f ound b y solving system (96). Constan ts C i, 1 cannot b e equal to zero since in that case, a ccording to (56), the ve lo cities V i w ould b ecome undetermined. Therefore, without loss of generality we can set C i, 1 = 1 and the n umber of free constan ts b ecomes N ( N − 2). Th us, the fo llo wing Prop osition is v alid: Prop osition 6.1 : Gener al solution of system ( 9 4) is determine d by solutions φ i k = det ˜ B i k r k + det ¯ B i k det B k , P k = det B ( P ) k det B k (97) 23 of N line ar systems (96), whe r e B k , ¯ B i k , ˜ B i k and B ( P ) k ar e matric es with eleme n ts ˜ b im k l = ¯ b im k l = b ( P ) m k l = b m k l =    1 for l = 2 − ξ l for l = 3 C m,l − 2 for l > 3 if l 6 = i + 1 l 6 = 1 (98) and b im k 1 = ˜ b im k 1 = ¯ b im k 1 = ǫ mk , ¯ b i +1 m k i = C i,N − 1 , ˜ b i +1 m k i = C i,N − 2 , b ( P ) m k 1 = C i,N − 2 r k + C i,N − 1 , wher e C m,l ar e arbitr ary c onstants such that det B k 6 = 0 . Remark: The set of constan ts C l,m for whic h det B k = 0 has Leb esque measure zero o r requires a ve ry special choice of t he parameters η k . The excep tional case is the followin g: the vec tors ξ , 1 and ǫ k are linearly dep enden t which yields, according to the deﬁnition (39), a set of equations for the sp ecial v alues η k . Th us, we ha v e prov ed that all elemen ts of the St¨ ac ke l matrix (5 5) dep end linearly on Riemann inv ariants and t hese elemen ts a re determined from the alg ebraic system (38) up to N ( N − 2) arbitr a ry constan ts remov able b y an appropriate c hange of the Riemann in v arian ts (for instance, b y a shift in the case N = 3). By Corollary 4.1, the existence of suc h a St¨ ac k el matrix aut o matically prov es t he semi-Hamiltonian and linearly-degenerate prop erties of the h ydro dynamic reductions (36), (3 8). No w, our main Theorem 3.1 is pro ved . 7 Riemann in v arian t s and c h aracteristic v elo cities: ex- plicit constr u ction The construction described in Sections 3 and 6 pro vides a pro of of the existence of Riemann in v ariants for system (36), (38) for arbitrary N . The Riemann in v ariants are found to parameterise system (36 ), (38) via t he sole St¨ ac k el matrix, whic h, by Corollary 4.1 , implies linear degeneracy and in tegrability of this system. Explicit represen tations fo r conserv at ion la w densities u i and transp o rt v elo cities v j in terms o f the Riemann in v arian ts are give n b y F era p on tov [11] for m ulae (57), (56) where the en tries φ n k of the St¨ ack el mat r ix (5 5) and t he functions P k ( r k ) are deﬁned b y form ula e (97) – (98). Using the functions φ n k one also obtains the generalised ho dogr a ph solutions (61). The outlined pro cedure, while pro viding general theoretical framew ork for the study of the ‘cold-gas’ reductions of the kinetic equation for a solito n gas, seems to b e not v ery conv enien t from the viewp oin t of practical calculations. It also inv olv es N ( N − 2) in termediate constan ts C l,m , whic h in tro duce an additional unnecessary complication. It is, th us, desirable to hav e more direct represen tatio ns for the Riemann in v ariants and c haracteristic v elo cities, whic h will a lso b e free from these intermediate ar bitr a ry constan ts. W e shall mak e use of the Theorem 3.1 and sho w that, once the linear degeneracy and in tegrability prop erties of syste m (36), (38) are established, explicit relations b et w een the Riemann inv ariants r and the conserv ed densities u can b e found b y a relativ ely straightfor- w ard calculation. The calculatio n will in v olve the prop erties o f the Lam ´ e co eﬃcien ts outlined in Section 4. First, without loss of generality w e c ho ose the follo wing normalization (see (53)) u k = ¯ H k , (99) 24 where ¯ H k ’s a re t he Lam´ e co eﬃcien ts (50). Now, using Theorem 3.1 w e assume that h ydro- dynamic type system (36), ( 3 8) can b e rewritten in a diagonal for m (44), so that u i = U i ( r ) and v i ( U ( r )) = V i ( r ). F o r con venie nce, in what follo ws we shall use small u ’s and v ’s only , assuming that u j = u j ( r ) ≡ U j ( r ), v j = v j ( r ) ≡ V j ( r ). T o obtain explicit formulae f or the Riemann inv ariants of the hyd ro dynamic reduction (36), (38) w e need ﬁrst to prov e its so-called “Egorov” prop ert y . Deﬁnition 7.1 (P avlo v & Tsarev 2003 [38]): Semi-Hamiltonian hydr o dynamic typ e sys- tem (42) is c al le d the Egor ov, if a sole p air of c onservation laws ∂ t a ( u ) = ∂ x b ( u ), ∂ t b ( u ) = ∂ x c ( u ) (100) exists. It w as prov ed in [38], that ∂ i a = ¯ H 2 i , ∂ i b = ˜ H i ¯ H i , ∂ i c = ˜ H 2 i , (101) (see (50) a nd (66) for the deﬁnitions of ¯ H i and ˜ H i ) while the corresp onding rotation co eﬃ- cien ts (62) b ecome symmetric, i.e. β ik = β k i , i 6 = k . (102) Another imp ortan t fact pro v en in [38] is that a ll comm uting ﬂo ws to a semi-Hamiltonia n Egoro v system are also Egor o v so commuting ﬂo w (48) p osses ses a similar pa ir of conserv ation la ws ∂ τ a ( u ) = ∂ x h ( u ), ∂ τ h ( u ) = ∂ x g ( u ) , where ∂ i h = H i ¯ H i , ∂ i g = H 2 i . (103) No w w e pro v e the followin g Lemma 7.1 : Hydro dynamic reductions (36), (38) a re Egoro v. Pro of : W e consider the sum of conserv ation la ws (36), (38) ∂ t  X u k  = ∂ x  X u k v k  = ∂ x " N X k =1 u k ξ k + X m 6 = k ǫ k m u m ( v m − v k ) !# (104) One can see that, since the matrix ǫ ik is symmetric, the last term in r.h.s. of (104 ) v anishes. Th us, (104) simpliﬁes to the form ∂ t  X u k  = ∂ x  X ξ k u k  . (105) Ho wev er, the ﬂux Σ ξ k u k of conserv ation la w (1 05) is nothing but the density of another conserv atio n law whic h can b e obtained b y the same summation but with t he sp ecial w eigh ts ξ i , i.e. ∂ t  X ξ k u k  = ∂ x  X ξ k u k v k  . Comparison with deﬁnition (100) implies that in our case a = X u m , b = X ξ m u m ≡ X u m v m , c = X ξ m u m v m , (106) 25 whic h completes the pro of. No w w e form ulate the follo wing Theorem 7.1 : The Riemann invaria n ts of N -c omp on e nt hydr o dynamic r e ductions (36), (38) c an b e found exp licitly a s r i = − 1 u i 1 + X m 6 = i ǫ im u m ! , i = 1 , . . . , N . (107) Pro of : F o r the sak e of completeness of our construction we ﬁrst sho w that t he linear degeneracy prop ert y (49) of sys tem (36), (38) readily follow s from the (already established) existence of the Riemann in v ariants r k . Indeed, diﬀeren tia ting (38) with resp ect to the Riemann in v ariant r i and taking into accoun t tha t (see (50), (99)) ∂ i ln u k = ∂ i v k v i − v k , i 6 = k , w e obtain the expression ∂ i v i = X m 6 = i ǫ im ( v m − v i ) ∂ i u m + X m 6 = i ǫ im u m ( ∂ i v m − ∂ i v i ) , whic h reduces, on using (5 2), to the form ∂ i v i 1 + X m 6 = i ǫ im u m ! = 0 . (108) Equation (108) can only b e satisﬁed if ∂ i v i = 0 for a ll i (otherwise the ﬁeld v ariables u m in the algebraic system (38) w ould cease to b e indep enden t). Th us system (36), (38) is indeed linearly degenerate. No w, diﬀerentiation of a lgebraic sys tem (38) with respect to the Riemann in v arian t r k yields ∂ k v i = X m 6 = i,k ǫ im u m ( ∂ k v m − ∂ k v i ) + X m 6 = i,k ǫ im ( v m − v i ) ∂ k u m + ǫ ik u k ( ∂ k v k − ∂ k v i ) + ǫ ik ( v k − v i ) ∂ k u k , whic h reduces, with an a ccoun t of (52) and the linear degeneracy prop erty , to ( v k − v i ) " 1 + X m 6 = i ǫ im u m ! ∂ k ln u i − X m 6 = i ǫ im ∂ k u m # = 0 . Since all c haracteristic v elo cities v k are distinct, the expression in square brac k ets must v anish for an y pair of indices i and k , i.e. we ha v e ∂ k ln u i = P m 6 = i ǫ im ∂ k u m 1 + P m 6 = i ǫ im u m , k 6 = i . (109) 26 In tegratio n of (109) yields X m 6 = i ǫ im u m + R i ( r i ) u i = − 1 , (110) where R i ( r i ), i = 1 , . . . , N ar e a rbitrary functions. W e now diﬀeren tia t e (11 0) with resp ect t o the Riemann in v ariants r i and r k , whic h gives , on using (99) and (62), X m 6 = i ǫ im β im + R ′ i ( r i ) + R i ( r i ) ∂ i ln ¯ H i = 0 (111) and X m 6 = i,k ǫ im β k m + R i ( r i ) β k i + ǫ ik ∂ k ln ¯ H k = 0 (112) resp ectiv ely . Substitution of (106) in to (1 01) giv es ¯ H i = X m 6 = i β im + ∂ i ln ¯ H i , ˜ H i = ξ i ¯ H i + X m 6 = i ( ξ m − ξ i ) β im . (113) By expressing ∂ i ln ¯ H i from the a b o ve ﬁrst equation, (111) and (112) reduce to the fo r m R i ( r i ) ¯ H i = R i ( r i ) X m 6 = i β im − X m 6 = i ǫ im β im − R ′ i ( r i ) , (114) ǫ im ¯ H m = ǫ im X n 6 = m β nm − X n 6 = i,m ǫ in β nm − R i ( r i ) β im . Substitution of the expressions R i ( r i ) ¯ H i and ǫ im ¯ H m in to (110 ) yields a set of constraints R ′ i ( r i ) = 1 , i.e. R i ( r i ) = r i + α i , where α i are arbitrary constan t s. Since any function o f a Riemann inv ariant is a Riemann inv ariant as w ell one can put without loss o f generalit y that R i ( r i ) = r i . Then (11 0) reduces to (107). The Theorem is prov ed. T aking in to accoun t R i ( r i ) = r i and eliminating ¯ H i from (114) w e arriv e at the linear algebraic system X m 6 = i,k ( r i ǫ k m − ǫ ik ǫ im ) β im + ( r i r k − ǫ 2 ik ) β ik = ǫ ik , i 6 = k (115) for the rotation co eﬃcien ts β ik , while (1 14) reduces (cf. the ﬁrst form ula in (113)) to ¯ H i = X m 6 = i  1 − ǫ im r i  β im − 1 r i . (116) Let us intro duce a matrix ǫ suc h that its oﬀ-diagonal co eﬃcien ts a re the a foremen tioned symmetric constants ǫ ik , while the diagona l co eﬃcien ts ǫ ii = r i . Theorem 7.2 : The r otation c o eﬃcients β ik satisfying line ar a lgebr aic system (11 5) ar e the oﬀ-diagon al c omp one n ts o f the matrix inverse to the m atrix − ǫ , i.e. N X m =1 ǫ im β k m = − δ ik . (117) 27 Pro of : W e intro duce the functions β ii ( r ) so that expression (63) could b e extended to the full set of indices, i.e. we will hav e ∂ i β j k = β j i β ik ∀ i, j, k . (118) It is easy to che c k that (118) is v alid for any curviline ar c o or dinate system asso ciate d with semi-Hamiltonian Egor ov l i n e arly de gener ate hydr o dynamic typ e system (see (49 ) and (101)), i.e. if and only if the rota t io n co eﬃcien ts β ik are symmetric (see (102)) and determined by (67), where the functions β ii ( r ) ≡ ∂ i ln ¯ H i . (119) Indeed, the ab ov e set of equations ( 1 18) for tw o distinct indices ( just tw o c hoices) r educes to the form ∂ k β j k = β j k β k k , ∂ i β k k = β 2 ik . (120) The ﬁrst part of these equations is nothing else but (67) while the second part is j ust the w ell-know n prop erty of any curvilinear co o r dinate net (see [10]) : the scalar p oten t ial V is determined b y its second deriv ativ es, i.e. ∂ 2 ik V = β ik β k i , k 6 = i . (121) Th us, in the Egoro v (symmetric) case, the ab o ve prop ert y (121) simpliﬁes to ∂ 2 ik V = β 2 ik , k 6 = i . (122) Comparing this formu la and the second form ula in (120), one can conclude that β k k = ∂ k V . If all indices in (120) coincide, the last nontrivial consequence g iv en b y ∂ k 1 β k k = − 1 (123) allo ws one to integrate (step-by -step) nonlinear system in partial deriv ativ es (1 18). Instead of this direct, but somewhat complicated pro cedure, w e shall use a more sophisticated but tec hnically muc h more simple appro ac h to the deriv ation o f general solutio n of system (118). First, let us in tro duce the com binations A ik = N X m =1 ǫ im β k m (124) (w e recall that ǫ ii = r i ). Then (1 1 5) reads as follo ws r i A ik = ǫ ik (1 + A ii ) , i 6 = k . (125) Diﬀeren tia t io n of (124) with resp ect to the R iemann in v ariants r i , r k , r j leads to the system ∂ i A ik = β ik (1 + A ii ) , ∂ k A ik = β k k A ik , ∂ j A ik = β j k A ij , i 6 = k . Compatibilit y conditions imply just one extra equation ∂ k A ii = β ik A ik , i 6 = k . 28 No w w e diﬀeren tia te (125) with resp ect to the Riemann inv arian t r j to obtain ( r i β j k − ǫ ik β ij ) A ij = 0 , i 6 = j 6 = k . Since expressions r i β j k − ǫ ik β ij cannot v anish iden t ically , w e ha v e the only p ossible choice : A ik = 0 for eac h pa ir of distinct indices, and A ii = − 1 (see (125)). Th us, we conclude that (124) reduces to the fo rm (117) (let us emphasize one more time that ǫ ii ≡ r i , while all the other ǫ j k = ǫ k j are constants). The matrix ǫ con tains N ( N − 1) / 2 arbitrary constan ts ǫ ik , then all comp onents o f the matrix β are par a meterised b y these N ( N − 1) / 2 arbitrary constants . On the other hand, (118) is an o v erdetermined system, where al l ﬁrst deriv a tiv es of β ik are expresse d via β j n only . Th us, a general solution of syste m in partial deriv ativ es (118) m ust dep end on N ( N + 1) / 2 arbitr ary constan ts, b ecause this system is written for N ( N + 1) / 2 functions β ik (these are N ( N − 1) / 2 symmetric o ﬀ-diagonal elemen ts, i.e. rotation co eﬃcien ts β ik ; and N diagonal comp onen ts β k k ). It means, that the inv erse matrix ǫ con tains extra N arbitrary constan ts α i whic h are nothing but the shifts of the Riemann in v arian ts r i lo cated on the diagonal (see the end of the pro of of Theorem 7.1). Then these N shift constan ts can b e remov ed without loss o f generalit y . The Theorem is pro v ed. In particular, for N = 3 w e hav e fro m (117) the explicit expressions for β ik : β 12 = r 3 ǫ 12 − ǫ 13 ǫ 23 r 1 r 2 r 3 − r 1 ǫ 2 23 − r 2 ǫ 2 13 − r 3 ǫ 2 12 + 2 ǫ 12 ǫ 13 ǫ 23 , β 13 = r 2 ǫ 13 − ǫ 12 ǫ 23 r 1 r 2 r 3 − r 1 ǫ 2 23 − r 2 ǫ 2 13 − r 3 ǫ 2 12 + 2 ǫ 12 ǫ 13 ǫ 23 , (126) β 23 = r 1 ǫ 23 − ǫ 12 ǫ 13 r 1 r 2 r 3 − r 1 ǫ 2 23 − r 2 ǫ 2 13 − r 3 ǫ 2 12 + 2 ǫ 12 ǫ 13 ǫ 23 ; β 11 = − r 2 r 3 + ǫ 2 23 r 1 r 2 r 3 − r 1 ǫ 2 23 − r 2 ǫ 2 13 − r 3 ǫ 2 12 + 2 ǫ 12 ǫ 13 ǫ 23 , β 22 = − r 1 r 3 + ǫ 2 13 r 1 r 2 r 3 − r 1 ǫ 2 23 − r 2 ǫ 2 13 − r 3 ǫ 2 12 + 2 ǫ 12 ǫ 13 ǫ 23 , β 33 = − r 1 r 2 + ǫ 2 12 r 1 r 2 r 3 − r 1 ǫ 2 23 − r 2 ǫ 2 13 − r 3 ǫ 2 12 + 2 ǫ 12 ǫ 13 ǫ 23 . Remark. Note that equation ( 1 17), despite of ha ving a simpler form tha n original equation (115), is more general as it deﬁnes al l (not only oﬀ-diagonal) comp onen t s β ik in terms of Riemann inv ariants. Th us, the rotation co eﬃcien t s β ik , i 6 = k satisfy b oth systems (115) and (118) and are completely deﬁned in terms of the matrix ǫ . As a by-product of the pro of of Theorem 7.2 we obtain t he follow ing imp ortant Corollary 7.1 Since system (118) describes rotation co eﬃcien ts β ik asso ciated with h ydro dynamic t yp e systems p osses sing simultane ously Egoro v and linear degeneracy prop- erties, we conclude t ha t our reduction (36), (38) of the kinetic equation (1 ) is the only (up to unessen tial transformatio ns) h ydro dynamic type system p ossess ing b oth these prop erties. No w, using (66), (99), (113), (116) a nd (119) we form ula t e the main result o f this Section: 29 A lgebr aic r elations (38) c an b e r e s o lve d in a p ar ametric form in terms of the Riemann invariants: u i = N X m =1 β im , v i = 1 u i N X m =1 ξ m β im , (127) wher e the symmetric c o eﬃc ients β ik ar e elements of the m atrix − ǫ − 1 (see (117)). As a matter of fact, the ﬁrst form ula in (1 27) represen ts the in version o f for m ula (107). As one can see, this in v ersion is rather nontrivial. In particular, for N = 3 we hav e fro m (127) the explicit expressions for conserv ation la w densities u i and characteris tic v elo cities v k u 1 = − r 2 r 3 + r 2 ǫ 13 + r 3 ǫ 12 − ǫ 12 ǫ 23 − ǫ 13 ǫ 23 + ǫ 2 23 r 1 r 2 r 3 − r 1 ǫ 2 23 − r 2 ǫ 2 13 − r 3 ǫ 2 12 + 2 ǫ 12 ǫ 13 ǫ 23 , u 2 = − r 1 r 3 + r 1 ǫ 23 + r 3 ǫ 12 − ǫ 12 ǫ 13 − ǫ 13 ǫ 23 + ǫ 2 13 r 1 r 2 r 3 − r 1 ǫ 2 23 − r 2 ǫ 2 13 − r 3 ǫ 2 12 + 2 ǫ 12 ǫ 13 ǫ 23 , (128) u 3 = − r 1 r 2 + r 1 ǫ 23 + r 2 ǫ 13 − ǫ 12 ǫ 13 − ǫ 12 ǫ 23 + ǫ 2 12 r 1 r 2 r 3 − r 1 ǫ 2 23 − r 2 ǫ 2 13 − r 3 ǫ 2 12 + 2 ǫ 12 ǫ 13 ǫ 23 , v 1 = ξ 1 ( ǫ 2 23 − r 2 r 3 ) + ξ 2 ( r 3 ǫ 12 − ǫ 13 ǫ 23 ) + ξ 3 ( r 2 ǫ 13 − ǫ 12 ǫ 23 ) ǫ 2 23 − r 2 r 3 + ǫ 13 ( r 2 − ǫ 23 ) + ǫ 12 ( r 3 − ǫ 23 ) , v 2 = ξ 2 ( ǫ 2 13 − r 1 r 3 ) + ξ 3 ( r 1 ǫ 23 − ǫ 12 ǫ 13 ) + ξ 1 ( r 3 ǫ 12 − ǫ 13 ǫ 23 ) ǫ 2 13 − r 1 r 3 + ǫ 13 ( r 3 − ǫ 13 ) + ǫ 23 ( r 1 − ǫ 13 ) , (129) v 3 = ξ 3 ( ǫ 2 12 − r 1 r 2 ) + ξ 2 ( r 1 ǫ 23 − ǫ 12 ǫ 13 ) + ξ 1 ( r 2 ǫ 13 − ǫ 12 ǫ 23 ) ǫ 2 12 − r 1 r 2 + ǫ 13 ( r 2 − ǫ 12 ) + ǫ 23 ( r 1 − ǫ 12 ) . One can observ e that formulae (129) do not coincide with represen tation (78), (76) obtained earlier for the same family of the characteristic v elo cities. The reason is that the tw o repre- sen tations cor r esp o nd to diﬀeren t c hoices of the Riemann in v ariants (w e recall one more time that a ny function of a Riemann in v arian t is a Riemann in v ariant as w ell). The relationship b et wee n these t w o equiv alen t sets of the Riemann inv ariants can b e obtained b y equating the c haracteristic v elo cities (78) and (129) (o r , alternatively , the densities (79) and (128)), whic h do not depend on a part icular normalization of the R iemann inv ariants. It is more con ve nien t, ho w ev er, to get the sough t relationship b y a substitution (1 2 8) in to (8 2), where w e replace r i with ˜ r i . As a result w e get ˜ r 1 = ( ǫ 13 − ǫ 12 )( ǫ 23 r 1 − ǫ 12 ǫ 13 ) ( ξ 2 − ξ 3 ) r 1 + ( ξ 3 − ξ 1 ) ǫ 12 + ( ξ 1 − ξ 2 ) ǫ 13 , ˜ r 2 = ( ǫ 12 − ǫ 23 )( ǫ 13 r 2 − ǫ 12 ǫ 23 ) ( ξ 3 − ξ 1 ) r 2 + ( ξ 1 − ξ 2 ) ǫ 23 + ( ξ 2 − ξ 3 ) ǫ 12 , (130) ˜ r 3 = ( ǫ 23 − ǫ 13 )( ǫ 12 r 3 − ǫ 13 ǫ 23 ) ( ξ 1 − ξ 2 ) r 3 + ( ξ 2 − ξ 3 ) ǫ 13 + ( ξ 3 − ξ 1 ) ǫ 23 . Here by ˜ r 1 , ˜ r 2 , ˜ r 3 w e denote the ‘old’ Riemann in v ariants as in Section 5. Note that the c hoice (130) of the R iemann in v ariants leads to the ho mo g eneous expres- sions (7 8 ) for the c haracteristic v elo cities, whic h mak es p o ssible the construction of the 30 similarit y solutions (85). Suc h a p o ssibilit y , how ev er, is unique to the case N = 3 a s for N > 3 the general rational substitution of the ty p e (1 30) will not allo w for the elimination of all inhomogeneous terms (unless one has a v ery sp ecial set of the co eﬃcien ts ǫ ij , ξ k ). As a consequenc e, the family of the similarity solutions (83) exists only for the case N = 3 whic h mak es this case sp ecial. 8 Comm uting h ydr o dynamic ﬂo ws 8.1 General explicit represen tation Comm uting ﬂows to semi-Hamiltonian linearly degenerate system (36 ), (38) are deﬁned in terms of the Riemann in v ariants by equations (48), (47 ). W e recall that, a ccording to Prop osition 4.1, only N − 2 of the commuting ﬂows are linearly degenerate (excluding the ‘trivial’ ﬂo ws sp eciﬁed by linear com binatio ns o f the constan t c haracteristic v elo cit y 1 and the c haracteristic v elo cit y v of the original ﬂow (44)). The general solution of the g eneralised ho dograph equations (47) sp ecifying comm uting ﬂows w a s obtained b y F erap on tov [11] in terms of the St¨ ack el matrix en tries (see Theorem 4.2). Here w e are in terested in a mor e explicit represen tation of the comm uting ﬂo ws for t he sp eciﬁc system (36), (38). F or that, instead of in tegrating system (47), w e ta k e a dv antage of the fact that our linearly degenerate system (3 6), (38) is Egoro v. In that case, the commuting ﬂo ws can b e found explicitly . W e ﬁrst observ e that an y conserv ation law densit y h for linearly degenerate h ydro dynamic t yp e system (3 6) can b e represen ted in the form (see (53) or ( 5 7)) h = N X k =1 u k P k ( r k ) , (131 ) with N a rbitrary functions P k ( r k ) of a single v ariable. Then w e mak e use of Lemma 8.1 (P a vlov & Tsarev 2003 [38]): Al l c ommuting ﬂows (48), (47) in the Egor ov c ase ar e sp e ciﬁe d by the expr essio n (se e (65), (101), (103)) W i = H i ¯ H i = ∂ i h ∂ i a . (132) Substituting (131 ) , (106) in to (132) and using the ﬁr st f orm ula from (113) we obtain an explicit represen tation for the c haracteristic v elo cities of the comm uting ﬂows (4 8 ), W i = P i ( r i ) + 1 ¯ H i P ′ i ( r i ) + X m 6 = i ( P m ( r m ) − P i ( r i )) β im ! . (133) W e recall that P k ( r k ), k = 1 , . . . , N a re arbitrar y functions and the dep endence of the rotation co eﬃcien ts β im on the Riemann in v arian ts is found by in vers ion of the matrix − ǫ (see (117)). If P k ( r k ) = 1, (13 3) reduces to W i = 1; if P k ( r k ) = ξ k , it reduces to t he second form ula in (127), i.e. to h ydro dynamic reduction (36), (38) itself. 31 8.2 generalised ho dograph metho d T aking into account the Com b escure transformation (65) and form ula (66 ) the generalised ho dograph solution (46) can b e represen t ed in a symmetric form x ¯ H i + t ˜ H i = H i ( r ) . (134) Since (see (127), (66), (99)) ¯ H i = N X m =1 β im , ˜ H i = N X m =1 ξ m β im , expression (13 4), with an accoun t of (65 ) , (133), assumes the fo rm N X m =1 ( x + ξ m t ) β k m = P ′ k ( r k ) + N X m =1 P m ( r m ) β k m . (135) Multiplying equation (135) through by the matrix ǫ and p erforming summation, P N k =1 ǫ ik [ . . . ] k , w e obtain, up on using (117), a general solution of the N -component linearly degenerate hy- dro dynamic reduction in a n implicit form (cf. (61) ) x + ξ i t = P i ( r i ) − r i P ′ i ( r i ) − X m 6 = i ǫ im P ′ m ( r m ) , i = 1 , 2 , ..., N , (136) where P i ( r i ), i = 1 , . . . , N , are a rbitrary functions. Note that under the re-para metrization P ′′ k ( ξ ) = − φ k ( ξ ) f ( ξ ) the generalised ho dogr a ph solution (13 6) b ecomes x + ξ i t = r i Z ξ φ i ( ξ ) f ( ξ ) dξ + X m 6 = i ǫ im r m Z φ m ( ξ ) f ( ξ ) dξ . (137) No w, comparison of ( 1 37) with the F erap onto v [11] solution (61) prov ides a direct w a y for the iden tiﬁcatio n of the en t r ies of the St¨ ac ke l ma t r ix (55). Also, for this c hoice of the St¨ ac k el matrix all constants C l,m (see Section 6) can b e expressed in terms of the co eﬃcien ts ǫ ij and ξ k b y (97), ( 9 8). F o r the particular c hoice of f ( ξ ) deﬁned as f ( ξ ) = p R K ( ξ ) (138) where R K ( ξ ) = K Y n =1 ( ξ − E n ) , 32 and E 1 < E 2 < · · · < E K are real constan ts ( K = 2 N + 1 if N is o dd and K = 2 N + 2 if N is ev en); and φ k ( ξ ) b eing arbitrary p olynomials in ξ of degrees less than N , system (136) describes quasip erio dic solutions of the form x + ξ i t = r i Z ξ φ i ( ξ ) dξ p R K ( ξ ) + X m 6 = i ǫ im r m Z φ m ( ξ ) dξ p R K ( ξ ) , i = 1 , 2 , ..., N , (139) The pro of of quasip erio dicity of solution (139) is analogo us to that for solution (8 6 ), (87), (88) obtained for N = 3. 8.3 Linearly degenerate comm uting ﬂo ws T o extract the fa mily of linearly degenerate comm uting ﬂo ws fro m general represen tation (133) we form ulate the follo wing Lemma 8.2 : F o r the lin e arl y de gener ate c omm uting ﬂows e ach function P i ( r i ) in (133) is line ar with r esp e ct to the c orr esp onding R iemann invariant r i . Pro of : The condition ∂ i W i = 0 of linear degeneracy of the comm uting ﬂo w implies, on using ( 6 7) and (113 ), that P ′′ i ( r i ) = 0. W e no w consider t he represen tation for the f amily o f linearly degenerate commuting ﬂo ws suggested b y t he form o f the kinetic equations ( 3 3), ( 3 4) for the KdV hierarc hy . Im- p ortantly , t he whole KdV kinetic hierarch y (33), (34 ) is ch aracterised by a single integral k ernel, G ( η , µ ) = ln | ( η − µ ) / ( η + µ ) | (whic h is consisten t with the fact that all equations o f the original ﬁnite-gap Whitham hierarch y a re asso ciated with the same Riemann surface). This suggests that there could exist a family of comm uting ﬂows to g eneral nonlo cal kinetic equation (1) ha ving the f orm f τ = ( ˜ sf ) x , ˜ s ( η ) = ˜ S ( η ) + 1 η ∞ Z 0 G ( η , µ ) f ( µ )[ ˜ s ( µ ) − ˜ s ( η )] dµ , (140) where ˜ S ( η ) is an arbitra r y function. Although veriﬁcation of comm uta tivit y of the kinetic equations (1) and (140) is b ey ond the scop e of the presen t pap er, it is clear that, if these equation do comm ute, this m ust b e manifested on the lev el of h ydro dynamic reductions a s w ell. Ha ving this in mind, we consider the N -compo nent hyd ro dynamic reductions to (1 4 0) obtained b y the familiar delta-functional ansatz (35) and try to see if they comm ute with the original reductions (36)–(39). First w e notice that equation (140) is, essen tia lly , the same kinetic equation (1) but with a diﬀeren t time v a riable and diﬀeren t “free soliton sp eed” function S ( η ). Now , since w e ha ve prov ed integrabilit y of the linearly degenerate hydrodynamic reductions (36)–(39) in a general f o rm, we automatically hav e that a nalogous N -compo nen t hydrodynamic reductions of (1 40) m ust a lso b e in tegrable linearly degenerate systems. It should b e noted that, since the function ˜ S ( η ) is arbitr a ry , the set { ˜ ξ 1 , . . . , ˜ ξ N } of its v alues ˜ ξ j = ˜ S ( η j ) can b e view ed as a set of N arbitrary nu m b ers, and the cor r esp onding ‘cold-gas’ hyd ro dynamic r eduction b ecomes (cf. (36)–(39)) u i τ = ( u i ˜ v i ) x , i = 1 , . . . , N , (141) 33 where t he v elo cities ˜ v i = − ˜ s i and the conserv ation law densities u i satisfy algebraic relations ˜ v i = ˜ ξ i + N X k 6 = i ǫ ik u k ( ˜ v k − ˜ v i ), ǫ ik = ǫ k i , (142) and ǫ ik are the same as in (3 9). According to Theorem 3.1, system (1 41), (142) can b e represen ted in the R iemann f orm r i τ = ˜ v i ( r ) r i x , i = 1 , 2 , ..., N ; (143) where the dep endence ˜ v i ( r ) of the c haracteristic v elo cities on the Riemann inv ariants is determined b y the same form ulae (127) with the only diﬀerence that, one no w replaces ξ j with ˜ ξ j , i.e. w e ha v e ˜ v i = 1 u i N X m =1 ˜ ξ m β im . (144) Indeed, represen tation (14 4) is a straigh tforw ar d consequence of (127) since the rot a tion co- eﬃcien ts β ij and Lam ´ e co eﬃcien ts ¯ H k do not dep end on the parameters ξ m (see (115), (116), the ﬁrst formula in (1 27), and normalisation (99)). It not diﬃcult to see that comm utativity relationships (see (47)) ∂ i ˜ v j ˜ v i − ˜ v j = ∂ i v j v i − v j , i, j = 1 , 2 , . . . , N , i 6 = j , (14 5) are satisﬁed iden tically . Th us, w e hav e prov ed the follow ing Lemma 8.3 : Line arly de gener ate sem i-Hamiltonian ﬂows ( 1 41), (1 4 2) and (36) , (38) c ommute for any N . In conclusion w e note that, although we ha v e pro v ed inte grabilit y o f the ‘cold-gas’ h y- dro dynamic reductions (36), (38) for a n arbitra r y c hoice of the functions S ( η ) and G ( η , µ ) in the original kinetic equation (1), one can expect that in tegrability of the full equation (1) w ould require some additiona l restrictions imp osed on the integral k ernel G ( η , µ ) (o ther than just symmetry). 9 Outlo ok and P ersp ectiv es Kinetic equation (1) ﬁrst arose as a con tinuum (thermo dynamic) limit of a semi-Hamiltonian h ydro dynamic type system (the KdV- Whitham syste m). This equation seems to b elong to an en tirely new class of integrable systems, whic h w e at presen t are unable to equip with the standard attributes suc h as a Lax pa ir , commuting ﬂows, Hamiltonian structures etc. This pap er make s the ﬁrst step tow ards the understanding of the in tegrable structure of equation (1) b y studying in detail the simplest class of its hy dro dynamic reductions a nd iden tifying them as the Egoro v, semi-Hamiltonian linearly degenerate h ydro dynamic ty p e systems. The a v ailability of an inﬁnite set of the aforementioned h ydro dynamic reductions is a strong evidence that the full equation ( 1) could constitute an integrable system in the sense y et to b e explored. While the studied ‘cold-ga s’ reductions turn o ut to b e in tegra ble fo r an arbitrary symmetric ‘interaction k ernel’ G ( η , µ ), in tegrabilit y of the full equation (1) will clearly require 34 some additional restrictions to b e imp osed o n this k ernel. Recen t results [12, 13], [18, 19] on the inte grabilit y of 2+1 hy dro dynamic t yp e systems and hydrodynamic c hains, whic h are close ‘relativ es’ of kinetic equations, suggest that these restrictions should b e determined by the condition of the existence , for an arbitr a ry N , of N -comp onen t hy dro dynamic reductions parameterised b y N arbitrary functions of a single v ariable. The most natural wa y to a t t ac k this problem is to study the asso ciated hydr o dynamic chain , i.e. an inﬁnite set of the m oment e quations (see e.g. [20]) for kinetic equation (1). Ho w ev er, due to the structure of the nonlo cal term in ( 1) the construction of this c hain is far from b eing a straigh tf orw ard task. The study o f t he moment equations for (1) is a lso imp ortan t in t he original con text of the description of macroscopic dynamics of soliton gases [4 6 ], [7 ]. Indeed , the kinetic description of a soliton gas reﬂects the part icle-like nature of solitons. A t the same time, one should remem b er that solitons represen t lo calized wa v es so the kinetic description of a soliton gas should b e complemen ted by the expressions f or the av eraged c haracteristics of the underlying ‘microscopic’ oscillatory w av e ﬁeld in terms of the distribution function f ( η , x, t ). Sa y , for the KdV equation (3 ) the expressions for the tw o ﬁrst momen ts of the w a ve ﬁeld ha v e t he form (see [6]) φ ( x, t ) = 4 Z ∞ 0 η f ( η , x, t ) dη , φ 2 ( x, t ) = 16 3 Z ∞ 0 η 3 f ( η , x, t ) dη (146) and are iden tical to those arising in the Lax-Lev ermore-V enakides t heory [31], [43 ], [4 5] with the crucial diﬀerence that t he dynamics of the distribution f unction f ( η , x, t ) is now go verne d b y kinetic equation (1), (2) rather than the N -phase av eraged Whitham equations (1 0) so (146) are ensemble aver ages . This pap er was concerned mostly with the structur e of the kinetic equation (1). A t the same time, b ehaviour of its solutions and the asso ciated evolution o f the dynamical (momen ts, amplitudes etc.) and probabilistic (proba bilit y densit y , correlat io n function etc.) c haracteristics of the underlying ra pidly oscillating w av e ﬁeld could b e of considerable in terest for a pplications. In this regard, we men tion an in teresting consideration fo llowing from our presen t study . In the original construction [6] describ ed in Section 3 the kinetic equation for t he KdV soliton gas w as obtained a s the thermo dynamic limit of the N -phase av eraged KdV-Whitham equations (10). These Whitham equations are genuinely nonline ar for an y N ∈ N [32], i.e. fo r a reasonably general class of initial conditions the mo dula t ion dynamics sp eciﬁed on a R iemann surface o f gen us N implies h ydro dynamic breaking at some t < ∞ accompanied b y the gro wth of the gen us N (see e.g. [5], [9], [22]). At the same time, t he ‘cold- gas’ h ydro dynamic reductions of the kinetic equations studied here are linearly degenerate, i.e. no breaking is exp ected and the n umber of gas comp onen ts do es not change during the ev olution (a simple example of suc h non-breaking ev olution for a t w o-comp onen t soliton gas w as considered in [7]). Of course, there is no con tr a diction betw een these tw o con trasting t yp es of b eha viour as the kinetic equation (28), (26) represen ts a singular limit as N → ∞ of the KdV-Whitham equations while their gen uine no nlinearit y prop erty is established only for ﬁnite N . Construction of phy sical solutions to linearly degenerate m ulti- comp onen t h ydro dynamic ty p e system (4) , (5) and study o f the asso ciated w av e ﬁeld dynamics of soliton gases in v arious in tegrable systems represen ts a separate in teresting mathematical problem, whic h could ﬁnd a pplications in the description of propagat io n and interaction of quasi-mono c hro matic soliton b eams in disp ersiv e dissipationless media. Another c hallenging problem is deriv ation of the 2+1 dimensional kinetic equation for 35 the soliton gas in the fra mew ork of the Kadom tsev - P etviash vili (KP-2) equation. This problem w ould require computing the thermo dynamic limit of the KP-Whitham equations asso ciated with g eneral algebraic (not necessarily h yp erelliptic) Riemann surfaces [25 ], [26]. Finally , w e would lik e to mention one more p erspectiv e arising fro m our study . T o our b est know ledge, no nlo cal kinetic equation (1) is the ﬁrst a v ailable example of a con tinuum limit of a semi-Hamiltonian h ydro dynamic t yp e system. The k ey p oin t of its deriv ation is that it is not suﬃcien t to simply tend the num b er of R iemann in v arian ts t o inﬁnit y but it is imp o rtan t to prescrib e a sp ecial scaling con tr o lling the distances b etw een neigh b oring in v ariants (in the case of a ve raged ﬁnite-gap dynamics, the widths of sp ectral bands and gaps – see (19)). W e b eliev e that a similar approac h could b e applied to a large class of semi-Hamiltonian hy dro dynamic type systems (not necessarily ar ising as the result o f the Whitham a v eraging). Of course, t he corresp onding thermo dynamic scaling (an analogue of distribution (19)) could b e diﬀeren t. Ac kno wledgments W e are grateful to V.E. Zakharov for his in terest in this work and a num b er o f enlight- ening comments . W e thank Y u. F edorov, E. F erap onto v, O. 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Kinetic equation for a soliton gas and its hydrodynamic reductions

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