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Internat. Math.Nachrichten Nr.233(2016), 1–24 The Camassa–Holm Equation and The String Density Problem Jonathan Eckhardt, Aleksey Kostenko and Gerald Teschl 7 1 Universita¨tWien 0 2 n a J 3 TheCamassa–Holm(CH)equation 1 ] u −u +2k u =2u u −3uu +uu , (1) h t xxt x x xx x xxx p - is an extensively studied nonlinear equation. It first appeared as an abstract bi- h Hamiltonian partial differential equation in an article of Fuchssteiner and Fokas [39] t a but did not receive much attention until Camassa and Holm [15] (see also [16]) de- m rived it as a nonlinear wave equation that models unidirectional wave propagation on [ shallow water and discovered its rich mathematical structure. In this context u(x,t) 1 representsthefluidvelocityinthexdirectionattimet andtherealconstantk isrelated v tothecriticalshallowwaterwavespeed. Regardingthehydrodynamicalrelevance,we 8 9 refer to the more recent articles [22, 52, 53]. Apart from this, the CH equation was 5 also foundin[25]as amodelfornonlinearwaves incylindricalhyperelasticrods. 3 0 Since its discovery, the literature on the CH equation has been growing exponentially . 1 (at the moment, thepaper by Camassa and Holm [15] has more than 2400 citations in 0 GoogleScholar, 1698 in Scopus and 1016 in MathSciNet) and it is impossibleto give 7 a comprehensive overview here. In fact, our main focus in this review paper lies on 1 : understandingtheCH equation and itsso-called conservativesolutionsviatheinverse v i scatteringtransform(IST)approach. ItwasnoticedbyCamassaandHolmthattheCH X equation is completely integrable in the sense that it enjoys a Lax pair structure and r a hencemaybetreatedwiththehelpoftheISTapproachinprincipal. Thecorresponding isospectralproblemisa Sturm–Liouvilleproblemoftheform 1 −y′′+ y=zyw (·,t), (2) 4 where w = u−u +k is known as the momentum. Using a simple change of vari- xx ables (and dropping the time dependence for a moment), this spectral problem can be ISSN 0020-7926 (cid:13)c 2016O¨sterr.Math.Gesellschaft transformed intothespectral problemforan inhomogeneousstring − f′′ =z f w . (3) Inthe1950’s,M.G.Kreindevelopeddirectandinversespectraltheoryforsuchstrings, e assuming that w is a nonnegative,locally finite measure on [0,L) for some L∈(0,¥ ]. Ofcourse,theoriginalmotivationofKreinwasfarfromapplicationstononlinearequa- tions (see, for instance, [42, Appendix 3]). Unlike in the case for strings, applications e to the CH equation make it necessary to deal not only with nonnegativebut also with real-valued (signed)Borel measures, thatis, withindefinitestrings. The structure of the present review is as follows: In the next section we give a brief historicalaccountontheCHequation. Section2thendiscussesmulti-solitonsolutions for (1) in the case when k =0; the so-called multi-peakons. In the next two sections wetouchupontheconceptsofglobalconservativesolutionsandgeneralizedindefinite strings. In Section 5, we overview recent progress in the understanding of the con- servative CH flow as a completely integrable nonlinear flow. In the final section we providean account onthelong-timebehaviorofsolutionsto theCH equation. 1 The Camassa–Holm equation Bi-Hamiltonianstructure Intherestofourpaper,wewillbemostlyconcentratedontheCauchyproblemforthe CH equation on the line, that is, (x,t)∈R×R (note that negativetimes are covered + by the transformation (x,t) → (−x,−t), which leaves (1) invariant). Then the CH equationcan bewritten intheequivalentform 1 1 u +uu +p =0, p(x,t)= e−|x−y| 2k u+u2+ u2 dy, (4) t x x 2ZR 2 x (cid:18) (cid:19) which is reminiscent of the 3-D incompressible Euler equations. The original deriva- tionof(1)wasobtainedbyapproximatingdirectlyintheHamiltonianforEuler’sequa- tionsintheshallowwaterregime(see[15,16])andhencetheCHequationinheritsthe Hamiltonianstructure d H [m] m =− 2k¶ +m¶ +¶ m 1 , m=u−u , (5) t ¶ m xx (cid:16) (cid:17) where theHamiltonianis givenby 1 H [m]:= u2+u2dx. (6) 1 2ZR x 2 This, in particular, implies that the H1 Sobolev norm of u is a conserved quantity. In fact, the CH equation is bi-Hamiltonian[39, 15]. Namely, it can also bewritten in the followingalternativeform: d H [m] 1 m =−(¶ −¶ 3) 2 , H [m]:= u3+uu2+2k u2dx. (7) t ¶ m 2 2ZR x The latter leads to an infinite numberof conserved quantities H [m], n∈Z, which are n defined recursivelyby d H [m] d H [m] (¶ −¶ 3) n+1 = 2k¶ +m¶ +¶ m n , n∈Z. ¶ m ¶ m (cid:16) (cid:17) H Schemes forthecomputationof can befoundin[36, 51,64, 76]. n Geometricformulation Equation(1)withk =0canbeinterpretedasthegeodesicflowonthegroupofdiffeo- morphismsofthelinewiththeRiemannianstructureinducedbytheH1 right-invariant metric. ThisresemblesthefactthattheEulerequationisanexpressionofthegeodesic flow in the group of incompressible diffeomorphisms (see [1, 28]). Equation (1) rep- resents the equations of motion in Eulerian coordinates, while the geometric interpre- tation corresponds to rewriting (1) in Lagrangian coordinates. This connection turned outto beveryuseful inthequalitativeanalysisofsolutionsoftheCH equation. Fork 6=0,theCHequationrepresentstheequationforgeodesicsontheBott–Virasoro group [72]. Let us also mention that the analogous correspondence for the Korteweg– deVries (KdV)equation u +u +6uu =0 (8) t xxx x wasestablishedin[75](seealso[57]). Infact,thereisaverycloseconnectionbetween the KdV and the CH equations. First of all, the Virasoro group (a one-dimensional extensionofthegroupofsmoothtransformationsofthecircle)servesasthesymmetry group fortheseequations[57]. On theotherhand, thereis aLiouvillecorrespondence between theCH and theKdV hierarchies [70], [63]. Dynamics ofsolutions In this subsection we assume for simplicity that k = 0 (note that the transformation u(x,t)→u(x−k t,t)+k reduces(1)tothiscase,butitdoesnotpreservespatialasymp- totics). TheSobolevspaces are thenatural phasespaces fortheCH equationsincetheHamil- tonian H given by (6) is exactly the H1 norm of the solution at time t. One of the 1 3 crucial differences between the CH and the KdV equations is the fact that the CH equation possesses both global solutions as well as solutions developing singularities infinitetime. Moreover,theblow-uphappensinawaywhichresembleswavebreaking to someextent. First ofall, let us mentionthattheCH equation(1)is locallywell-posed inHs forany s > 3/2 (the first result was obtained by Escher and Constantin [18] and for further improvements see [66], [77]). The problem with global well-posedness stems from thefact thattheSobolevnormsHs arenotcontrolledbytheconservationlawsifs>1 and hence one cannot extend local solutions automatically to the whole line and in fact, the blow-up can occur in finite time. The singularity formation was first noticed by Camassa and Holm [15]. Moreover, it was shown in [18] that for any even initial datau ∈H3(R)with u′(0)<0 thecorrespondingsolutiondoes notexistglobally. In 0 0 particular,thisresultshowsthatinitialdatawitharbitrarysmallHs normmayblowup in finite time. On the other hand, it was shown in [19] that the encountered blow-up looks as follows: thesolutionremains bounded but its slopebecomes vertical in finite time, which resembles a breaking wave. Let us also mention that in certain situations it is possible to prove global existence. Namely, it was noticed in [18] that solutions are globalforu ∈Hs withs>3/2ifthecorrespondingmomentumw =u −u′′ isa 0 0 0 0 positivefinitemeasure. In contrast to classical solutions, weak solutions to the CH equation (4) are global although they are not necessarily unique anymore. In [82], Xin and Zhang proved the existenceofglobalweaksolutionsforanyu ∈H1(R). Itturnsoutthatthepositivityof 0 thecorrespondingmomentumplaysacrucialrolefortheuniquenessofweaksolutions. Namely, it was proved by Constantin and Molinet [24] that for u ∈H1(R) such that 0 the corresponding momentumw is a positivefinite measure on R, a weak solutionto 0 (4)existsandisuniqueforalltimes. Moreover,inthiscaseuiscontinuouswithvalues inH1(R)andthequantitiesH ,H andH areconservedalongthetrajectories. Infact, 0 1 2 thepositivityofw providesacriterionfortheuniquenessofweaksolutions: McKean 0 [69], [71] proved that a weak solution exists and is unique if u ∈C¥ (R)∩H1(R) 0 is such that the set S := {x ∈ R : w (x) < 0} lies wholly to the right of the set − 0 S := {x ∈ R : w (x) > 0}. Hence, either the forward or backward CH flow blows + 0 up in finite timeif bothsets S and S havea nonzero Lebesgue measure. Forfurther + − detailsand references werefer thereader toasurveybyMolinet[73]. The Lax pair The presence of infinitely many integrals of motion established in [39] indicates that theCHequationmightbecompletelyintegrable. ThelatterwasconfirmedbyCamassa and Holm[15]byfindingthecorrespondingLaxpair. Indeed,theCH equationcan be 4 formulatedas thecompatibilityconditionbetween 1 −y + y=zyw , w =u−u +k (9) xx xx 4 and 1 1 y = u y− +u y , (10) t x x 2 2z (cid:18) (cid:19) that is, y =y holds if and only if u satisfies (1). Let us also mention that the CH xxt txx equationgivesa counterexampleto thePainleve´ integrabilitytest[41]. The spectral problem (9) is a Sturm–Liouville problem. It very much resembles the 1-D Schro¨dinger spectral problem, −y′′+qy = zy, which serves as the isospectral problemfortheKdVequation(8). However,thespectralparameterzisinthe”wrong” place. Ofcourse,underadditionalsmoothnessandpositivityassumptions(forexample k >0,w ∈C2(R)and w >0 on R),theLiouvilletransformation ¥ w (s) f(x)=w (x)1/4y(x), x(x)=x− −1ds, (11) Z k x r converts(9)intothe1-D Schro¨dingerform k w (x,t) w (x,t)−k 5k w (x,t)2 − f′′+Q(·,t)f =z f, Q(x,t)= xx − − x . (12) 4 w (x,t)2 4w (x,t) 16 w (x,t)3 Hence,asfortheKdVflow,inthiscaseonecanapplythewell-developedinversescat- teringtheoryfor1-DSchro¨dingerequationsinordertointegratetheCHflowusingthe ISTapproach(see,e.g.,[3],[17],[20],[21]). Thesametrickcanbeusedtoinvestigate the CH equation on the circle [23]. However, the direct and inverse spectral theory for(9)withouttheseadditional(positivityandsmoothness)assumptionshasnotbeing developedandwepostponeitsfurtherdiscussionto Section3. Solitons Oneofthemostinterestingfeatures oftheCH equationisthepresenceofsolitonsand the simplicity of their interaction when k =0 (see Section 2). For k >0, solitons are smoothbutthereisnoclosedformevenforaone-solitonsolutionoftheCH equation. However, the Liouville correspondence (11)–(12) allows to obtain a detailed descrip- tionofmulti-solitonsolutions. Namely,asfortheKdVequation,multi-solitonprofiles are reflectionless potentials for (12) and one can employ this fact and the Liouville transform in order to get various representations of multi-soliton solutions (see, e.g., [54], [67], [68], [79]). Unfortunately, it is a difficult task to invert (11)–(12) and this fact(atleastpartially)explainstheabsenceofaclosedformformulti-solitonsolutions. Finally, notice that the one-soliton solution is a traveling wave and a complete de- scription of all weak traveling wave solutions to the CH equation (peakons, cuspons, stumpons,etc.) isgivenin[65](seealso [62]). 5 2 Multi-peakons and the moment problem Solitondynamics In thedispersionlesscasek =0, thetravelingwavesolutioncalled peakonis givenby u(x,t)= pe−|x−pt+c|, where pandcarerealparameters. Ithasapeakatx (t)= pt−canditsheightisequal 0 to its speed (a positivepeak travels to theright and a negativepeak travels to the left). Since it obviously has a discontinuous first derivative at x , it has to be interpreted as 0 a suitable weak solution of (4) (see [4, 18, 24, 44]). It was noticed by Camassa and Holm [15] that the multi-soliton solution to the CH equation with k = 0 is simply a linearcombinationofpeakons N u(x,t)= (cid:229) p (t)e−|x−qn(t)|, (13) n n=1 where thecoefficients p andq satisfythesystemofordinarydifferentialequations n n N N q′ = (cid:229) p e−|qn−qk|, p′ = (cid:229) p p sgn(q −q )e−|qn−qk|. (14) n k n n k n k k=1 k=1 ThissystemisHamiltonian,that is, dq ¶ H(p,q) dp ¶ H(p,q) n n = , =− , (15) dt ¶ p dt ¶ q n n withtheHamiltoniangivenby H(p,q)= 1 (cid:229)N p p e−|qn−qk| = 1kuk2 . (16) 2 n k 4 H1(R) n,k=1 Before we proceed further, let us mention that the Hamiltonian system (15)–(16) is a special caseoftheCalogero–Françoisesystemsintroducedin [13,14] H(p,q)= 1 (cid:229)N p p G(q −q ), G(x)=a+b cos(n x)+b sin(n |x|), (17) n k n k + − 2 n,k=1 where a, b , b and n are arbitrary constants. Clearly, a=0, b =1 and b =n =i + − + − gives (16). Let us also mention that a = 0, b = coth(1/2), b = 1 and n = i gives + − rise to periodic multi-peakons (see [5, 23]); the limiting case G(x)=a+b|x|+cx2 is associated totheHunter–Saxtonequation(see[48, 49]). 6 Theright-handsidein (14)isnotLipschitzifq −q is closeto zero and henceinthis n k case,onecannotgetexistenceanduniquenessofsolutionsof(14)byusingthestandard arguments. However, if we know in advance that all the positions stay distinct, then theright-handsidein(14)becomesLipschitzandthusthePicardtheoremapplies. Let us also mention that the Calogero–Françoise flows are completely integrable in the Liouvillesense, that is, there exist N integrals of motion in involution [14]. However, the classical Arnold–Liouville theorem is not applicable since the Hamiltonians (17) are notcontinuouslydifferentiablewheneverb 6=0. − One of the most prominent features of multi-peakons is the fact that almost all qual- itative properties of solutions to the CH equation can be seen just by considering multi-peakons,i.e.,finitedimensionalreductions(15)–(16)oftheinfinitedimensional Hamiltoniansystem(5)–(6). Forexample,thebehaviorofmulti-peakonsolutionscru- ciallydependsonwhetheralltheheights p ofthesinglepeaksareofthesamesignor n not. Noticethatthecorrespondingmomentumis simplygivenby N w (·,t)=2 (cid:229) p (t)d . (18) n qn(t) n=1 So, w isapositivemeasurepreciselywhen alltheheightsarepositiveandinthiscase, all the positions q of the peaks stay distinct, move to the right and the system (14) n allows a unique global solution [18, 24, 44]. Otherwise, some of the positions q of n the peaks will collide eventually, which causes the corresponding heights p to blow n up in finite time [15]. All this happens in such a way that the solution u in (13) stays uniformly bounded but its derivative develops a singularity at the points where two peaks collide. Let usdemonstratethisbyconsideringtheinteractionoftwopeakons. Example1 (Two peakons) Consider the case N =2 and assume that q (0)<q (0). 1 2 Introducing the new variables Q = q −q as well as P = p −p and noting that 2 1 2 1 Q>0 ina vicinityofzero,we can rewritethesystemas P2−P2 Q′ =P(1−e−Q), P′ = 0 e−Q, 2 where P ≡ p (t)+p (t)isa constantofmotion. Noticealsothat 0 1 2 P2+P2 P2−P2 P2−P2 2H(p,q)= 0 + 0 e−Q =P2− 0 (1−e−Q)≡2H2. (19) 2 2 0 2 0 Therefore, weget P2+P2 P2−2H2 P′ =2H2− 0 , Q′ =2P 0 0. (20) 0 2 P2−P2 0 7 Onethen easilyobtains (P(0)+h )eh0t+(P(0)−h ) P(t)=h 0 0 , h = 4H2−P2, (21) 0(P(0)+h0)eh0t−(P(0)−h0) 0 0 0 q and eh0t−P0+h0P(0)−h0 eh0t−P0−h0P(0)−h0 Q(t)=Q(0)+log P0−h0P(0)+h0 P0+h0P(0)+h0 . (22) (cid:12)(cid:12)(cid:16)1−P0+h0P(0)−h0 (cid:17)1(cid:16)−P0−h0P(0)−h0 eh0(cid:17)t (cid:12)(cid:12) (cid:12) P0−h0P(0)+h0 P0+h0P(0)+h0 (cid:12) (cid:12) (cid:16) (cid:17)(cid:16) (cid:17) (cid:12) Clearly,P is discontinuouso(cid:12)nlyif (P(0)−h )(P(0)+h )>0 andinthisc(cid:12)ase (cid:12) 0 0 (cid:12) 1 P(0)−h P(t)→−¥ as t →t× := log 0 . (23) h P(0)+h 0 0 (cid:18) (cid:19) Noticethat (P(0)−h )(P(0)+h )=P(0)2+P2−4H2 =(P(0)2−P2)e−Q 0 0 0 0 0 ispositiveonlyif p (0)p (0)<0. Hencetherearetwodistinctcases. If p (0)p (0)> 1 2 1 2 0, i.e., both peaks are of the same sign, then the solution is global and peaks never collide. Ifwehaveapeakon-antipeakoninteraction,thentheblowuphappensatt=t× givenby(23)andinthiscaseP(t)=p (t)−p (t)tendsto−¥ andQ(t)=q (t)−q (t) 2 1 2 1 tendsto zeroast approachest×. Complete integrabilityand the Stieltjes momentproblem It was noticed by Beals, Sattinger and Szmigielski [4] that similar to the finite Toda latticeontheline[74],themulti-peakonflowcanbesolvedbyusingthesolutionofthe Stieltjes moment problem [80]. More precisely, consider the corresponding spectral problem(9)withthemomentw givenby (18)(we omitthetimedependence): −y′′+1y=zyw , w =2 (cid:229)N p d . (24) 4 n qn n=1 Withoutlossofgenerality,wecan assumethat p 6=0forall n∈{1,2,...,N}and n −¥ <q <q <...<q <¥ . 1 2 N Somefunctionyis asolutionofthedifferentialequation(24)ifitsatisfies 1 −y′′+ y=0 (25) 4 8 away fromthepoints{q ,...,q }, togetherwiththeinterfaceconditions 1 N y(q +) 1 0 y(q −) n = n , n∈{1,...,N}. (26) y′(q +) −2zp 1 y′(q −) n n n (cid:18) (cid:19) (cid:18) (cid:19)(cid:18) (cid:19) The set of all values z ∈ C for which there is a nontrivial solution of the differential equation (24) that lies in H1(R) is referred to as the spectrum s of the spectral problem (24). Note that in this case, the solution in H1(R) of this differential equation is unique up to scalar multiples. Since the measure w has a compact support, for every z∈C one has spatially decaying solutions f ±(z,·) of (24) with f ±(z,x)=e∓2x for all x near ±¥ . In particular, note that f (·,x) and f ′ (·,x) are real polynomialsfor each ± ± fixed x∈R. TheWronskideterminantofthesesolutions W(z)=f (z,x)f ′ (z,x)−f ′ (z,x)f (z,x), z∈C (27) + − + − is independent of x∈R and vanishes at some point l ∈C if and only if the solutions f (l ,·)andf (l ,·)arelinearlydependent. Asaconsequence,oneseesthatthespec- − + trum s is precisely the set of zeros of the polynomialW and, moreover, the spectrum s of (24) consistsof exactly N real and simpleeigenvalues (see, e.g., [4]). Associated witheach eigenvaluel ∈s isthequantity 1 1 := |f ′ (l ,x)|2dx+ |f (l ,x)|2dx>0, (28) g l ZR − 4ZR − which isreferred toas the(modified)normingconstant(associatedwithl ). Thecentralroleintheinversespectraltheoryfor(24)isplayedbytheWeyl–Titchmarsh m-function,whichisdefined by 1W(f (z,x),e−x/2) W(f (z,x),e−x/2) M(z)= lim − + = lim − + , z∈C\R. (29) x→−¥ z W(f (z,x),ex/2) x→−¥ zW(z) + M isa Herglotz–Nevanlinnafunctionand itadmitsthepartialfraction expansion: g M(z)= (cid:229) l , z∈C\R. (30) l −z l ∈s Ontheotherhand,takingintoaccountthefactthatf (z,·)solvesthedifferenceequa- + tion (24), it is also possible to write down a finite continued fraction expansion for M in termsofw : 1 zM(z)−1= , z∈C\R, (31) 1 −l + 0 1 m z+ 1 .. 1 . + 1 −l + N−1 1 m z− N l N 9 where q 1 q q m =8p cosh2 n , l = tanh n+1 −tanh n . (32) n n n 2 2 2 2 Hereby, wesetq =−¥ and(cid:16)q (cid:17) =¥ fors(cid:16)implic(cid:16)ityofn(cid:17)otation.(cid:16) (cid:17)(cid:17) 0 N+1 In the case when all p are positive, the classical result of Stieltjes [80] (see also [55, n §13]) recovers the coefficients (32) in terms of the spectrum s and the corresponding norming constants {g l }l ∈s . For that purpose, one needs to consider the Laurent expansionofM(z)at infinity(takingintoaccount thepartialfraction expansion(30)): M(z)−1 =− (cid:229)¥ sk , |z|→¥ ; s = 1+(cid:229) l ∈s g l , k=0, (33) z k=0zk+1 k ( (cid:229) l ∈s l kg l , k∈N. IntroducingtheHankeldeterminants D = s k , D = s k , (34) 0,k i+j i,j=0 1,k i+j+1 i,j=0 (cid:12) (cid:12) (cid:12) (cid:12) theformulasofStieltjesread a(cid:12)s fol(cid:12)lows (cid:12) (cid:12) D 2 D 2 0,n 1,n−1 m = , l = , n∈{1,...,N}. (35) n D D n−1 D D 1,n−1 1,n 0,n−1 0,n Since sk, k ∈ N are the moments of a nonnegative measure r = d 0+(cid:229) l ∈s g l d l , the Hankel determinants D are positive for all n ∈ {0,...,N} (see [40], [80]). On the 0,n otherhand,theHankeldeterminantsD arepositiveforalln∈{0,...,N}ifandonly 1,n if the support of r is contained in [0,¥ ), i.e., the spectrum s consists only of positive eigenvalues,whichisfurtherequivalenttothepositivityofthemeasurew . Introducing thetimedependence 1 g˙l = 2l g l , l ∈s , one then can integrate the multi-peakon flow by using the Stieltjes solution of the moment problem (32)–(35) if all p are positive. Moreover, the formulas (31)–(35) n remaintrueuntilthedenominatorsin(35)becomezero,thatis,onecanexploitStieltjes solution of the moment to integrate the multi-peakon flow in the general case (see [4] for further details). One of the important observations made in [4] is that one of the Hankel determinantsD vanishesexactly whentwo adjacentpeakons collide. 1,n 3 Generalized indefinite strings Thedirectandinversespectraltheoryforthespectralproblem(9)isofvitalimportance for investigating the CH equation by using the IST approach. Assume for a moment 10