INTEGRABLE EVOLUTION EQUATIONS ON SPACES OF TENSOR DENSITIES AND THEIR PEAKON SOLUTIONS*

Transcription

1 INTEGRABLE EVOLUTION EQUATIONS ON SPACES OF TENSOR DENSITIES AND THEIR PEAKON SOLUTIONS* JONATAN LENELLS, GERARD MISIO LEK, AND FERIDE TIĞLAY Abstract. We study a family of equations defined on the space of tensor densities of weight λ on the circle and introduce two integrable PDE. One of the equations turns out to be closely related to the inviscid Burgers equation while the other has not been identified in any form before. We present their Lax pair formulations and describe their bihamiltonian structures. We prove local wellposedness of the corresponding Cauchy problem and include results on blow-up as well as global existence of solutions. Moreover, we construct peakon and multi-peakon solutions for all λ, 1, and shock-peakons for λ = 3. We argue that there is a natural geometric framework for these equations that includes other well-known integrable equations and which is based on V. Arnold s approach to Euler equations on Lie groups. Contents 1. Introduction. A family of equations on the space of λ densities 3 3. Lax pairs 5 4. Hamiltonian structures and conserved quantities Bihamiltonian structure of µdp Hamiltonian formulation of µ-equations Orbits in F λ 8 5. The Cauchy problem for the µdp equation Local wellposedess in Sobolev spaces Blow-up of smooth solutions Global solutions Local wellposedness for any λ Peakons Traveling waves Periodic one-peakons Green s function Multi-peakons Poisson structure Multi-peakons for µch Shock-peakons for µdp The µburgers equation 7.1. Burgers equation and the L -geometry of Diff s Diff s /S Lifespan of solutions, Hamiltonian structure and conserved quantities of the µburgers equation 1 8. Multidimensional µch and µdp equations 3 Appendix A. Proof of Theorem 6. 4 Appendix B. Proof of Theorem * TO APPEAR IN COMM. MATH. PHYS. 1

2 JONATAN LENELLS, GERARD MISIO LEK, AND FERIDE TIĞLAY Appendix C. Conservation of H for µdp 7 References 8 1. Introduction Integrability of an infinite-dimensional dynamical system typically manifests itself in several different ways such as the existence of a Lax pair formulation or a bihamiltonian structure, the presence of an infinite family of conserved quantities or at least the ability to write down explicitly some of its solutions. In this paper we introduce and study two nonlinear partial differential equations and show that they possess all the hallmarks of integrability mentioned above. The first of these equations we shall refer to as the µburgers µb equation 1 µb u txx 3u x u xx uu xxx =, and the second as the µdp equation µdp µu t u txx + 3µuu x 3u x u xx uu xxx =, where µu = u dx. Here ut, x is a spatially periodic real-valued function of a time variable t and a space variable x S 1 [, 1. Both of these equations belong to a larger family that also includes the Camassa-Holm equation [CH] see also [FF] CH u t u txx + 3uu x u x u xx uu xxx =, the Hunter-Saxton equation [HuS] HS u txx + u x u xx + uu xxx =, the Degasperis-Procesi equation [DP] DP u t u txx + 4uu x 3u x u xx uu xxx =, as well as the µ-equation which was derived recently in [KLM] we will refer to it as the µch equation in this paper µch u txx µuu x + u x u xx + uu xxx =, all of which are known to be integrable. One of the distinguishing features of the CH and DP equations which makes them attractive among integrable equations is the existence of so-called peakon solutions. In fact, CH and DP are the cases λ = and λ = 3, respectively, of the following family of equations 1.1 u t u txx + λ + 1uu x = λu x u xx + uu xxx, λ Z, with each equation in the family admitting peakons see [DHH] although only λ = and λ = 3 are believed to be integrable see [DP]. One of our results will show that each equation in the corresponding µ-version of the family 1.1 given by 1. µu t u txx + λµuu x = λu x u xx + uu xxx, λ Z, also admits peakon solutions. The choices λ = and λ = 3 yield the µch and µdp equations, respectively, and as with 1.1, we expect that these are the only integrable 1 This equation is mentioned in Remark 3.9 of [HoS] and Remark 3. of [Lu] as the high-frequency limit of the Degasperis-Procesi equation, see DP below. Our terminology will be explained in Section 7. In [KLM] the authors refer to µch as the µhs equation.

3 INTEGRABLE EVOLUTION EQUATIONS AND PEAKONS 3 CH m t = um x u x m µch DP m t = um x 3u x m µdp HS µb Figure 1. The λ = and λ = 3 families of equations. members of the family 1.. Moreover, we will show that the µdp equation admits shock-peakon solutions of a form similar to those known for DP, see [Lu]. In Section, we present a natural setting in which all the equations above can be formally described as evolution equations on the space of tensor densities of different weight λ over the Lie algebra of smooth vector fields on the circle. In Section 3 we present Lax pairs for µdp and µb, establishing their integrability. In Section 4 we describe the Hamiltonian structure of the equations in 1.. In particular, we consider the bihamiltonian structure of the µdp equation together with the associated infinite hierarchy of conservation laws. In Section 5 we study the periodic Cauchy problem of µdp; we prove local wellposedness in Sobolev spaces and show that while classical solutions of µdp break down for certain intial data, the equation admits global solutions for other data. In Section 6 we construct multi-peakon solutions of 1. as well as shock-peakons of µdp. In Section 7 we discuss the µb equation and its properties and present a geometric construction to explain its close relation to the inviscid Burgers equation. Finally, in Section 8 we consider generalizations of the µch and µdp equations to a multidimensional setting. Left open is the question of physical significance, if any, of the two equations µdp and µb. It is possible that they may play a role in the mathematical theory of water waves as CH, see [CH] or find applications in the study of more complex equations as e.g. HS, see [ST]. We do not pursue these issues here. Our approach draws heavily on [KM] and [KLM] and the present work is in some sense a continuation of those two papers.. A family of equations on the space of λ densities Perhaps the simplest way to introduce the equations that are the main object of our investigation is by analogy with the known cases. We shall therefore first briefly review Arnold s approach to the Euler equations on Lie groups and then describe a more general set-up intended to capture the µdp and the µb equations. In a pioneering paper Arnold [A] presented a general framework within which it is possible to employ geometric and Lie theoretic techniques to study a variety of equations ODE as well as PDE of interest in mathematical physics see also more recent expositions in [KM] or [KW]. Arnold s principal examples were the equations of motion of a rigid body in R 3 and the equations of ideal hydrodynamics. The formal set-up is the following. Consider a possibly infinite-dimensional Lie group G the configuration space of a physical system with Lie algebra g. Choose an inner product, on g essentially, the kinetic energy of the system and, using right- or left- translations, endow G with the associated right- or left- invariant Riemannian metric. The motions of the system can now be studied either through the geodesic equation defined by the metric on G equivalently, the geodesic flow on its tangent or cotangent bundles or else directly on the Lie algebra g using Hamiltonian

4 4 JONATAN LENELLS, GERARD MISIO LEK, AND FERIDE TIĞLAY reduction. 3 The equation that one obtains by this procedure on g is called the Euler or Euler-Arnold equation. Using the inner product it can be reformulated as an equation on the dual algebra g as follows. Let, be the natural pairing between g and g and let A : g g denote the associated inertia operator 4 determined by the formula Au, v = u, v for any u, v g. The Euler equation on g reads E m t = ad A 1 m m, m = Au g, where ad : g Endg is the coadjoint representation of g given by.1 ad um, v = m, [u, v] for any u, v g and m g. ad u is the infinitesimal version of the coadjoint action Ad : G g g of the group G on the dual algebra g. In our case G will be the group DiffS 1 of orientation-preserving diffeomorphisms of the circle whose Lie algebra g is the space of smooth vector fields T e DiffS 1 = vects 1. The dual vect S 1 is the space of distributions on S 1 but we shall consider only its regular part which can be identified with the space of quadratic differentials F = { mxdx : m C S 1 } with the pairing given by mdx 1, v x = mxvx dx see e.g. Kirillov [K]. The coadjoint representation of vects 1 on the regular part of its dual space is in this case precisely the action of vects 1 on the space of quadratic differentials. By a direct calculationm using.1 we have. ad u x mdx = um x + u x m dx and the Euler equation E on g takes the form.3 m t = ad A 1 m m = um x u x m, m = Au. Equivalently, when rewritten on vects 1, equation.3 becomes.4 Au t + u x Au + uau x = so that with an appropriate choice of the inertia operator A which is equivalent to picking an inner product on T e Diff 1 x for CH,.5 A = µ x for µch, for HS, x one recovers the CH, µch and HS equations 5 as Euler equations on vects 1 ; these results can be found in [M1], [KM] and [KLM]. We now want to extend this formalism to include the DP equation as well as the two equations of principal interest in this paper: µdp and µb. Admittedly, our construction does not have the same beautiful geometric interpretation as that of Arnold s and hence perhaps is not completely satisfactory. 3 We will be making use of both structures in this paper. 4 In infinite-dimensional examples A is often some positive-definite self-adjoint pseudodifferential operator acting on the space of smooth tensor fields on a compact manifold. 5 Note that in the case of the HS equation the inertia operator is degenerate, see [KM], Section 4, for details.

5 INTEGRABLE EVOLUTION EQUATIONS AND PEAKONS 5 The main point is to replace the space of quadratic differentials with the space of all tensor densities on the circle of weight λ. 6 Recall that a tensor density of weight λ respectively λ < on S 1 is a section of the bundle λ T S 1 respectively λ T S 1 and set { } F λ = mxdx λ : mx C S 1. There is a well-defined action of the diffeomorphism group DiffS 1 on each density module F λ given by.6 F λ mdx λ m ξ x ξ λ dx λ F λ, ξ DiffS 1, which naturally generalizes the coadjoint action Ad : DiffS 1 Aut F on the space of quadratic differentials. The infinitesimal generator of the action in.6 is easily calculated,.7 L λ u x mdx λ = um x + λu x m dx λ, and can be thought of as the Lie derivative of tensor densities. It represents the action of vects 1 on F λ which for λ = coincides with the algebra coadjoint action on F that is L u x = ad u x. If we think of.7 as defining a vector field on the space F λ then we can consider the equation for its flow.8 m t = um x λu x m in analogy with.3. The substitution m = Au transforms.8 into an equation on the space of quadratic differentials.9 Au t + λu x Au + uau x = which is the λ-version of.4. Setting λ = 3 and choosing suitable inertia operators A as above we obtain the DP, µdp and µb equations. More precisely, the specific choices of A are in parallel with those made for the λ = family in.5, that is 1 x for DP,.1 A = µ x for µdp, for µb. x More generally, letting A = µ x in.9 for any λ Z, we find the equations in Lax pairs An elegant manifestation of complete integrability of an infinite dimensional dynamical system is the existence of a Lax pair formalism. It is often used as a tool for constructing infinite families of conserved quantities. This formalism has been quite extensively developed in recent years for the CH and DP equations, see e.g. [BSS, CM, DHH, LS]. Our next result describes the Lax pair formulations for the µdp and µburgers equations. Theorem 3.1. The µdp and the µb equations admit the Lax pair formulations { ψ xxx = λmψ, 3.1 ψ t = 1 λ ψ xx uψ x + u x ψ, 6 We refer to [O], [OT] or [GR] for basic facts about the space of tensor densities.

6 6 JONATAN LENELLS, GERARD MISIO LEK, AND FERIDE TIĞLAY where λ C is a spectral parameter, ψt, x is a scalar eigenfunction and m = Au with A = µ x or A = x as defined in.1. Proof. This is a straightforward computation which shows that the condition of compatibility ψ t xxx = ψ xxx t of the linear system 3.1 is equivalent to the equation µdp or µb when m is given by m = µu u xx or m = u xx, respectively. 4. Hamiltonian structures and conserved quantities In this section we describe the bihamiltonian structure of µdp see Section 7 for the bihamiltonian structure of the µburgers equation as well as the Hamiltonian structure for the family of µ-equations 1. for any λ Z. In the last subsection we consider the orbits of DiffS 1 in the space of tensor densities of weight λ and give a geometric interpretation of one of the conserved quantities Bihamiltonian structure of µdp. Recall that DP admits the bihamiltonian formulation [DHH] δh m t = J δm = J δh δm, where H = 9 m dx and H = 1 6 u 3 dx, and the Hamiltonian operators are J = m /3 x m 1/3 x x 3 1m 1/3 x m /3 and J = x 4 x 1 x with m = 1 xu. Similarly, the µdp equation admits the bihamiltonian formulation δh m t = J δm = J δh δm, where now the Hamiltonian functionals H and H are 4.1 H = 9 m dx and H = the operators J and J are given by 3 µu A 1 x u u3 J = m /3 x m 1/3 3 x m 1/3 x m /3 and J = 3 xa = 5 x dx, and m = Au with A = µ x. The fact that J and J form a compatible bihamiltonian pair is a consequence of Theorem in [HW]. It can be verified directly that H and H, as well as H 1 = 1 u dx are conserved in time whenever u is a solution of µdp see Appendix C for details of this calculation in the case of H. Using the standard techniques we can now construct an infinite sequence of conservation laws H 1, H, H 1, H,... see Figure. As in the case of the CH and DP equations the H n s are nonlocal and not easy to write down explicitly for n 3, while they are readily computable recursively in terms of m and its derivatives for negative n. In fact, the first negative flow in the µdp hierarchy is µdp 1 : m t = 79m 17/3 616m 5 x 13mm xx m 3 x + 36m m xm xxx 675m mm xxxx 8m xx mx + 7m 3 3mm xxxxx 5m xx m xxx.

7 INTEGRABLE EVOLUTION EQUATIONS AND PEAKONS 7 µdp 3 δh δm m t = m x u 3mu x J J δ 1 δm u dx = δh 1 δm m t = m x J J δ δm 9 mdx = δh δm m t = J J δ δm m 1/3 dx = δh 1 δm µdp 1 J J δ δm 1 m x dx = δh m 7/3 δm Figure. Recursion scheme for the µdp equation. Remark 4.1. One peculiarity of the µdp and DP equations is that the recursion operator J 1 J maps δh n /δm to δh n+ /δm and not to δh n+1 /δm. Thus, the family 4. is divided into two families consisting of the H n s with even and odd values of n, respectively. One would like to define an operator J 1 which satisfies J = J 1 J 1 J 1 and such that J 1 and J form a bihamiltonian pair with the corresponding Hamiltonians H 1 and H. However, it appears difficult to find an expression for such a J Hamiltonian formulation of µ-equations. Although we expect that µch and µdp are the only equations among the family of µ-equations in 1. which admit a bihamiltonian structure we can nevertheless provide one Hamiltonian structure for any λ 1. Indeed, if we set 4.3 J = 1 λ m x + λm x 3 x and 4.4 H = λ λ 1 λ 1mx + λm x m dx

8 8 JONATAN LENELLS, GERARD MISIO LEK, AND FERIDE TIĞLAY then equation 1. is equivalent to m t = J δh δm. In order to see that J is a Hamiltonian operator we first rewrite it in the form and refer to Theorem 1 of [HW]. J = m λ 1/λ x m 1/λ x 3 m 1/λ x m λ 1/λ 4.3. Orbits in F λ. From Section we know that for any value of λ a solution m of the Cauchy problem 7 for equation 1. with initial data m belongs to the orbit of m in F λ under the DiffS 1 -action defined in.6. It is therefore tempting to exploit this setting to study the family in 1. as in the case of the coadjoint action when λ =. We present here two results in this direction leaving a detailed investigation of the geometry of the corresponding orbits for general λ for a future work. We let L ξ mdx λ = m ξ x ξ λ dx λ denote the action of an element ξ DiffS 1 on mdx λ in F λ. The following two propositions reflect the fact that m 1/λ dx transforms as a one-form. The first proposition shows the conservation of H 1 = m 1/λ dx under the flow of 1.. Proposition 4.. The map m dx λ H 1 [m] = is invariant under the action.6 of DiffS 1 on F λ. m 1/λ dx : F λ R + Proof. This is a straightforward change of variables [ H 1 Lξ m dx λ ] = m ξ x ξ λ 1/λ 1 dx = m 1/λ ξ dξ = H 1 [m], since ξ is a smooth orientation-preserving circle diffeomorphism and m dx λ F λ. In many respects the classification of orbits in F λ resembles that of coadjoint orbits of DiffS 1, cf. [GR]. In order to state the next result we denote by M λ the set of those elements mdx λ in F λ such that mx for all x S 1. Proposition 4.3. The orbit space M λ /DiffS 1 is in bijection with the set R\{}. More precisely, the map 4.5 mdx λ sgnmh 1 λ : M λ R\{}, H 1 = m 1/λ dx, is constant on the DiffS 1 -orbits in M λ F λ and induces a one-to-one correspondence between orbits in M λ and R\{}. Proof. That the map is constant on the orbits follows from Proposition 4.. To see that it is surjective we note that for any nonzero real number a the element adx λ is mapped to a. In order to prove injectivity we observe that the orbit through an arbitrary element mdx λ of M λ contains an element of the form adx λ where a is a constant. Indeed, given mdx λ we define ξ DiffS 1 by ξx = 1 H 1 x m 1/λ dx. 7 See Section 5.4 for results on wellposedness of the equations 1. for any λ.

9 INTEGRABLE EVOLUTION EQUATIONS AND PEAKONS 9 Then L ξ sgnmh λ 1dx λ = sgnmh λ 1 x ξ λ dx λ = mdx λ. Thus, the action of ξ 1 maps m to sgnmh 1 λ dxλ where sgnmh 1 λ constant. R\{} is The stabilizer in DiffS 1 of a constant element adx λ in M λ a R consists exactly of the rigid rotations and is isomorphic to S 1. It follows from Proposition 4.3 that the stabilizer of any point mdx λ M λ is conjugate to S 1 within DiffS 1. In particular, each orbit in M λ is of the form DiffS 1 /S The Cauchy problem for the µdp equation In this section we turn to the periodic Cauchy problem for µdp. Rather than aiming at the strongest possible theorems we present here only basic results that display the interesting behavior of solutions. We start with local existence, uniqueness and persistence theorems for the µdp equation in Sobolev spaces. Next, we describe the breakdown of smooth solutions and prove a global existence result for the class of initial data with non-negative momentum density. At the end of this section we briefly discuss local wellposedness of the whole family 1. of µ-equations. In order to prove these results we will need Sobolev completions of the group of circle diffeomorphisms 8 DiffS 1 and its Lie algebra of smooth vector fields vects 1. We denote by H s = H s S 1 the Sobolev space of periodic functions H s S 1 = { v = n ˆvne πinx : v H s = n Λs vn < where s R and the pseudodifferential operator Λ s = 1 x s/ is defined by Λ s vn = 1 + 4π n s/ˆvn. In what follows we will also make use of another elliptic operator 5.1 Λ µ : H s S 1 H s S 1, Λ µv = µv v xx whose inverse can be easily checked to be x Λ µ vx = x vx dx + 1 x y x 1 vz dzdy + x x y } vy dydx vz dzdydx Local wellposedess in Sobolev spaces. We begin with the µdp equation. Our proof follows with minor changes the approach of [M] to the CH equation based on the original methods in [EM] developed for the Euler equations of hydrodynamics. We include it for the sake of completeness. The two above papers and [KLM] will be our main references for most of the basic facts we use in this section. We can write the Cauchy problem for µdp in the form 5.3 u t + uu x + 3µu x Λ µ u = 5.4 u = u. 8 Although no longer a Lie group, Diff s S 1 retains the structure of a topological group for a sufficiently high Sobolev index; see below.

10 1 JONATAN LENELLS, GERARD MISIO LEK, AND FERIDE TIĞLAY Note that applying Λ µ to both sides of 5.3 we obtain the equation in its original form given in the Introduction. Theorem 5.1 Local wellposedness and persistence. Assume s > 3/. Then for any u H s T there exists a T > and a unique solution u C T, T, H s C 1 T, T, H s 1 of the Cauchy problem which depends continuously on the initial data u. Furthermore, the solution persists as long as ut, C 1 stays bounded. Our strategy will be to reformulate as an initial value problem on the space of circle diffeomorphisms Diff s S 1 of Sobolev class H s. It is well known that whenever s > 3/ this space is a smooth Hilbert manifold and a topological group. We will then show that the reformulated problem can be solved on Diff s S 1 by standard ODE techniques. Let u = ut, x be a solution of µdp with initial data u. Then its associated flow t ξt, x, i.e. the solution of the initial value problem ξt, x = ut, ξt, x, ξ, x = x is at least for a short time a C 1 smooth curve in the space of diffeomorphisms starting from the identity e Diff s S 1. Differentiating both sides of this equation in t and using 5.3 we obtain the following initial value problem ξ = 3µ ξ ξ 1 x Λ µ ξ ξ 1 ξ, x = x, ξ, x = u x ξ =: F ξ, ξ In this section we make repeated use of a technical result which we state here for convenience for a proof refer to [M], Appendix 1. Lemma 5.. For s > 3/ the composition map ξ ω ξ with an H s function ω and the inversion map ξ ξ 1 are continuous from Diff s S 1 to H s S 1 and from Diff s S 1 to itself respectively. Moreover, 5.8 ω ξ H s C1 + ξ s H s ω H s with C depending only on inf x ξ and sup x ξ. It will also be convenient to introduce the notation A ξ = R ξ A R ξ 1 for the conjugation of an operator A on H s S 1 by a diffeomorphism ξ Diff s S 1. The right hand side of 5.6 then becomes 5.9 F ξ, ξ = Λ µ,ξ x,ξhξ, ξ where hξ, ω = 3ω ω ξ 1 dx. Proof of Theorem 5.1. From Lemma 5. we readily see that F maps into H s S 1. We aim to prove that F is Fréchet differentiable in a neighborhood of the point e, in Diff s H s S 1. To this end we compute the directional derivatives ξ F ξ,ω and ω F ξ,ω and show that they are bounded linear operators on H s which depend continuously on ξ and ω. We use the formulas 5.1 ξ Λ µ,ξ v = Λ µ,ξ [ v ξ 1 x, Λ µ]ξ Λ µ,ξ 5.11 ξ x,ξ v = [ v ξ 1 x, x ]ξ 9 Here dot indicates differentiation in t variable.

11 INTEGRABLE EVOLUTION EQUATIONS AND PEAKONS 11 and 5.1 ξ h ξ,ω v = 3 ω ω ξ 1 x v ξ 1 dx along with 5.9 to get ξ F ξ,ω v = 3 { v ξ 1 Λ µ xω ξ 1 ω ξ 1 dx which with the help of the identity + Λ µ x v ξ 1 x ω ξ 1 ω ξ 1 dx } Λ µ x ω ξ 1 ω ξ 1 x v ξ 1 dx ξ 5.13 Λ µ x = 1 + µ can be simplified to ξ F ξ,ω v = 3 { v ω ω ξ 1 dx v ω ξ 1 dx + Λ µ,ξ x,ξ v x,ξ ω ω ξ 1 dx } Λ µ,ξ x,ξω ω ξ 1 x v ξ 1 dx. On the other hand we have ω h ξ,ω v = 3v ω ξ 1 dx + 3ω v ξ 1 dx and similarly ω F ξ,ω v = 3Λ µ,ξ x,ξv ω ξ 1 dx 3Λ 1 µ,ξ x,ξω v ξ 1 dx. In order to show that v ξ F ξ,ω v is a bounded operator on H s it is sufficient to estimate the sum ξ F ξ,ω v L + x ξ F ξ,ω v ξ 1 H s 1. Using Cauchy-Schwarz, Lemma 5. and the formulas above both of these terms can be bounded by a multiple of v H s ω H s which gives the estimate ξ F ξ,ω v H s C v H s ω H s where C depends only on inf x ξ and sup x ξ. A similar argument yields the bound for the other directional derivative ω F ξ,ω v H s C v H s ω H s. It follows that F is Gâteaux differentiable near e, and therefore we only need to establish continuity of the directional derivatives. Continuity in the ω-variable follows from the fact that the dependence of both partials on this variable is polynomial. It thus only remains to show that the norm of the difference ξ F ξ,ω v ξ F e,ω v H s is arbitrarily small whenever ξ is close to the identity e Diff s S 1, uniformly in v and ω H s S 1. Once again, it suffices to estimate the terms 5.14 ξ F ξ,ω v ξ F e,ω v L + x ξ F ξ,ω v ξ F e,ω v H s 1

12 1 JONATAN LENELLS, GERARD MISIO LEK, AND FERIDE TIĞLAY We proceed as above. Using formula 5. and Lemma 5. together with the fact that for any r > we have a continuous embedding H r S 1 C r 1/ S 1 with a pointwise estimate gx gy g H r x y r 1/, we can bound both terms in 5.14 by C ω H s v H s ξ 1 e H s + ξ 1 e s 3/ Continuity of ξ ω F ξ,ω follows from a similar estimate. In this way we obtain that F is continuously differentiable near e, and applying the fundamental ODE theorem for Banach spaces see e.g. Lang [L] establish local existence, uniqueness and smooth dependence on the data u of solutions ξt and ξt of Local wellposedness of the original Cauchy problem follows now at once from u = ξ ξ 1 and the fact that Diff s is a topological group whenever s > 3/. In order to complete the proof of Theorem 5.1 we need to derive a C 1 bound on the solution u. The standard trick is to use Friedrichs mollifiers J ɛ with < ɛ < 1 to derive the inequality d dt J ɛu H s = Λ s J ɛ u x Λ s J ɛ u dx 3 µu. Λ J ɛ u x Λ s Λ µ J ɛ u dx u C 1 u H s + µu x Λ s Λ µ J ɛ u L Λ s J ɛ u L u C 1 u H s where the first estimate is standard see e.g. [T], Chapter 5 and the second follows easily with the help of 5. and Passing to the limit with ɛ and using Gronwall s inequality yields the persistence result. 5.. Blow-up of smooth solutions. A family of examples displaying a simple finite-time breakdown mechanism of a µdp solution is described by the following result. Theorem 5.3. Given any smooth periodic function u with zero mean there exists T c > such that the corresponding solution of the µdp equation stays bounded for t < T c and satisfies u x t as t T c. Proof. Let u be the solution of µdp with initial data u. Differentiating the equation with respect to the space variable we obtain u tx + uu xx + u x + 3µu xλ µ u =. Since xλ µ = µ 1, using conservation of the mean, we find that 5.15 u tx + uu xx + u x = 3µu u µu = by the assumption on the data u. If we let ξt denote the flow of u as in 5.5 then and setting w = x ξ/ x ξ we find that x ξ = ux ξ x ξ w t = x ξ x ξ x ξ x ξ = u xt + u xx u ξ. With the help of these formulas we rewrite 5.15 in the form w t + w =

13 INTEGRABLE EVOLUTION EQUATIONS AND PEAKONS 13 and, since w, x = u x x, solve for w to get wt, x = 1 t + 1/u x x. Furthermore, our assumptions are such that it is always possible to find a point x S 1 such that u x x is negative. Setting T c = 1/u x x we conclude that the solution w must blow-up in the L norm as expected. More sophisticated break-down mechanisms for µdp can be demonstrated but we will not pursue them in this paper Global solutions. Our global result for µdp is obtained under a sign assumption on the initial data of the type that has been used previously in the studies of equations such as CH, µch and DP. What follows is again a basic result. Theorem 5.4. Let s > 3. Assume that u H s S 1 has non-zero mean and satisfies the condition Λ µu or. Then the Cauchy problem for µdp has a unique global solution u in CR, H s S 1 C 1 R, H s 1 S 1. Proof. From the local wellposedness and persistence result in Theorem 5.1 we know that the solution u is defined up to some time T > and that in order to extend it we only need to show that the norm u x t, remains bounded. On the one hand, given any periodic function wx differentiating the formula in 5. we readily obtain the estimate 5.16 x w Λ µw L 1. On the other hand, recall from Section that for any solution u of µdp with initial condition u H s S 1 the expression m = Λ µu is a tensor density in F 3 S 1 and hence from the Diff s S 1 -action formula in.6 we get the following pointwise conservation law 5.17 d dt Λ µu ξ x ξ 3 = u txx uu xxx + 3µuu x 3u x u xx ξ x ξ 3 =, where t ξt Diff s S 1 is the associated flow i.e. ξ = u ξ and ξ = e. Thus, for any x S 1 as long as the solution exists it must satisfy Λ µu t, ξt, x x ξt, x 3 = Λ µ u x and therefore by the assumption on u it follows that Λ µut, x for any x S 1 as long as it is defined. The above inequality together with the conservation of the mean and 5.16 gives x ut, Λ µut L 1 = Λ µut, x dx = µu < and we conclude that the solution u must persist indefinitely.

14 14 JONATAN LENELLS, GERARD MISIO LEK, AND FERIDE TIĞLAY 5.4. Local wellposedness for any λ. For completeness we state here a local wellposedness result for the family 1. of µ-equations for any λ. Using the substitution v = Av = Λ µu and the fact that, as in the case of µdp, the mean of the solution u is conserved, we can write the Cauchy problem for 1. in the form 5.18 u t + uu x + Λ µ x λµuu + 3 λ u x = 5.19 u = u. Theorem 5.5 Local wellposedness. Let s > 3/. For any u H s S 1 there exist a T > and a unique solution u C T, T, H s C 1 T, T, H s 1 of which depends continuously on the initial data u. Furthermore, the solution persists as long as ut, C 1 stays bounded. The proof of this theorem is analogous to Theorem 5.1 and we omit it. In the previous section we proved a global existence result for µdp. Here, we formulate a similar result for any positive λ. For a fixed λ > let u = ut, x be any smooth solution of the problem and let ξt Diff s S 1 denote the corresponding flow of diffeomorphisms of the circle. As before, the argument rests on the pointwise conservation law obeyed by the solutions. Namely, using the formula for the action of Diff s S 1 on λ-densities F λ S 1 in.6 we find the analogue of 5.17 to be d Λ dt µu ξ x ξ λ = Λ µu t uu xxx + λµuu x λu x u xx ξ x ξ λ = so that as long as ut, x exists and therefore we have Λ µu t, ξt, x x ξt, x λ = Λ µ u x, x S 1 Theorem 5.6. Fix λ >. Assume that u H s S 1 with s > 3 has non-zero mean and satisfies the condition Λ µu or. Then the Cauchy problem has a unique global solution in C R, H s S 1 C 1 R, H s 1 S Peakons The CH and DP equations famously exhibit peakon solutions, see e.g. [CH, DHH]. In this section we show that each of the µ-equations 1. parametrized by λ Z also admits peakons. Let us point out here that the fact that µch admits peaked solutions went unnoticed in [KLM], so that the present discussion provides an extension of the results of that paper. We will investigate existence of traveling-wave solutions of the form ut, x = ϕx ct. This will lead us to expressions for the one-peakon solutions. Subsequently, we will analyze the more general case of multi-peakons.

15 INTEGRABLE EVOLUTION EQUATIONS AND PEAKONS Traveling waves. We first consider equation 1. where µu is replaced by a parameter ν R, i.e. 6.1 u txx + λνu x = λu x u xx + uu xxx. After analyzing traveling-wave solutions of this equation we will impose the two conditions µu = ν and periodu = 1 to find which of these are traveling waves of equation 1.. We assume that λ, 1. Substituting ut, x = ϕx ct into equation 6.1 we find cϕ xxx + λνϕ x = λϕ x ϕ xx + ϕϕ xxx. We write this in terms of f := ϕ c and integrate to find 6. λνf + a = λ 1 fx + ff xx, where a in an integration constant. We multiply equation 6. by f λ f x and integrate the resulting equation with respect to x. The outcome is νf λ + a λ 1 f λ 1 + b = f λ 1 fx, where b is another constant of integration. This equation can be written as 6.3 ϕ x = νϕ c + a λ 1 + b ϕ c λ 1. A solution of this equation yields the shape of a traveling wave within an interval where ϕ is smooth, and gluing such smooth wave segments together at points where ϕ = c produces the complete collection of traveling wave solutions of 6.1. The analysis proceeds along the same lines and yields similar results as for CH or as for HS if ν = cf. [Le]. Here we choose to consider the peaked solutions. For completeness we point out that the equation analogous to 6.3 when λ = or λ = 1 is 6.4 ϕ x = { a + bϕ c, λ =, νϕ c + aln ϕ c + b, λ = Periodic one-peakons. The periodic peakons arise when b = in Indeed, setting b = and replacing a by a new parameter m via a = λ 1c mν in 6.3, we find ϕ x = νϕ m. Solving this equation, we find the peakon solution ut, x = ϕx ct where [ 6.5 ϕx = m + ν ] c m c m x for x,, ν ν and ϕ is extended periodically to the real axis. The parameters are required to satisfy c m/ν. The period of ϕ is fixed by the condition that ϕ = c at the peak. Equation 6.5 defines a periodic peaked solution of 6.1 for any choices of the real parameters m, c, ν. In order to determine which of these solutions are solutions with period one of the µ-equation 1. we have to impose the conditions meanϕ = ν and periodϕ = 1. A computation shows that meanϕ = m + c m 3 and periodϕ = 1 We henceforth assume ν.

16 16 JONATAN LENELLS, GERARD MISIO LEK, AND FERIDE TIĞLAY c m ν. Therefore, ut, x = ϕx ct where ϕ is given by 6.5 is a period-one solution of equation 1. if and only if c m = 1, ν c m ν, and ν = m + c m. 3 Solving these equations we find m = 3c/6 and ν = 1c/13. following result. This leads to the Theorem 6.1. For any c R and λ, 1, equation 1. admits the peaked periodone traveling-wave solution ut, x = ϕx ct where 6.6 ϕx = c 6 1x + 3 for x [ 1, 1 ] and ϕ is extended periodically to the real line. We note that the one-peakon solutions of 1. are the same for any λ, that they travel with a speed equal to their height, and that µϕ = 1c/ Green s function. For the construction of multi-peakons it is convenient to rewrite the inverse of the operator Λ µ = µ x in terms of a Green s function an explicit formula for the inverse Λ µ was previously given in equation 5.. We find Λ µ mx = where the Green s function gx is given by gx x mx dx, 6.7 gx = 1 13 xx 1 + for x [, 1 S 1, 1 and is extended periodically to the real line. In other words, gx x = x x x x , x, x [, 1 S 1. Note that gx has the shape of a one-peakon 6.6 with c = 13/1 and peak located at x =. In particular, µg = Multi-peakons. The construction of multi-peakons for the family of µ-equations 1. is similar to the corresponding construction for the family 1.1. Just like for CH and DP, the momentum m = Λ µu of an N-peakon solution is of the form 6.8 mt, x = p i tδx q i t, where the variables {q i, p i } N 1 evolve according to a finite-dimensional Hamiltonian system. Indeed, assuming that m is of the form 6.8, the corresponding u is given by u = Λ µ m = p i tgx q i t. We assign the value zero to the otherwise undetermined derivative g, so that { 6.9 g, x =, x := x 1, < x < 1.

17 INTEGRABLE EVOLUTION EQUATIONS AND PEAKONS x x.5 x Figure 3. The periodic Green s functions gx and g µ x corresponding to the operators A = 1 x and A = µ x, respectively, and their difference gx g µ x. This definition provides a naive way to give meaning to the term λu x m in the PDE.8. Substituting the multi-peakon Ansatz 6.8 into the µ-equation 1. then shows that p i and q i evolve according to q i = p j gq i q j, ṗ i = λ 1 p i p j g q i q j. j=1 A more careful analysis reveals that this computation is in fact legitimate and yields the following result. Note that the peakons fall outside the result of Theorem 5.5 since for a peakon u / H s S 1 for s > 3/. Theorem 6.. The multi-peakon 6.8 satisfies the µ-equation 1. in the weak form 5.18 in distributional sense if and only if {q i, p i } N 1 evolve according to 6.1 q i = uq i, ṗ i = λ 1p i {u x q i } where {u x q i } denotes the regularized value of u x at q i defined by 6.11 {u x q i } := p j g q i q j and g x is defined by 6.9. Proof. See Appendix A. j=1 The formulas for the peakons of the µ-equations 1. along with the expressions for the previously known peakons of the CH and DP family 1.1 are summarized in Table 6.4. At any particular time, the multi-peakon is a sum of Green s functions for the associated operator A, see Figure 3. Since the inertia operator for the µ-equations A = µ x is different from that of the family 1.1 A = 1 x, the new class of µ-peakons have a different form with respect to the CH and DP peakons. For a given family of equations the Green s function is the same for all λ, but the time evolution of the positions q i and momenta p i of the peaks of course depends on λ and is for both families given by the system Poisson structure. The equations 6.1 take the Hamiltonian form with respect to the Hamiltonian j=1 q i = {q i, H }, ṗ i = {p i, H }, λ H = λ 1 p i,

18 18 JONATAN LENELLS, GERARD MISIO LEK, AND FERIDE TIĞLAY Equation family Green s function x S 1 [, 1 Multi-peakon ut, x 1.1 gx = coshx 1/ N sinh1/ p itgx q i t 1. gx = 1 x N p itgx q i t Table 1. The periodic multi-peakon solutions of the family 1.1 which includes CH and DP and of the corresponding µ-family 1. which includes µch and µdp when λ, 1. and the Poisson structure λ 1 {p i, p j } = p i p j G q i q j, {q i, p j } = λ 1 λ λ p j G q i q j, {q i, q j } = 1 λ Gqi q j, where Gx = x gx dx. In order to make sense of Gq i q j although G is not periodic, we restrict q i, q j to be local coordinates on S 1 satisfying say < q i, q j < 1. Then 1 < q i q j < 1 and Gq i q j is well-defined. This Poisson structure can be derived from the Poisson structure corresponding to J in 4.3 by noting that {mx, my} J = 1 λ [ Gx ymx xm x y λg x ym x xmy + λg x ymxm x y λ G x ymxmy ]. Indeed, let m x = i p iδx q i be a multi-peakon of the form 6.8. Let φ i : S 1 R resp. ψ i : S 1 R be a function which takes the value 1 resp. x in a small neighborhood of x = q i and the value elsewhere. Then, for mx = i p iδx q i sufficiently close to m, we obtain φ i xmxdx = p i, ψ i xmxdx = q i p i. S 1 S 1 We now find {p i, p j } from the computation λ 1 {p i, p j } = φ i xφ j y{mx, my} J dxdy = p i p j G q i q j. S 1 S 1 λ We then compute {q i p i, p j } = ψ i xφ j y{mx, my} J dxdy S 1 S 1 = λ 1 λ p i p j G q i q j + λ 1 q i p i p j G q i q j. Since {q i p i, p j } = q i {p i, p j }+{q i, p j }p i and we know the value of {p i, p j }, this gives the expression for {q i, p j }. Finally, the expression for {q i, q j } is obtained by calculating {q i p i, q j p j } = ψ i xψ j y{mx, my} J dxdy S 1 S 1 and comparing the result with the expression for {q i p i, q j p j } = q i q j {p i, p j } + q i {p i, q j }p j + q j {q i, p j }p i + {q i, q j }p i p j obtained by substituting in the expressions for {p i, p j }, {p i, q j }, and {q i, p j }. λ

19 INTEGRABLE EVOLUTION EQUATIONS AND PEAKONS Multi-peakons for µch. For µch λ = the equations 6.1 for {q i, p i } are canonically Hamiltonian with respect to the Hamiltonian h = 1 p i p j gq i q j, and describe geodesic flow on T N R N here T N = S 1 S 1 denotes the N-torus with respect to the metric g ij with inverse given by g ij = gq i q j. In this case the reduction to multi-peakon solutions can be elegantly understood in terms of momentum maps using the ideas of [HM]. We consider q i, p i as an element of T T N T N R N. Then the map J sing : T T N vects 1 : q i, p i p i δx q i is an equivariant momentum map for the natural action of DiffS 1 on T T N. Since momentum maps are Poisson, J sing maps the solution curves in T T N which satisfy 6.1 to solution curves in vects 1 which satisfy.3. Note that h = J sing H is the pull-back of the Hamiltonian H on vects 1 defined by H = 1 umdx Periodic two-peakon for µch. In the case of two peaks for the µch equation N =, λ =, the system 6.1 is given explicitly by q 1 = p 1 g + p gq 1 q, q = p 1 gq q 1 + p g, ṗ 1 = ṗ = p 1 p g q 1 q. Letting Q = q q 1 and P = p p 1 and using that we find the system where α = h 13 1 H 16 h = 13 H p 1p Q Q, Q = 1 P Q Q, P α = Q Q 1 Q sgnq,. This leads to the following equation for Q: QQ Q = Q + αq Q Q 1 sgnq. The solution to this ODE can be expressed in terms of inverses of elliptic functions Shock-peakons for µdp. In addition to the peakon solutions the DP equation admits an even weaker class of solutions with jump discontinuities called shockpeakons, see [Lu]. We will show here that µdp also allows shock-peakon solutions. We will seek the solutions of the form mt, x = p i tδx q i t + s i tδ x q i t.

20 JONATAN LENELLS, GERARD MISIO LEK, AND FERIDE TIĞLAY The corresponding u is 6.1 u = p i gx q i + s i g x q i. Substituting this into the equation m t = um x λu x m, we find formally q i = uq i, ṗ i = λ 1 s i u xx q i p i u x q i, ṡ i = λ s i u x q i. The following theorem states that in the case of λ = 3 this is the case of µdp, this formal computation can be rigorously justified provided that the u x and u xx terms are appropriately regularized. The reason that such weak solutions can be made sense of in the case of µdp is that the term 3 λ u x in the weak formulation 5.18 is absent when λ = 3. Theorem 6.3. The shock-peakon 6.1 satisfies the weak form 5.3 of µdp in distributional sense if and only if {q i, p i, s i } N 1 evolve according to 6.13 where {u x q i } = q i = uq i, ṗ i = s i {u xx q i } p i {u x q i }, ṡ i = s i {u x q i }, p j g q i q j + j=1 and g x is defined by 6.9. Proof. See Appendix B. s j, {u xx q i } = j=1 p j, j=1 7. The µburgers equation In this section we discuss the µb equation and its properties. We introduced this equation in Section as the λ = 3 analogue of the Hunter-Saxton equation. Here we take a different view and employ Riemannian geometric techniques to display its close relationship with the inviscid Burgers equation Burgers equation and the L -geometry of Diff s Diff s /S 1. The constructions in this subsection are well known. First, recall that the inviscid Burgers equation B u t + uu x = is related to the geometry of the weak Riemannian metric on Diff s S 1 which on the tangent space T ξ Diff s S 1 at a diffeomorphism ξ is given by the L inner product 7.1 V, W L = V xw x dx where V, W T ξ Diff s. It is not difficult to show that this metric admits a unique Levi-Civita connection whose geodesics ηt in Diff s S 1 satisfy the equation η η = η = u t + uu x η = 11 In fact, the geodesic equation is readily obtained from the first variation formula for the energy functional Eη = 1 b a ηt L dt of the L metric 7.1 on Diff s S 1.

21 INTEGRABLE EVOLUTION EQUATIONS AND PEAKONS 1 and hence correspond to classical solutions of the Burgers equation. Here ηt is simply the flow of ut, x so that ηt, x = ut, ηt, x and hence the second equality in 7. follows at once from the chain rule. Consider next the homogeneous space Diff s = Diff s /S 1 where S 1 denotes the subgroup of rotations. For s > 3/ this space is also a smooth Hilbert manifold whose tangent space at the point [e] in Diff s S 1 corresponding to the identity diffeomorphism can be identified with the periodic Sobolev functions with zero mean { } T [e] Diff = HS s 1 = w H s S 1 : wx dx =. It is useful to think of Diff s S 1 as the space of probability densities on the circle and view the map π : Diff s Diff s as a submersion given by the pull-back [ξ] = πξ = ξ dx where dx is the fixed probability measure with density 1 see e.g. [KY]. The fibre through ξ Diff s S 1 consists of all diffeomorphisms that are obtained from ξ by a composition on the left with a rotation and thus is just a right coset of the rotation subgroup S 1 which acts on Diff s S 1 by left translations. Furthermore, the L metric 7.1 is preserved by this action and π becomes a Riemannian submersion with each tangent space decomposing into a horizontal and a vertical subspace T ξ Diff s = P ξ T ξ Diff s L Q ξ T ξ Diff s which are orthogonal with respect to 7.1. The two orthogonal projections P ξ : T ξ Diff s T πξ Diff s and Q ξ : T ξ Diff s R are given explicitly by the formulas 7.3 P ξ W = W W x dx and Q ξ W = W x dx. Finally, as with any Riemannian submersion 1, note that a necessary and sufficient condition for a curve ηt in Diff s S 1 to be an L geodesic satifying 7. and hence correspond to a solution of Burgers equation is that P η η η = Q η η η =. We are now ready to introduce the µb equation in this set-up. 7.. Lifespan of solutions, Hamiltonian structure and conserved quantities of the µburgers equation. We first characterise local in time smooth solutions. Theorem 7.1. A smooth function u = ut, x is a solution of the µb equation u txx + 3u x u xx + uu xxx = if and only if the horizontal component of the acceleration of the associated flow ηt in Diff s S 1 is zero i.e. P η η η =. In fact, given any u H s S 1 the flow of u has the form ηt, x = x + t u x u + ηt, for all sufficiently small t. Proof. Integrating the µb equation with respect to the x-variable we get 7.4 u tx + u x + uu xx = and integrating once again gives u t + uu x = ct u x + uu xx dx = 1 See e.g. O Neill [ON] for details on Riemannian submersions.

22 JONATAN LENELLS, GERARD MISIO LEK, AND FERIDE TIĞLAY where ct is a function of the time variable only. Since the mean of a µb solution need not be preserved in time we obtain the equation u t + uu x = µu t. On the other hand, if ηt is the flow of u in Diff s S 1, that is ηt, x = ut, ηt, x, η, x = x then from 7. and 7.3 we compute the horizontal component of the acceleration of ηt in Diff s S 1 to be P η η η = η η dx = 1 u t + uu x η ut + uu x η dx. Using these formulas and integrating by parts we find that the equation P η η η = is equivalent to ut + uu x η = η = η dx = = η η 1 dx u t + uu x dx = u t dx which establishes the first part of the theorem. In order to prove the second statement it suffices to observe that from 7.6 we have in particular which immediately implies ηt, x = η η 1 t, x dx = ηt, ηt, x ηt, = x + t u x u for any x 1 and any t for which η is defined. Remark 7.. It is easy to verify that equation 7.5 is also equivalent to η P η η = which can be interpreted as saying that the horizontal component of the velocity of the flow ηt of the µb equation is parallel transported along the flow. Remark 7.3. A similar construction can be used to derive the µb equation from the right-invariant L metric on Diff s S 1 defined by V, W ξ = V ξ 1 xw ξ 1 x dx where V, W T ξ Diff s and ξ Diff s. In this case the corresponding orthogonal projections are P ξ W = W W dξ = W Q ξ W and the geodesic equation in Diff s S 1 reads η η = η + η x η x η 1 =. Proceeding as in the proof of Theorem 7.1 we then get = P η η η = u t + 3uu x µu t η which after rescaling the dependent variable yields 7.5. It is therefore not surprising to find that the µb equation shares a number of properties with the Burgers equation. Corollary 7.4. Suppose that ut, x is a smooth solution of the µb equation and let u, x = u x. 13 In the case when the mean µu is independent of time 7.5 becomes the standard inviscid Burgers equation. The presence of the mean is also the reason for our terminology.

23 INTEGRABLE EVOLUTION EQUATIONS AND PEAKONS 3 1 The following integrals are conserved by the flow of u 1 pdx u µu = u µu p dx, p = 1,, There exists T c > such that u x t as t T c. 3 The µb equation has a bi-hamiltonian structure given by the Hamiltonian functionals H = 9 m dx, H = 1 u 3 dx, 6 and the operators J = m /3 x m 1/3 3 x m 1/3 x m /3, J = 5 x. Proof. The first statement follows by direct calculation. Regarding the second, it suffices to consider the equation in 7.4 and apply the argument that was used in the proof of Theorem 5.3. The third statement follows from a straightforward computation establishing δh J δm = J δh δm = um x 3u x m where m = xu and our earlier observation from section 4 that J and J are compatible. 8. Multidimensional µch and µdp equations In this final section we briefly consider possible generalizations of the µch and µdp equations to higher dimensions. Let XT n be the space of smooth vector fields on the n-dimensional torus T n comprising the Lie algebra of DiffT n. Let A be a self-adjoint positive-definite operator defining an inner product on the Lie algebra. Given a vector field u XT n we define the corresponding momentum density by m = Au. The EPDiff equation for geodesic flow on DiffT n is given by 8.1 m t + L u m =, where L u m denotes the Lie derivative of the momentum one-form density m in the direction of u see [HM]. Identifying T n R n /Z n and letting x i, i = 1,,..., n denote standard coordinates on R n, this can be written as m i + u j m i t x j + m u j j x i + m idivu = where m = m i dx i d n x and u = u i / x i. The CH equation is the 1+1-dimensional version of the EPDiff equation 8.1 when A = 1. It is natural to define a multidimensional generalization of µch as equation 8.1 with the operator A defined by Au = µ u = u d n x u. T n Similarly, using higher-order tensor densities we arrive at multidimensional versions of DP and µdp. Let m = m i dx i d n x d n x. Then equation 8.1 becomes m i 8. + u j m i t x j + m u j j x i + m idivu =. In dimensions this reduces to m t + m x u + 3mu x =.

24 4 JONATAN LENELLS, GERARD MISIO LEK, AND FERIDE TIĞLAY Hence, when A = 1, equation 8. can be viewed as a multidimensional DP equation, while if A = µ it can be viewed as a multidimensional µdp equation. Appendix A. Proof of Theorem 6. In this appendix we prove Theorem 6., that is, we show that the multi-peakons defined in 6.8 are weak solutions in the distributional sense of the µ-equation 1. if and only if {q i, p i } N 1 evolve according to 6.1. Equation 1. in weak form reads 14 A.1 u t + 1 u x + λµug u + 3 λ g u x =, where g is the Green s function defined in 6.7. This is to be satisfied in the space of distributions D R S 1, i.e. S 1 uφ t + 1 u φ x + λµug uφ x + 3 λ g u xφ x dxdt =, for all test functions φt, x Cc R S 1. Let g j := gx q j and g j := g x q j. We specify the value of g at x = by setting g := as in equation 6.9. We will need the following identities which can be verified by direct computation A. A.3 A.4 A.5 g g j = 1 3 g g j =1 g j, g j 13 1 g j, g i g j g i g j =g jq i g i + g iq j g j + gq i q j 13 1 g i + gq j q i 13 g 1 j, g ig j =g i + g j + g q i q j g i g j + gq i q j 3. These identities hold pointwise for all x, q i, q j S 1 except A.5 which holds for all x, q i, q j S 1 unless x = q i = q j ; a fact that does not matter in what follows. Let u = i p ig i be a peakon. As explained in the appendix of [Lu], we can compute the left-hand side of A.1 as a distribution in the variable x without having to involve test functions explicitly. This yields u t = Using A. we find Moreover, ṗi g i p i q i g i, u x = λµug u = λ 3 u x = p i p j g i g j. p i p j g j 13 g 1 j. p i p j g ig j, 14 Here denotes convolution: for two functions f, g : S 1 [, 1 R, we have f gx = fx ygydy.