## Abstract

The Degasperis–Procesi equation (DP) is one of several equations known to model important nonlinear effects such as wave breaking and shock creation. It is, however, a special property of the DP equation that these two effects can be studied in an explicit manner with the help of the multi-peakon ansatz. In essence, this ansatz allows one to model wave breaking as a collision of hypothetical particles (peakons and antipeakons), called henceforth collectively multi-peakons. It is shown that the system of ordinary differential equations describing DP multi-peakons has the Painlevé property which implies a universal wave breaking behaviour, that multipeakons can collide only in pairs, and that there are no multiple collisions other than, possibly simultaneous, collisions of peakon–antipeakon pairs at different locations. Moreover, it is demonstrated that each peakon–antipeakon collision results in creation of a shock thus making possible a multi-shock phenomenon.

## 1. Introduction

The Degasperis–Procesi (DP) equation [1],
1.1belongs to a class of one-dimensional wave equations that have attracted considerable attention over the past decade, following the most studied equation in this class, namely the Camassa–Holm (CH) equation [2]
Both these equations can be derived from the governing equations for water waves under the assumption of moderate amplitude [3,4]. What makes them special is that, on the one hand, both are Lax-integrable, on the other hand, both exhibit wave breaking phenomena not captured by the linear theory or the shallow water, small amplitude, theory such as the Korteweg–de Vries equation. For the CH equation, the most relevant study of the breakdown of solutions was carried out by McKean [5,6]. In these works, it was argued that the breakdown of the CH waves is controlled by a kind of caricature of the higher dimensional vorticity, namely *m*=*u*−*u*_{xx} (see Majda & Bertozzi [7]). One of the fascinating aspects of both the CH and DP equations is the existence of special solutions, peakons which play the role of basic building blocks of the underlying full theory. Peakons are a simple superposition of exponential terms
for which the function *m* referred to earlier is . Were we to take the analogy with vorticity at its face value, *m* for peakons could be viewed as a collection of point vortices, situated at *x*_{j}(*t*), with strengths *m*_{j}(*t*) for *j*=1,…,*n* respectively, initially ordered in some fixed way, say *x*_{1}(0)<*x*_{2}(0)<⋯<*x*_{n}(0). The case of CH peakons shows that if the strengths {*m*_{k}(0):1≤*k*≤*n*} are not of the same sign, then *collisions* can occur, meaning that *x*_{j}(*t*_{c})=*x*_{j+1}(*t*_{c}) for some *j* and sometime *t*_{c}. Each collision is accompanied by a blow-up of *m*_{j}(*t*_{c}) and *m*_{j+1}(*t*_{c}) resulting in the derivative *u*_{x}(*t*_{c}) becoming unbounded even though *u*(*t*_{c}) remains bounded, in fact, continuous. Thus, peakons can be used to test ideas about wave breaking, the advantage being that the peakon dynamics is described by a finite system of ordinary differential equations (ODEs; see §2). The complete analysis of the CH peakon collisions was carried out in Beals *et al.* [8,9] with the help of explicit formulae.

The case of the DP equation is superficially similar to the case of CH. However, deeper analysis shows a remarkable number of new features. For example, the associated spectral problem, termed a cubic string in reference [10], is not self-adjoint, and this has the immediate consequence that the inverse problem is far more involved. The peakon problem in the case of positive measure, that is when all weights *m*_{j} are positive, was solved explicitly in reference [10]. However, the generalization to the case of a signed measure *m* is not straightforward, because the spectral data break up into several types depending on the degeneracy of the spectrum, in contrast to the CH case where the spectrum remains real and simple.

Similar to the CH case, the presence of a DP peakon collision signals an occurrence of wave breaking; in the DP context, the connection between wave breaking and peakon collisions was studied earlier by Lundmark [11] for the case *n*=2 and further by us [12] for *n*=3.

The DP equation, however, in contrast to the CH equation, also admits shock solutions (see [13,14] for a general, very thorough discussion). It was Lundmark who introduced the concept of shockpeakons
for which
1.2and showed that the solution describing a collision of two peakons has a unique entropy extension to shockpeakons. He also hypothesized that this might be a general phenomenon valid also for *n*>2. Intuitively speaking, the collisions of DP peakons signal both the wave breaking and the onset of shocks.

Let us briefly describe our strategy. Because, in general, the spectral problem for DP peakons is quite involved, we choose instead to concentrate on analytic properties of *x*_{j}(*t*),*m*_{j}(*t*) as functions of *t* insofar as their behaviour can be inferred from the inverse problem without explicitly solving it. Then, we use the ODEs describing peakons time evolution to refine information about the details of collisions of peakons. More concretely, the plan of the paper is as follows: we review basic facts about the DP equation in §2; in §3, we discuss the inverse problem for peakons of both signs generalizing the uniqueness result known from the pure peakon case [10] and use this result to establish analytic properties of positions *x*_{j} and masses *m*_{j}. We prove that each *x*_{j}(*t*) must be a holomorphic function at *t*_{c}, whereas *m*_{j}(*t*) is, in general, only meromorphic (theorem 3.5). Then, we perform singularity analysis of the ODEs describing peakons (2.6) and prove their Painlevé property with the help of theorems 3.5 and 3.6. In §4, which is the main section of the paper, we analyse the singular behaviour at the time of collisions and establish a universal singularity pattern, according to which, in the leading term, only the time of the blow-up depends on the initial conditions, whereas the residue remains universal. This fact is proved in theorem 4.5. We furthermore rule out triple collisions in theorem 4.7, and give an example of a simultaneous collision, in different positions, of two peakon–antipeakon pairs; finally, in §5, we apply results of §4 to prove theorem 5.1 stating that the distributional limit of colliding peakons is, indeed, a shockpeakon. More precisely, we prove that the distributional limit of *m* at the collision point *t*_{c} produces shockpeakon data (1.2) with positive shock strengths *s*_{j} thus allowing a unique entropy weak extension (see theorem 5.1 and corollary 5.2), which settles in positive Lundmark's conjecture.

Important questions of stability and general analytic results dealing with DP peakons and the DP wave breaking have been addressed [15–18]. A considerable amount of work has been also carried out on adapting numerical schemes to deal with the DP equation; we just mention a few: an operator splitting method of Feng & Liu [19], or numerical schemes discussed by Coclite *et al.* [20] and Hoel [21].

## 2. Basic facts about the Degasperis–Procesi equation

The nonlinear equation 2.1often written as 2.2was introduced by Degasperis & Procesi [1].

Formal integrability for the DP equation was established by Degasperis *et al.* [22] through the construction of a Lax pair and a bi-Hamiltonian structure. In particular, it was shown in reference [22] that the DP equation admits the Lax pair:
2.3Moreover, one can impose additional boundary conditions provided they do not violate the compatibility of these equations. One such a pair of boundary conditions was introduced in reference [10]:
2.4where it was also shown that the spectrum of this boundary value problem will remain time invariant (isospectral deformation). It suffices for our purposes to restrict our attention to the case in which *m* is a finite discrete (signed) measure. Thus, for the remainder of the paper, we will use the *multi-peakon* ansatz
2.5where *x*_{1}(0)<*x*_{2}(0)<⋯<*x*_{n}(0) and *m*_{i}(0) can have both positive and negative values. This ansatz produces . Moreover, with the proper interpretation of weak solutions to equation (2.1), we can easily check that *u* is a weak solution to (2.1) provided *x*_{i}(*t*),*m*_{i}(*t*) satisfy the following ODEs
2.6aand
2.6bwhere is the average of *f* at the point *x*. We will refer to the coefficients *m*_{j} as *masses* to emphasize their role in the spectral problem. We also need a bit of terminology regarding the phenomenon of breaking. We will say that a *collision* occurred at sometime *t*_{c} if *x*_{i}(*t*_{c})=*x*_{i+1}(*t*_{c}) for some *i*. We can make this concept more geometric by introducing a configuration space in which to study peakon solutions, namely the sector *X*={**x**∈**R**^{n} |*x*_{1}<*x*_{2}<⋯<*x*_{n}}. Then, a collision corresponds to the solution *x*_{i} hitting the boundary of *X*.

A very useful property of equations (2.6) is the existence of *n* constants of motion. This follows readily from theorem 2.10 in reference [10].

### Lemma 2.1

*M*_{p}(1≤*p*≤*n*), *given by*,
*are n constants of motion of the system of equations (2.6), where* *is the set of all p-element subsets I*={*i*_{1}<…<*i*_{p}} *of* {1,…,*n*}.

## 3. Inverse problem for multi-peakons

The boundary value problem (2.3) and (2.4) can be transformed to a finite interval boundary value problem, the cubic string problem. Indeed, following Lundmark & Szmigielski [10], the change of variables (Liouville transformation)
3.1maps the DP spectral problem into the cubic string problem:
3.2aand
3.2bwhere *g* is the transformation of the measure *m* induced by the Liouville transformation (3.1). Furthermore, as one can explicitly check, *g* is also a finite signed measure and its support does not include the endpoints if the original measure *m* is a finite signed measure. More concretely, in this paper,
3.3with weights *g*_{i}∈**R**. The inverse problem is studied with the help of two Weyl functions.

### Definition 3.1

Let *ϕ*(*y*;*z*) denote the solution to the initial value problem (3.2a) with initial conditions *ϕ*(−1;*z*)=*ϕ*_{y}(−1;*z*)=0, *ϕ*_{yy}(−1;*z*)=1. The Weyl functions are ratios:

These two functions encode spectral information required to solve the inverse problem. It is easy to verify that in the case of (3.3) both *W*(*z*) and *Z*(*z*) are rational functions that make inversion algebraic. However, in contrast to the pure peakon case *g*_{i}>0, the spectrum of the boundary value problem (3.2a) is, in general, complex and not necessarily simple. This makes the inversion more challenging. Regardless of the complexity of the spectrum, the Weyl functions undergo a simple evolution under the DP flow. Indeed, using the second member of the Lax pair given by (2.3), one can find the time evolution of *W*(*z*) and *Z*(*z*). To wit, using results from theorem 2.3 in reference [12], we obtain the following characterization of the time evolution of *W*(*z*) and *Z*(*z*).

### Theorem 3.2

*Let*
*be the partial fraction decomposition of W(z)/z, where d*_{j} *denotes the algebraic degeneracy of the j*th *eigenvalue. Then, the DP time evolution implies*

(

*1*)*where**is a polynomial in t of degree d*_{j}−*k*.(

*2*).(

*3*)

We immediately have the following.

### Corollary 3.3

*Under the DP flow, the Weyl functions W*,*Z are entire functions of time*.

The uniqueness result below plays a major role in the solution to the inverse problem.

### Theorem 3.4

*Suppose* *is the map that associates with the cubic string problem (3.2) with a finite signed measure g, the Weyl functions W(z),Z(z). Then, Φ is injective*.

### Proof.

The proof relies on remarks made in reference [23]. We will construct a recursive scheme to solve the inverse spectral problem; given *W* and *Z* obtained from the map *Φ*, we will reconstruct the finite, signed measure *g* whose Weyl functions are *W* and *Z*. More precisely, we will show that the *y*_{j} and *g*_{j} in equation (3.3) are uniquely determined from *W*(*z*),*Z*(*z*). First, we recall that *W* and *Z* are constructed from solutions to the initial value problem (see definition 3.1)
3.4aand
3.4bMasses *g*_{j} are situated at *y*_{j}, 1≤*j*≤*n* and for convenience let us set *y*_{0}=0, *y*_{n+1}=1 and denote by *l*_{j}=*y*_{j+1}−*y*_{j} the length of the interval (*y*_{j},*y*_{j+1}). Then, on each interval (*y*_{j},*y*_{j+1}), the solution to (3.4) takes the form
where notation *f*(*a*±) denotes the right-hand or left-hand limits at *a*. Then, the condition of crossing *y*_{j+1} is continuity of *ϕ* and *ϕ*_{y} and the jump condition *ϕ*_{yy}(*y*_{j+1}+)−*ϕ*_{yy}(*y*_{j+1}−)=−*zg*_{j+1}*ϕ*(*y*_{j+1}). We establish, for example, by an easy induction,
3.5valid for 0≤*j*≤*n*, with the convention that for *j*=0 the product equals 1 and there is no remainder. Likewise,
3.6valid for 1≤*j*≤*n*.

For 0≤*j*≤*n*, we define (*w*_{2j},*z*_{2j})=(*ϕ*_{y}/*ϕ*,*ϕ*_{yy}/*ϕ*)|_{y=yj+1−} and (*w*_{2j−1},*z*_{2j−1})=(*ϕ*_{y}/*ϕ*_{yy},*ϕ*/*ϕ*_{yy})|_{y=yj+}. These quantities are essentially the left-hand and the right-hand analogues of Weyl functions introduced in definition 3.1 and correspond to shorter strings terminating at *y*_{j+1} with no mass at the endpoint, or terminating at *y*_{j} but with the mass *g*_{j} at the end. Equation (3.4) implies that the sequence (*w*_{2j},*z*_{2j},*w*_{2j−1},*z*_{2j−1}) satisfies the recurrence relations
3.7and
3.8the iteration starts at *w*_{2n}=*W*(*z*),*z*_{2n}=*Z*(*z*) and terminates at *w*_{−1},*z*_{−1}. Moreover, based on equations (3.5) and (3.6), we easily establish
3.9which implies that the quantities {*l*_{j},*g*_{j}} are determined in each step from the large *z* asymptotics of terms known from the previous step. Indeed, if we denote by *a*^{(m)} the coefficient of *z*^{−m} in the expansion of a holomorphic function *a*(*z*) at , then we obtain the recovery formulae
3.10Thus, we proved that given a pair of Weyl functions *W*(*z*),*Z*(*z*) obtained from a cubic string problem (3.4) with a finite signed measure *g*, there exists a unique solution to the recurrence relations (3.7) subject to (3.9) and thus a unique cubic string corresponding to *W*(*z*),*Z*(*z*). □

We are now ready to state the main theorem of this section.

### Theorem 3.5

*Let* {*x*_{j}(*t*),*m*_{j}(*t*)}, *j*=1,…,*n be the positions and masses of the peakon ansatz* (*2.5*) *corresponding to an arbitrary signed measure* *satisfying peakon equations* (*2.6*) *on the time interval* (0,*t*_{c}) *and suppose that a collision occurs at t*_{c}. *Then, the positions x*_{1}(*t*)…,*x*_{n}(*t*) *are analytic functions at t*_{c}, *whereas the masses m*_{1}(*t*)…*m*_{n}(*t*) *are given by meromorphic functions at t*_{c}.

### Proof.

Given the initial conditions and {*m*_{1}(0),*m*_{2}(0),…,*m*_{n}(0)}, we set up the string problem (3.2a) after mapping *m*(0) to *g*(0). This produces the Weyl functions *W*(0),*Z*(0), which under the peakon flow evolve as entire functions of time in view of corollary 3.3. We then set up the recursive scheme (3.7) with *W*(*t*),*Z*(*t*) as inputs. At each stage of recursion, only rational operations are involved, and because the recursion is finite, the formulae (3.10) result in functions meromorphic in *t*. Thus, all *g*_{j},*y*_{j} are meromorphic in *t*. For *t*<*t*_{c}, all distances *l*_{j}>0 and at *t*_{c} some *l*_{i} vanishes but all *l*_{j} remain finite, because this is a finite string. Hence, *l*_{j}(*t*) is regular at *t*_{c} hence analytic there. For a signed measure *g*, there are no restrictions on individual *g*_{j}, so, in general, *g*_{j} remains meromorphic at *t*_{c}. Mapping back to the real axis is afforded by ; hence, positions of individual masses are given by . The only singular points of this map are for *y*_{j}=±1 which means the end of the string or, after mapping the problem back to the real axis, . However, based on results in reference [12], none of the masses can escape to in finite time. So, ((*y*_{j}+1)/(*y*_{j}−1) is in the domain of analyticity of and, hence, the *x*_{j}s are analytic at *t*_{c}. The relation between the measures *m* and *g* appearing in equations (2.3) and (3.2a) is given by which implies the claim, because *g*_{j} is meromorphic and *y*_{j} analytic. □

Theorem 3.5 establishes that the only singular points of solutions to the peakon ODE system (2.6a) and (2.6b) are poles. Because the inverse problem argument is valid for a fixed ordering *x*_{1}<*x*_{2}<⋯<*x*_{n} of masses, the analytic continuation of masses and positions into the complex domain in *t* will satisfy equations (2.6a) and (2.6b) in which are replaced with sgn(*k*−*i*), e^{−sgn(k−i)(xk−xi)}, respectively, to be consistent with the original ordering. It is for these equations that we note the *absence of movable critical points* also known as *Painlevé property* [24,25]. To facilitate the statement of the theorem 3.5. of this section, we set *X*_{i}=e^{xi}, 1≤*i*≤*n* and rewrite the system (2.6a) and (2.6b) in new variables {*m*_{i},*X*_{i}}.

### Theorem 3.6 (Painlevé property)

*The system of differential equations*
*has the Painlevé property.*

### Proof.

First, we observe (using the variables appearing in the proof of theorem 3.5) that *X*_{i}=((*y*_{i}+1)/(*y*_{i}−1)), hence *X*_{i} are meromorphic in *t* because so are *y*_{i}. The formulae for *X*_{i} and *m*_{i} obtained from the inverse problem are meromorphic in *t* in the complex plane **C** and depend on 2*n* constants (spectral data consisting, in the generic case, of *n* positions of poles and *n* residues of the Weyl function *W*), which for the cubic string problem, in view of the ordering condition, are confined to an open set in **R**^{2n} by continuity of the inverse spectral map. Relaxing that condition results in a solution depending on 2*n* arbitrary constants that comprises a general solution which is meromorphic in *t* in the whole complex plane **C**. □

### Remark 3.7

For an excellent tutorial on the Painlevé property and its ties to integrability, see Hone [26], where interested readers can also find an extensive collection of literature on the subject. In addition, it is worth mentioning that one of the topics discussed in Hone [26] is the weak Painlevé property of the full DP equation. This by no means contradicts theorem 3.6, which refers only to that specific system of ODEs, depending on the specific choice of variables.

In the remainder of the paper, we will apply theorem 3.6 to investigate the singularity structure arising at the time of collisions of peakons. We point out that our analysis addresses the singularity structure of the peakon ansatz *u*(*x*,*t*) in the *t* space. For analysis of singularity formation in the space variable *x* for more general initial data, the reader is referred to Coclite *et al.* [27].

## 4. Blow-up behaviour

We now proceed to establish several theorems on peakon collisions for DP equation. To begin with, we recall the definition of a peakon collision briefly discussed in the Introduction. We call *t*_{c} the *collision* time if there exists some *i* such that
4.1where *x*_{i}(*t*)s are the position functions in the ansatz (2.5). Equivalently, we say that the *i*th peakon collides with the (*i*+1)th peakon at the time *t*_{c}. If there exist more than two position functions being identical at *t*_{c}, then we will say that a *multiple collision* happens at *t*_{c}.

In this section, we describe the behaviour of the peakon dynamical system (2.6) in the neighbourhood of a collision time *t*_{c}.

To this end, we need to study a special skew-symmetric *n*×*n* real matrix *A*_{n} given by
4.2whose entries satisfy *a*_{ij}=*a*_{ji} and
4.3The following propositions hold for such a matrix.

### Lemma 4.1

*There exists a matrix P with* *such that P*^{T}*A*_{n}*P*=*B*_{n}, *where*
or

### Proof.

The conclusion is trivial for *n*=1,2. We assume the conclusion to hold for *n*−2; to show that it holds for *n* we divide *A*_{n} into four block submatrices
4.4where . Let us set , then a direct computation shows that
In view of condition (4.3), *B* can be written as (**a**,*a*_{n−1 n}**a**), where **a**=(*a*_{1 n−1},*a*_{2 n−1},…,*a*_{n−2 n−1})^{T}. It is now elementary to verify that *BC*^{−1}*B*^{T}=0. By the induction hypothesis, there exists a matrix *P*_{2} with such that , hence if we set
the conclusion follows. □

### Corollary 4.2

*If n*=2*k, then* . *If n*=2*k*+1, *then the rank of A*_{2k+1} *is* 2*k*.

### Lemma 4.3

*Let E*=(1,1,…,1),*n*=2*k*+1 *and all entries satisfy* 0<*a*_{ij}≤1. *Then, the rank of the matrix* *is* 2*k*+1.

### Proof.

Let *n* be any odd number. It suffices to show that the determinant of the matrix
is positive. For *n*=3, direct computation shows that
We assume now that the conclusion holds for *n*−2. We will show that it also holds for *n*. First, we divide into four submatrices by
where
and **b**=(*a*_{2 n−1},*a*_{3 n−1},…,*a*_{n−2 n−1})^{T}. Because *C* is invertible, we can factor into the product of upper and lower block triangular matrices as follows:
Hence, .

Direct computation shows that all the entries of vanish except the first row that equals , therefore
where we used that *n* is odd. Finally, in view of equation (4.3), we can replace the matrix in the second determinant by an upper triangular matrix by performing appropriate column additions, obtaining
□

By using the lemmas 4.1 and 4.3, we can obtain the property of *m*_{i}(*t*) at the time of blow-up.

### Theorem 4.4

*If m*_{i} *blows up at some t*_{0}, *then m*_{i} *has a pole of order 1 at t*_{0}.

### Proof.

Because *m*_{i}s are meromorphic in *t* we can assume that the leading term in the Laurent series of *m*_{i} around *t*_{0} is *C*_{i}/(*t*−*t*_{0})^{αi}, *C*_{i}≠0. If the conclusion does not hold, then
Set *S*={*i*_{j}:*α*_{ij}=*α*}={*i*_{1},…,*i*_{k}}, where *i*_{1}<*i*_{2}<⋯<*i*_{k} and *k* is at least 2 by virtue of lemma 2.1 with *p*=1. Comparing the leading term of both sides of (2.6b) with *i*_{j}∈*S*, one can see the leading term on the left-hand side is −*αC*_{ij}/(*t*−*t*_{0})^{α+1}, whereas the leading term on the right-hand side is
Because 2*α*>*α*+1, the coefficient of (*t*−*t*_{0})^{−2α} must be zero, which leads to a homogeneous linear equation *A*_{k}*C*=0, where *A*_{k}=(sgn(*j*−*l*)*a*_{jl}) is a *k*×*k* skew–symmetric matrix with *a*_{jl}=e^{xij−xil}(1<*j*<*l*) and *C*=(*C*_{i1},…,*C*_{ik})^{T}. Additionally, one can also find
by comparing the leading term in *M*_{1}. It is clear that *A*_{k} satisfies (4.3) and the condition in lemma 4.3. Hence, *C* must be zero according to corollary 4.2 and lemma 4.3, which leads to a contradiction. □

In the proof above, we use only the *m*_{i}(*t*)*s* that are meromorphic. However, we can get stronger conclusions if we also take into account that the *x*_{i}(*t*)*s* are holomorphic.

### Theorem 4.5

*If m*_{j1},…,*m*_{jk} (1≤*j*_{1}<⋯<*j*_{k}≤*n*) *blow up at t*_{c} *and all other m*_{i} *remain bounded, then the following conclusions hold*.

(

*1*)*k must be even*.(

*2*)*t*_{c}*must be a collision time. Moreover, for all odd l such that*1≤*l*<*k*,*the peakon with label j*_{l}*must collide with the peakon with label j*_{l+1}.(

*3*)*The leading term of m*_{js}(*t*)*in the Laurent series around t*_{c}*must have the form*(−1)^{s}/2(*t*−*t*_{c})*for all*1≤*s*≤*k*.

### Proof.

Assume that *m*_{j1},*m*_{j2},…,*m*_{jk} blow up at *t*_{c}. Because *M*_{1}=*m*_{1}+*m*_{2}+⋯+*m*_{n} is conserved, *k* is at least 2. Moreover, by theorem 4.4, the leading term in each *m*_{ij}'s Laurent series has the form *C*_{j}/(*t*−*t*_{c}). Hence, by equations (2.6b), the coefficients *C*=(*C*_{1},…,*C*_{k})^{T} satisfy the linear equations
4.5where the matrix satisfies (4.2) and (4.3). Likewise, comparing the leading terms of both sides of (2.6a) with the subscript *j*_{s} (1≤*s*≤*k*), one finds
4.6where the matrix . Now, we prove that the theorem holds for *k*=2. In this case, (4.5) and (4.6) reduce to
where *a*_{12}=e^{xj1−xj2}. Direct computation shows that the solution of the equations exists iff *a*_{12}=1, and the solution is . Because *a*_{12}=1 is equivalent to *x*_{j1}(*t*_{c})=*x*_{j2}(*t*_{c}), we conclude that the peakon with label *j*_{1} collides at *t*_{c} with the peakon with label *j*_{2}. Suppose now the conclusions are valid for *k*−2. We will show that they also hold for *k*. Let us use the same block decomposition as in equation (4.4), obtaining
and
where **a**=(*a*_{1 k−1},*a*_{2 k−1},…,*a*_{k−2 k−1})^{T}. Let us now combine the last two rows of (4.5) and (4.6), writing them collectively as
The latter expression can subsequently be easily reduced to
which implies the condition for the existence of the solution to be , hence *a*_{k−1 k}=1. The latter condition indicates the collision of *m*_{jk} with *m*_{jk−1}. Furthermore, the solution for the last two components of *C* is then and , which proves the sign statement for the last two components. Substituting into (4.5) and (4.6) and denoting the first *k*−2 components of *C* by **C**, we obtain the following equations:
The first two equations hold by the induction hypothesis. To show that the third equation holds automatically if the induction hypothesis is satisfied, we observe that as the result of collisions (*j*_{1}th mass collides with *j*_{2}th mass, etc.) **a**^{T}=(*a*_{1 k−1},*a*_{1 k−1},*a*_{3 k−1},*a*_{3 k−1},…,*a*_{k−3 k−1},*a*_{k−3 k−1}); hence, indeed, the last equation follows from the induction hypothesis. □

The following amplification of item 2 in the above theorem is automatic.

### Corollary 4.6

*Suppose m*_{j1},…,*m*_{jk} (1≤*j*_{1}<⋯<j_{k}≤*n*) *blow up at t*_{c} *and all other m*_{i} *remain bounded. Then, for all odd l such that* 1≤*l*<*k, the peakon with label j*_{l} *must collide at t*_{c} *with the peakon with label j*_{l}+1 (*its neighbour to the right*).

So far, we have established that when the masses become unbounded the collisions must occur. The converse also turns out to be valid.

### Theorem 4.7

*For all initial conditions for which M*_{n}≠0, *the following properties are valid:*

(

*1*)*If the ith peakon collides with another peakon/peakons at t*_{c},*m*_{i}(*t*)*must blow up at t*_{c}.(

*2*)*For all i, m*_{i}(*t*)*cannot change its sign. In particular, m*_{i}(*t*)≠0*for t*<*t*_{c}.(

*3*)*There are no multiple collisions*.(

*4*)*The distance between colliding peakons has a simple zero at t*_{c}.

### Proof.

Without loss of generality, we can suppose that the labelling is chosen so that *x*_{1}(*t*_{c})=⋯=*x*_{k1}(*t*_{c})<*x*_{k1+1}(*t*_{c})=*x*_{k1+2}(*t*_{c})=⋯=*x*_{k2}(*t*_{c})<⋯<*x*_{kl}(*t*_{c}) for all colliding peakons at distinct positions *x*_{k1}(*t*_{c})<*x*_{k2}(*t*_{c})<⋯<*x*_{kl}(*t*_{c}). Let us now denote the set indexing all colliding peakons by *I*. Because *M*_{n}=*m*_{1}*m*_{2}(1−e^{x1−x2})^{2}⋯*m*_{n−1}*m*_{n}(1− e^{xn−1−xn})^{2} is conserved and non-zero, it is clear that some of the masses must become unbounded. Let us denote the set of labels of those masses which blow up at *t*_{c} by *J*. By theorem 4.4, any such mass corresponds to a colliding peakon; thus, *J*⊂*I*. Moreover, any such a mass has a simple pole at *t*_{c}. On the other hand, for each pair of colliding peakons with adjacent indices *j* and *j*+1, *t*_{c} is a zero of (1−e^{xj−xj+1})^{2} of order bounded from below by 2. Thus, the order of the zero of all such exponential factors appearing in *M*_{n} is bounded from below by , where denotes the cardinality of *I*. Hence, because all unbounded masses have poles of order 1, to ensure that *M*_{n}≠0. The maximum of *l* occurs when the masses collide in pairs, hence and thus . This proves that *J*=*I* because *J*⊂*I* and thus (1) is proven. To prove (3), we return to the inequality above which now reads , implying . Because the right-hand side is the maximum of *l*, follows, which, in turn, implies that all collisions occur in pairs; hence we infer the absence of multiple collisions. To prove (4), we note that for *M*_{n} to remain bounded the order of the zero of all exponential factors has to be exactly ; hence each factor (1−e^{xj−xj+1})^{2} has a zero of order exactly equal 2.

This concluded the proof of (1), (3) and (4). In order to prove (2), we suppose that for some *i*, *m*_{i}(*t*) changes its sign, then there exists some *t*_{0} for which *m*_{i}(*t*_{0})=0 while all *m*_{j} remain bounded because *t*_{0}<*t*_{c}. Hence
This contradicts *M*_{n}≠0. □

### Remark 4.8

It is now not difficult to verify that the constants of motion *M*_{1},…,*M*_{n} can be extended up until the collision time *t*_{c} by using lemma 2.1, followed by theorems 4.5 and 4.7.

### Corollary 4.9

*If m*_{j} *and m*_{j+1} *collide at t*_{c}>0, *then m*_{j}>0 *and m*_{j+1}<0 *before the collision*.

### Proof.

Because collisions occur only in pairs, the leading terms in *m*_{j} and *m*_{j+1}'s Laurent series must be ∓1/2(*t*−*t*_{c}), respectively. This implies
The conclusion holds because in view of theorem 4.7 *m*_{j} and *m*_{j+1} cannot change their signs. □

The following proposition shows that the *simultaneous collisions* (several peakon–antipeakon pairs collide at distinct locations at the common time *t*_{c}) can happen. We indicate below how certain symmetric initial conditions will lead to simultaneous collisions. To this end, we consider equations (2.6) for *n*=4 and a special choice of initial conditions.

### Lemma 4.10

*If the initial conditions satisfy*
4.7a*and*
4.7b*then m*_{1}(*t*)=−*m*_{4}(*t*),*m*_{2}(*t*)=−*m*_{3}(*t*),*x*_{1}(*t*)−*x*_{2}(*t*)=*x*_{3}(*t*)−*x*_{4}(*t*) *will hold for all* 0<*t*<*t*_{c}.

### Proof.

Consider the following ODEs
then direct computation shows that {*x*_{1},*x*_{2},*x*_{3},*x*_{2}+*x*_{3}−*x*_{1},*m*_{1},*m*_{2},−*m*_{2},−*m*_{1}} satisfy the system of ODEs (2.6) for *n*=4. □

The following is then immediate (figure 1).

### Corollary 4.11

*If the initial conditions (4.7) hold and the peakon–antipeakon pair* (*m*_{1},*m*_{2}) *collides at t*_{c}, *then so does* (*m*_{3},*m*_{4}) *and vice versa*.

## 5. Collisions and shocks

In this section, we investigate the behaviour of *m* and *u* at the time of collision(s). We start with *m* and observe that because the collision of peakons occurs in pairs it is sufficient to study a fixed colliding pair *m*_{j},*m*_{j+1}.

### Theorem 5.1

*If m*_{j} *collides with m*_{j+1} *at time t*_{c}>0 *and the position x*_{c}, *then*
in .

### Proof.

For an arbitrary ,
Using corollary 4.9, we can write
around *t*_{c}. Hence
where in the last step we used equation (2.6a). The conclusion now follows from the definition of *δ* and *δ*′. □

Because *m*=*u*−*u*_{xx}, we have the immediate corollary.

### Corollary 5.2

*Suppose* *is a multi-peakon at t*=0 *for which M*_{n}≠0 *and such that at t*_{c} *one, or several of its peakon–antipeakon pairs collide. For any colliding pair k*,*k*+1, *let us denote* *respectively. Then*,
*The shock strengths are given by*
*and they satisfy the (strict) entropy condition* ([11]) *s*_{k}(*t*_{c})>0 .

### Proof.

It suffices to prove the claim if there is only one colliding pair; the general case follows easily, because masses collide pairwise. For *t*<*t*_{c}, the measure evolves as , where *x*_{k}(*t*),*m*_{k}(*t*) satisfy equations (2.6a) and (2.6b), respectively. Suppose now that the pair *j*,*j*+1 collides at the point *x*_{c}. Then, by theorem 5.1 To prove that *s*_{j}(*t*_{c})≥0, we write and observe
which implies the entropy condition *s*_{j}(*t*_{c})≥0 in view of the ordering assumption *x*_{j}(*t*)<*x*_{j+1}(*t*). The strict inequality follows from item (4) in theorem 4.7. □

The following amplification of the previous theorem brings the issues of the wave breakdown and a shock creation sharply into focus. To put our result into the proper perspective, we first briefly review the well-posedness result for proved by Coclite & Karlsen [13], section 3. We state only the core result pertinent to our paper, leaving other aspects, including the precise definition of the entropy condition, required for the stability and uniqueness of the DP equation, to an interested reader.

### Theorem 5.3 (Coclite–Karlsen)

*Let* . *Then, there exists a unique entropy weak solution to the Cauchy problem u*|_{t=0}=*u*_{0} *for the DP equation* (2.1).

It is then proven by Lundmark [11] that the shockpeakon ansatz
is an entropy-weak solution provided the shock strengths *s*_{j}≥0. This sets the stage for the next theorem. First, we will equip the space with the norm , where, in general, *V* _{f}(*U*) denotes the variation of *f* over an open subset .

### Theorem 5.4

*Given a multi-peakon solution u*(*x,t*) *defined on* *then*

(

*1*)*for all*0≤*t*<*t*_{c},(

*2*)*u*(⋅,*t*)*converges in*∥⋅∥_{L1}*to the shockpeakon**where the Laurent expansion of m*_{i}(*t*)*around t*_{c}*is written as**with the proviso that C*_{i}=0*if the ith mass is not involved in a collision and**for a colliding peakon*,*for a colliding antipeakon, respectively*.(

*3*)*as t*→*t*^{−}_{c}.

### Proof.

We start with the case *n*=2. Then, *u*(*x*,*t*)=*m*_{1}(*t*) e^{−|x−x1(t)|}+ *m*_{2}(*t*) e^{−|x−x2(t)|} and *x*_{1}(*t*_{c})=*x*_{2}(*t*_{c})=*x*_{c}. According to theorem 4.5, we have
where are analytic around *t*_{c}. It is clear that
By the mean value theorem, we find that
where 0<*θ*_{j}<1, *j*=1,2,3. Hence, we have the pointwise limit
Let us define
and consider the integral
Then, the first and the last term of the right-hand side converge to zero as owing to Lebesgue's dominated convergence theorem.

Observe that the second term satisfies
where *s*∈(*t*,*t*_{c}) and *y*∈(*x*_{1}(*t*),*x*_{2}(*t*)). Because and are bounded, and
as *t*→*t*^{−}_{c}, we have that *v*(*x*,*t*) converges to *v*(*x*,*t*_{c}) in the sense of *L*^{1}, which shows that the conclusion holds for *n*=2.

In general, because collisions can occur only in pairs, we can assume that *m*_{j1}(*t*),*m*_{j1+1}(*t*),*m*_{j2}(*t*),*m*_{j2+1}(*t*),…,*m*_{jk}(*t*),*m*_{jk+1}(*t*) blow up at *t*_{c} and all the other *m*_{i} remain bounded. It is clear that *m*_{i}(*t*) e^{−|x−xi(t)|} lies in and converges to *m*_{i}(*t*_{c}) e^{−|x−xi(tc)|} in *L*^{1} if *m*_{i}(*t*) remains bounded at *t*_{c}. Meanwhile, according to the proof above, we can easily see that
whose limit is
as *t*→*t*^{−}_{c}, which proves the *L*^{1} convergence. Let us now compute the variation *V* _{u(⋅,tc)}. Because *u*(*x*,*t*_{c}) is piecewise smooth, its distributional derivative, which is a Radon measure, equals , where means the classical derivative. One easily checks that the jump if the peakons *m*_{i} and *m*_{i+1} collided at *x*_{i}(*t*_{c}); otherwise, [*u*](*x*_{j}(*t*_{c}))=0. Thus, . We will now compute the total variation *V* _{u(⋅,t)}. For ease of notation, we will write *u* instead of *u*(⋅,*t*) in the remainder of the proof.

Let us denote , , while, as above, *x*_{j}=*x*_{j}(*t*), 1≤*j*≤*n*, denotes the positions of peakons. We then write the total variation . We will now compute the limit *t*→*t*^{−}_{c} for each *V* _{u}(*x*_{i},*x*_{i+1}). There are three cases to consider:

(

*1*)*V*_{u}(*x*_{0},*x*_{1}) and*V*_{u}(*x*_{n},*x*_{n+1}),(

*2*)*V*_{u}(*x*_{i},*x*_{i+1}) when*m*_{i}is not colliding with*m*_{i+1}, and(

*3*)*V*_{u}(*x*_{i},*x*_{i+1}) for a colliding peakon–antipeakon pair.

In all the three cases, as long as *t*<*t*_{c}, the distributional derivative *u*_{x} is the same as and we will drop the superscript from the notation; moreover, *u*_{x} is continuous on (*x*_{i},*x*_{i+1}) and bounded on [*x*_{i},*x*_{i+1}], hence implying . From the peakon ansatz, (2.5), *u*_{x}=±*u* on (*x*_{0},*x*_{1}) and (*x*_{n},*x*_{n+1}), whereas on (*x*_{i},*x*_{i+1}). Thus, in the first case, and an analogous result holds for *V* _{u}(*x*_{n},*x*_{n+1}). Because *u*(*x*,*t*) is exponentially bounded in *x* and continuous in *t*, we can take the limit obtaining .

Now, consider the interval (*x*_{i},*x*_{i+1}) for the non-colliding peakon pair *m*_{i} and *m*_{i+1}. The colliding peakon pairs are either in or , both of which remain bounded at *t*_{c}, because for colliding peakons expressions *m*_{j} e^{±xj}+*m*_{j+1} e^{±xj+1} have finite limits at *t*_{c}, implying that which is continuous on [*x*_{j}(*t*_{c}),*x*_{j+1}(*t*_{c})], implying that |*u*_{x}| is uniformly bounded, hence by Lebesgue's dominated convergence theorem .

Finally, in the third case, when *m*_{i} collides with *m*_{i+1} and the interval under consideration is [*x*_{i},*x*_{i+1}], we can split . Because there are no triple collisions, all the remaining colliding peakon pairs are in or , which are bounded, and hence these terms do not contribute in the limit *t*→*t*_{c} for which *x*_{i}(*t*_{c})=*x*_{i+1}(*t*_{c}). Thus, . The last limit can be computed explicitly using theorem 4.5 giving the final result
Thus, by taking the limit *t*→*t*^{−}_{c} in the sum we obtain , which concludes the proof. □

### Remark 5.5

Even though we showed the *L*^{1} convergence of a multi-peakon to a shockpeakon *u*(*x*,*t*_{c}) which is also in , as well as the convergence of respective variations, this does not imply the convergence in the BV topology. Indeed, consider any point *x*_{0} for which *u*(*x*_{0},*t*)→*u*(*x*_{0},*t*_{c}) and take another arbitrary point *x*. Denote *u*(*t*,*x*)−*u*(*t*_{c},*x*)=*f*(*t*,*x*). Then, |*f*(*t*,*x*)|≤|*f*(*t*,*x*_{0})|+ |*f*(*x*,*t*)−*f*(*x*_{0},*t*)|≤|*f*(*t*,*x*_{0})|+*V* _{f(⋅,t)}. Because *f*(*t*,*x*_{0}) goes to 0 as *t*→*t*^{−}_{c}, would imply not only the pointwise convergence of *f*(*t*,*x*) to 0, hence *u*(*x*,*t*)→*u*(*x*,*t*_{c}), but also the uniform convergence on any compact set, thus contradicting the existence of discontinuities at the points of collisions of peakons. This is an instance of the *L*^{1} topology providing more appropriate framework for the BV functions, which ultimately reinforces viewing as a subspace of ; for another, similar, case the reader might consult theorem 1 on p. 331, section 4.1, in Giaquinta *et al.* [28].

In the body of the proof of theorem 5.4, we identified the jump [*u*](*x*_{i}(*t*_{c}) to be equal *x*_{i+1}(*t*_{c})−*x*_{i}(*t*_{c}) at any collision point. Thus, *s*_{i} from Lundmark's shockpeakon ansatz equals and, as already indicated in corollary 5.2, *s*_{i}>0. This allows us to conclude with.

### Corollary 5.6

*The L*^{1} *limit of a multi-peakon u*(⋅,*t*) *as* *is a shockpeakon in the sense of Lundmark and as such it admits a unique entropy weak extension past the collision point t*_{c}.

## Funding statement

This work was supported by the National Natural Science Funds of China (NSFC11271285 to L.Z.) and the Natural Sciences and Engineering Research Council of Canada (NSERC163953 to J.S.).

## Acknowledgements

We thank Hans Lundmark for numerous perceptive comments. J.S. thanks the Centro Internacional de Ciencias (CIC) in Cuernavaca (Mexico) for hospitality and F. Calogero for making the stay so enjoyable and productive. We also thank the Department of Mathematics and Statistics of the University of Saskatchewan for making the collaboration possible.

- Received June 6, 2013.
- Accepted July 23, 2013.

- © 2013 The Author(s) Published by the Royal Society. All rights reserved.