Equations with Peakon Solutions in the Negative Order Camassa-Holm Hierarchy

The negative order Camassa-Holm (CH) hierarchy consists of nonlinear evolution equations associated with the CH spectral problem. In this paper, we show that all the negative order CH equations admit peakon solutions; the Lax pair of the N-order CH equation given by the hierarchy is compatible with its peakon solutions. Special peakon-antipeakon solutions for equations of orders −3 and −4 are obtained. Indeed, for N ≤ −2, the peakons of N-order CH equation can be constructed explicitly by the inverse scattering approach using Stieltjes continued fractions.The properties of peakons forN-order CH equation whenN is odd are much different from the CH peakons; we present the case N = −3 as an example.


Introduction
The Camassa-Holm (CH) equation [1,2], where (, ) may be interpreted as a horizontal fluid velocity, retains higher order terms of derivatives in a small amplitude expansion of incompressible Euler's equations for unidirectional motion of waves at the free surface under the influence of gravity.The CH equation can also arise in the modeling of the propagation of shallow water waves over a flat bed [3], capturing stronger nonlinear effects than the classical nonlinear dispersive Benjamin-Bona-Mahoney and Korteweg-de Vries equations, in particular, tests ideas about wave breaking [3][4][5].Mathematically, the CH equation possesses bi-Hamiltonian structure, Lax pair, and peaked soliton solutions (peakons), which were described by a finite dimensional Hamiltonian system [1,2], the integrability of which was established in the framework of the -matrix approach [6] and the explicit multipeakons were expressed in terms of the orthogonal polynomials associated with classical moment problem [4].Since the rediscovery by Camassa and Holm in 1993, a large number of studies related to CH equation have been developed; see [7][8][9][10][11][12] and references therein for other topics.We remark that the peakon interactions were a key ingredient in the development of the theory of continuation after blow-up of global weak solutions of the CH equation, as developed in the papers [13][14][15].
In this paper, we consider equations in the negative CH hierarchy associated with the CH spectral problem with the aim of obtaining more peakon equations and properties of the negative -order CH equation.Equation ( 1) can be obtained 2 Advances in Mathematical Physics as the compatibility condition for an overdetermined system [1,4]   = ( + 1) , Starting from the first equation in (2), one can derive integrable hierarchies of nonlinear evolution equations, which contain the well-known Dym-type equation and CH equation [7,8,12].Qiao [12] derived the negative order CH hierarchy and the positive order CH hierarchies via the spectral gradient method, and Alber et al. [7,8] gave the hierarchy by the method of generating equations, both of which based on the assumption that the potential  is a smooth function.
We extend the differential operator  = −(  +   ) to  ∞ (R) ∩ (R), the space of continuous and piecewise smooth functions on R, where  is a discrete measure, and show that all negative order CH equations admit peakons of the form Besides, we show that the Lax integrability is preserved in the peakon case; some constants of motion for the peakon dynamical systems of -order CH equations are obtained.The remainder of this paper is organized as follows.In Section 2, we describe the negative order CH hierarchy using the method of finite power expansion with respect to spectral parameter for the purpose of this paper.In Section 3, we derive the negative order CH hierarchy for discrete potential.In Section 4, we prove that the equations of orders −3, −4, . . .admit multipeakons and the Lax pair of the order CH equation given by the negative order CH hierarchy is compatible with its peakons.In Section 5, we give some examples for peakon solutions of the −3-order equation and the −4-order equation.In Section 6, we give some remarks on the work of this paper and that in [34,35].

Negative Order Camassa-Holm Hierarchy
To make a self-contained discussion on equations in the negative order CH hierarchy, we now make a description for the negative order CH hierarchy in the following way.
Consider the CH spectral problem with potential  and spectral parameter .The equation above is equivalent to where Φ = (,   )  .
The negative order CH hierarchy is the following linear system of differential equations where The compatibility condition of ( 6) is   −   + [, ] = 0, which should hold for all .Equating like powers of , we obtain and the evolution of matrix Solving the recursion (8), we will obtain the evolution of  from (9).Let Formally, we have where  −1  is an integral operator.Set L =  −1 , then Remark 1.In general, the differential operator  is not invertible; (12) just gives an integrodifferential operator formally.In Section 3, we shall see that L −1 is actually an operator on the function space  ∞ (R) ∩ (R) when  is a finite discrete measure.
Remark 4. Choosing other  −1 ∈ Ker , for example,  −1 =  2 , the first equation in our negative order CH hierarchy is an integrodifferential equation; one can see examples in [12].

Hierarchy Associated with Discrete Potential
In the description of the negative order CH hierarchy, we have tacitly assumed that the potential  in ( 5) is a smooth function.In the remainder, we suppose that  is a finite discrete measure given by where    fl ( −   ) is a Dirac delta distribution supported at the point   .In this case, to define   ( < −2) from ( 14), distributional calculus will be needed, we take derivatives as distributional derivatives, and   will be used to denote distributional derivative with respect to .Besides, we extend the definition of as follows: where ⟨  (  )⟩ is the average of  at   .As we shall see in Section 4, (19) makes the distributional Lax pair of the order CH equation compatible with its peakons.
The distributional derivatives we need in this paper can be calculated from the following lemma on piecewise smooth functions.
In this section, we show that   ( < −2) can be defined by formula (14) recursively for  given by (18).
The following proposition gives the well-defined  −2 .
Denote the unique solution to ( 21) and ( 22) by  −2 , and with Lemma 5 and ( 19), we have the following theorem.
Theorem 7.For  given by (18),   ( < −2) is well defined by (14), and Proof.According to formula (14),   =  +1 ; in the distribution sense, we have the following equation formally: Thus, with the definition (19) of ,   can be defined recursively.We now prove this theorem by induction.First, Proposition 6 shows that  −2 is a continuous function in  ∞ (R).Consider the following problem: where  is given by (18).With (19) and Lemma 5 at hand, (31) is equivalent to Therefore,  ∈  ∞ (R) satisfies (31) if and only if −(1/2)  +2  = 0 when  ̸ =   and Advances in Mathematical Physics 5 Hence, where   ,   ,   satisfy The asymptotic condition (32) shows that  0 =  0 = 0 and Thus, (36) defines a function  ∈  ∞ (R) satisfying the asymptotic condition; using the first condition in (34), we can define a unique continuous function f as we have done in the proof of Proposition 6.Thus,  −3 is well defined by ( 14) and  −3 ∈  ∞ (R) ∩ (R); we have so far proved the conclusion for the case  = −3.

Peakons and Lax Integrability
The -order CH equation can be written formally as where  = − −2 and   given by ( 14).The −2-order equation, that is, CH equation ( 1), admits peakons (3), where   ,   obey the following Hamiltonian system: In Section 3, we define   by ( 14) for discrete potential; based on this, we have the following theorem.
Theorem 9.The -order CH equation given by Definition 2 admits peakons taking the form (3).
Proof.Substituting (3) into the second equation in (40), we obtain By Proposition 6 and Theorem 7,  above makes   a function in  ∞ (R) ∩ (R), which is expressed by elementary functions composed of power functions and exponential functions.In the sense of distribution, the -order CH equation ( 40) is equivalent to the following dynamic system: for  = 1, . . ., .Hence, for the initial condition (43) has a unique solution locally, which completes the proof.
We shall see that the Lax pair of the -order CH equation (40) given by ( 6) is compatible with its peakons.Recall that for the case  = −2 the same result had been used in [4]; we will prove the compatibility just for the case  < −2.
In the remainder of this section,   denotes the distributional derivatives in , the subscripts of functions denoting the usual partial derivatives, and for simplicity, we will write ∑ instead of ∑  =1 .We first present a lemma on the calculus of piecewise smooth function.
Theorem 11.With  and  given by ( 3) and ( 42) and   defined by ( 14) and (19), the Lax pair (47) satisfies the compatibility condition:     Φ =     Φ if and only if Proof.Computing the distributional derivatives of Φ, comparing the coefficients of    and the regular parts in (47) leads to Next we compute     Φ and     Φ; using (50) we have Thus,     Φ =     Φ is equivalent to Computing [Φ(  )] by using Lemma 10 and relations (50) and ( 46), we have that (53) is equivalent to The second component of Φ, that is,   , has jump discontinuities at   .By (50) and (51), we have Therefore, The first row of (56) holds naturally for that the first element of Φ is zero; by the continuity of  at the point   , the second row holds if and only if ṁ = ⟨ , (  )⟩  , which completes the proof.
Theorem 11 implies that the Lax pair of the -order CH equation is compatible with its peakons when extending the definition of  by (19).Using this result, we can also obtain some constants of motion for the peakon ODEs (43).

Examples of Peakon Solutions
In this section, we present some special peakon solutions for the -order CH equation in the cases  = −3, −4 by integrating the ODEs (43).
Remark 3. Note that  1 −  2 ≡  2 < 0 for all  ∈ R; this peakon-antipeakon pair can not collide; that is, (70) gives a peakon-antipeakon solution globally.Besides, (70) is a superposition of two traveling waves (with constant amplitudes and the same constant speeds), which is an unusual feature for two-peakon solutions compared with the CH equation ( 1).
For the case  = −4, the peakon ODEs (43) can be written as where  is an arbitrary nonzero constant.Thus, the −4-order CH equation admits the peakon solution (73).