Research Article | Open Access
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 -order CH equation given by the hierarchy is compatible with its peakon solutions. Special peakon-antipeakon solutions for equations of orders and are obtained. Indeed, for , the peakons of -order CH equation can be constructed explicitly by the inverse scattering approach using Stieltjes continued fractions. The properties of peakons for -order CH equation when is odd are much different from the CH peakons; we present the case as an example.
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 , capturing stronger nonlinear effects than the classical nonlinear dispersive Benjamin-Bona-Mahoney and Korteweg-de Vries equations, in particular, tests ideas about wave breaking [3–5]. 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  and the explicit multipeakons were expressed in terms of the orthogonal polynomials associated with classical moment problem . Since the rediscovery by Camassa and Holm in 1993, a large number of studies related to CH equation have been developed; see [7–12] 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 [13–15].
In the past two decades, peakons have become a hot subject in the field of integrable system and some relevant branches of mathematics. Some other partial differential equations admitting peakons have been reported [16–21]; these equations introduced many challenging problems, including existence, uniqueness, stability, and breakdown of solutions, for part of which one can see [22–30]. We recall that the CH (periodic) peakons [31, 32] and Degasperis-Procesi peakons  are stable in the sense that their shape is stable under small perturbations, which shows the peakons are detectable.
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 as the compatibility condition for an overdetermined system [1, 4]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  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 , the space of continuous and piecewise smooth functions on , where is a discrete measure, and show that all negative order CH equations admit peakons of the formBesides, 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 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 -order equation and the -order equation. In Section 6, we give some remarks on the work of this paper and that in [34, 35].
2. 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 towhere
The negative order CH hierarchy is the following linear system of differential equationswhereThe compatibility condition of (6) is , which should hold for all . Equating like powers of , we obtainand the evolution of matrix Solving the recursion (8), we will obtain the evolution of from (9).
LetFormally, we havewhere is an integral operator. Set , 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 is actually an operator on the function space when is a finite discrete measure.
In terms of the components, (8) shows thatwhere the superscripts denote the position of the entry in the matrix; that is, is the entry of .
With the discussion above, we can define the -order CH equation as follows.
Definition 2. The -order CH equation is the zero curvature equation , where and is a -valued Laurent polynomial of with the lowest degree term and the highest degree term (i.e., -term) with nonvanishing entry.
Remark 4. Choosing other , for example, , the first equation in our negative order CH hierarchy is an integrodifferential equation; one can see examples in .
3. 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 bywhere is a Dirac delta distribution supported at the point . In this case, to define 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 to 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.
Lemma 5. Suppose is a piecewise smooth function and has discontinuities at , then we have where and are ordinary partial derivatives; denotes the jump of at .
In the remainder, we take the convention that , .
The following proposition gives the well-defined .
Proposition 6. For given by (18), there exists a unique continuous function in solving the following problem:
Proof. According to Lemma 5, (21) is equivalent to Therefore, satisfies (21) if and only if when andThus, we can set A straightforward calculation translates (24) into the following relation:The asymptotic condition (22) shows that , , and substituting into (26) yields Hence, Define a function on by by the first condition in (24), , which completes the proof.
Proof. According to formula (14), ; 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 is a continuous function in . Consider the following problem:where is given by (18). With (19) and Lemma 5 at hand, (31) is equivalent to Therefore, satisfies (31) if and only if when andHence, where satisfyThe asymptotic condition (32) shows that and Thus, (36) defines a function satisfying the asymptotic condition; using the first condition in (34), we can define a unique continuous function as we have done in the proof of Proposition 6. Thus, is well defined by (14) and ; we have so far proved the conclusion for the case .
Next, we assume that is well defined by (14) and . Consider the following problem:where is given by (18). Similarly, we can obtain a unique continuous function in solving (38) and (39).
By induction, for any , is well defined by (14) and , which completes the proof.
Remark 8. In Proposition 6 and the proof of Theorem 7, we have chosen different exponential decay conditions at negative infinity for different ; however, we can replace them by boundary condition: ; see Remark 13.
4. Peakons and Lax Integrability
Proof. Substituting (3) into the second equation in (40), we obtainBy Proposition 6 and Theorem 7, above makes a function in , 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 . 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 the same result had been used in ; we will prove the compatibility just for the case .
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 . We first present a lemma on the calculus of piecewise smooth function.
Lemma 10. Given smooth functions , suppose and are piecewise smooth functions with jumps at , then
For peakon solutions (3), is given by (42). By Proposition 6 and Theorem 7, given by (14) are continuous functions in and ,The derivatives in (15) should be replaced by corresponding distributional derivatives. Thus, (6) can be rewritten as the following well-defined distributional Lax pair:whereNote that and when ; it is easy to see that .
Proof. Computing the distributional derivatives of , comparing the coefficients of and the regular parts in (47) leads toNext we compute and ; using (50) we have Thus, is equivalent toComputing by using Lemma 10 and relations (50) and (46), we have that (53) is equivalent to The second component of , 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 is zero; by the continuity of at the point , the second row holds if and only if , 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).
The Lax pair for the -order CH equation (40) given by the negative order CH hierarchy is equivalent toWhen is given by (42), the spatial part leads toDefine , , thenwhereTaking derivative of with respect to , using , the second equation in (57), and (58), we obtain certain matrix such thatSince with is single-pair matrix and is oscillatory , so is nonsingular. Hence, the compatibility condition of (59) and (61) leads to , where is arbitrary positive integer. Therefore, is a constant of motion for the peakon ODEs of the -order CH equation.
Remark 13. When , , for any , the peakon ODEs (43) has a constant of motion: ; therefore,By (62), for given by (18) and , there is a unique defined by (14) and approaches to zero as . Indeed, only minor modifications are needed. Replacing (22), (32), and (39) by , which is equivalent to , then the condition (62) implies the existence and uniqueness of corresponding boundary problems, and defined in Section 3 are exactly the solutions to the modified problems.
5. Examples of Peakon Solutions
In this section, we present some special peakon solutions for the -order CH equation in the cases by integrating the ODEs (43).
The -order CH equation can be written as the following integrodifferential equation:For , the ODEs (43) can be written as
Example 2. When , for , in the neighborhood of , (64) is simplified asGiven the initial condition, Note that is a constant of motion for system (67); thus, (67) admits the following solution:where , , are constants and , . Therefore, the -order CH equation (63) admits the following peakon-antipeakon solution:where , are given by (69).
Remark 3. Note that for all ; 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 , the peakon ODEs (43) can be written as
Example 5. When , let ; in the neighborhood of , (71) is simplified as