Advances in Mathematical Physics
Volume 2018 (2018), Article ID 1679625, 11 pages
https://doi.org/10.1155/2018/1679625

Research Article

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

Fengfeng Dong and Lingjun Zhou

School of Mathematical Sciences, Tongji University, Shanghai 200092, China

Correspondence should be addressed to Fengfeng Dong; 1433494@tongji.edu.cn

Received 26 December 2017; Accepted 1 February 2018; Published 28 February 2018

Academic Editor: Andrei D. Mironov

Copyright © 2018 Fengfeng Dong and Lingjun Zhou. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

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.

1. 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 [3–5]. 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 [7–12] 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 [33] 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 [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 , 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 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 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 to

where

The negative order CH hierarchy is the following linear system of differential equations

where

The compatibility condition of (6) is , 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 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 that

where the superscripts denote the position of the entry in the matrix; that is, is the entry of .

Let , then . Define the following recursion sequence (called Lenard’s sequence in [12]):

Substituting (14) into (13) and (9), we have

and the nonlinear evolution equation

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 3. Note that

, and the -order CH equation is given by (16) with . The -order equation is (1), where by setting . For , the -order CH equation is an integrodifferential equation in general.

Remark 4. Choosing other , for example, , the first equation in our negative order CH hierarchy is an integrodifferential equation; one can see examples in [12].

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 by

where 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 , .

In this section, we show that can be defined by formula (14) recursively for given by (18).

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 and

Thus, 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.

Denote the unique solution to (21) and (22) by , and with Lemma 5 and (19), we have the following theorem.

Theorem 7. For given by (18),    is well defined by (14), and .

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 and

Hence,

where satisfy

The 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

The -order CH equation can be written formally as

where and given by (14). The -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 , 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 [4]; 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:

where

Note that and when ; it is easy to see that .

Theorem 11. With and given by (3) and (42) and defined by (14) and (19), the Lax pair (47) satisfies the compatibility condition: if and only if

Proof. Computing the distributional derivatives of , comparing the coefficients of and the regular parts in (47) leads to

Next we compute and ; using (50) we have

Thus, is equivalent to

Computing 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.

Remark 12. Replacing in (47) by or , the compatibility condition of (47) is also (43).

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 to

When is given by (42), the spatial part leads to

Define , , then

where

Taking derivative of with respect to , using , the second equation in (57), and (58), we obtain certain matrix such that

Since with is single-pair matrix and is oscillatory [36], 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.

Remark 14. When , , particularly, when , we have . Thus, for any -order equation, the corresponding two-peakon ODEs have a constant of motion: , which equals the Hamiltonian of CH peakon ODEs [1, 4].

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 1. When , (64) is simplified as

Imposing the initial condition , , we have

where is an arbitrary nonzero constant. Thus, (63) admits the peakon solution (66).

Example 2. When , for , in the neighborhood of , (64) is simplified as

Given 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 4. When , (71) is simplified as

Imposing the initial condition , , we have

where is an arbitrary nonzero constant. Thus, the -order CH equation admits the peakon solution (73).

Example 5. When , let ; in the neighborhood of , (71) is simplified as

Let

Obviously, is a constant of motion for system (74); for , we have that ; then ; thus (74) admits the following solution:

where are arbitrary constants. Exchanging the place of and , we can obtain another solution to (76). Hence, the -order CH equation admits symmetric peakon-antipeakon solution

where , are given by (76) and can be exchanged.

Remark 6. It is easy to find that peakon-antipeakon given by (77) will collide at the position as . This phenomenon is similar to the CH equation (1).

6. Concluding Remarks

In this paper, we extend the negative order CH hierarchy to the case where the potential is a finite discrete measure, showing that the -order CH equation admits peakons and the peakons are compatible with corresponding Lax pairs obtained from negative order CH hierarchy. We have proved the existence of multipeakons of (40), and peakon-antipeakon solutions for - and -order CH equation were given.

We would like to remark that, due to Remark 12, the peakon ODEs (43) can be solved explicitly by the inverse scattering approach using Stieltjes continued fractions, and the process is similar to that in [4]. Specifically, the -order CH equation admits peakon solutions (3) with

where

, , , and

When is even, the properties of peakons of the -order CH equation are similar to the CH equation [4], while for the odd cases, there are some new features (see Example 2 and Remark 3 for peakon-antipeakon pair); further studies on multipeakons given above will be needed.

The CH equation (1) can be viewed as isospectral deformation of the string equation [4, 34]. Recently, we found the work [34, 35] of Szmigielski et al., in which they studied the isospectral deformation of mass density of the classical string problem. The -order CH equation (16) we derived using method of finite power expansion with respect to spectral parameter may be interpreted as limit of small parameter; one can see [35, Example ] for details, where vanishing condition was imposed on associated spectral problem in their setting, which is implied by the fact that holds for ; see Remark 13.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This work was supported by the National Natural Science Foundation of China NSFC11301398 and the Fundamental Research Funds for the Central Universities.

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