Research Article  Open Access
Symmetry, Pulson Solution, and Conservation Laws of the HolmHone Equation
Abstract
In this paper, we focus on the HolmHone equation which is a fifthorder generalization of the CamassaHolm equation. It was shown that this equation is not integrable due to the nonexistence of a suitable Lagrangian or biHamiltonian structure and negative results from Painlevé analysis and the WahlquistEstabrook method. We mainly study its symmetry properties, travelling wave solutions, and conservation laws. The symmetry group and its onedimensional optimal system are given. Furthermore, preliminary classifications of its symmetry reductions are investigated. Also we derive some solitary pattern solutions and nonanalytic firstorder pulson solution via the ansatzbased method. Finally, some conservation laws for the fifthorder equation are presented.
1. Introduction
In the study of shallow water waves, Camassa and Holm derived a nonlinear dispersive shallow water wave equation [1]which is called CamassaHolm (CH) equation now. The here denotes the fluid velocity at time in the direction [2]. Eq. (1) admits biHamiltonian structure and it is completely integrable [3, 4]. What is more, the equation has infinite conservation laws [5] as well as a spike solitary wave solution where is an arbitrary constant. The solitary wave curve of the solution has a cusp at the peak, and the first derivative of the cusp is not continuous. Accordingly, it is called a peakon [6, 7].
With the further researches on the CH equation, a lot of findings about the equation have been obtained and it is impossible to give a comprehensive overview here. For example, Eq. (1) represents the equation for geodesics on the BottVirasoro group and owns the geometric interpretation [8]. The CH equation possesses both global solutions and solutions developing singularities in finite time and the blowup happens in a way which resembles wave breaking to some extent [9, 10]. The welldeveloped inverse scattering theory can also be used to integrate the CH flow [11]. In [12], it was shown that the wellknown CH equation is included in the negative order CH hierarchy and a class of new algebrogeometric solutions of the CH equation was presented. Moreover, the time evolution of travelingwave solutions and the interaction of peaked and cusped waves were numerically studied [13].
Recently, one of the latest trends is that researchers are trying to generalize the CH equation to higher order, which is also the subject of this paper. In fact, some higherorder CH equations are well considered. For example, Wazwaz studied the nonlinear fourthorder dispersive variants of the generalized CH equation by using sinecosine method [14]. The existence of global weak solutions was established for a higherorder CH equation describing exponential curves of the manifold of smooth orientationpreserving diffeomorphisms of the unit circle in the plane [15]. Nonsmooth travelling wave solutions of a generalized fourthorder nonlinear CH equation were studied in [16]. In [17], the CH model was extended to fifth order and some interesting solutions were obtained including explicit single pseudopeakons, twopeakon, and peakon solutions.
Especially, Holm and Hone discussed a fifthorder partial differential equation (PDE)which is a generalization of the integrable CH equation [18]. We will call it the HolmHone equation in the following. This fifthorder PDE admits exact solutions in terms of an arbitrary number of superposed pulsons. The pulsons of Eq. (3) are weak solutions with discontinuous second derivatives at isolated points. Numerical simulations show that the pulsons are stable and they dominate the initial value problem and scatter elastically [19]. These characteristics are reminiscent of solitons in integrable systems. However, after demonstrating the nonexistence of a suitable Lagrangian or biHamiltonian structure [20] and obtaining negative results from Painlevé analysis [21] and the WahlquistEstabrook method [22], they asserted that the HolmHone equation is not integrable. Consequently, most effective methods for integrable systems are not applicable to this equation. However, the Lie symmetry approach originated from Norwegian mathematician Sophus Lie [23] systematically unifies and extends wellknown ad hoc techniques to construct explicit solutions for differential equations, no matter the equations are integrable or not.
Actually, there are several important applications of symmetry analysis in the investigation of differential equations [24–27]. Since some solutions of partial differential equations asymptotically tend to solutions of lowerdimensional equations obtained by symmetry reduction, some of these special solutions illustrate important physical phenomena. In particular, the exact solutions arising from symmetry methods can often be used effectively to study properties such as asymptotics and blowup. Besides, the explicit solutions found by symmetry methods can play important roles in the design and testing of numerical integrators, and these solutions provide an important practical check on the accuracy and reliability of such integrators.
This paper is organized as follows. We first present the common form of the infinitesimal generators of the HolmHone equation. Then we establish the optimal system of onedimensional subalgebras and consider the similarity reductions of the equation. Furthermore, the travelling wave solutions are investigated. At last, some conservation laws of the equation are also derived.
2. Lie Symmetries for the HolmHone Equation
In this section, we investigate the Lie symmetries and similarity reductions of the HolmHone equation through the classical methods. The infinitesimal generators corresponding to the oneparameter transformation group are presented. Furthermore, the onedimensional optimal system of the group is derived with the help of the adjoint representation among the vector fields. Finally, the reduced equations are given through the similarity transformation.
2.1. Infinitesimal Generators
We introduce the infinitesimal form of the single parameter transformation groupwhere is the infinitesimal group parameter and , , are the infinitesimals of the transformation for the independent and dependent variables, respectively. The vector field associated with the above group of transformations can be written as Using the software package GeM [28], we obtainwhere , , are arbitrary constants. The infinitesimal generators of the corresponding Lie algebra are given by
In order to obtain the group transformation which is generated by the infinitesimal generators for , we need to solve the firstorder ordinary differential equations Exponentiating the infinitesimal symmetries we get the oneparameter groups generated by for as Accordingly, if is a solution of the HolmHone equation, so are the functions
2.2. OneDimensional Optimal System
In general, the Lie group has infinite subgroups; it is not usually feasible to list all possible group invariant solutions to the system. We need an effective systematic approach to classify these solutions, so the optimal 1dimensional subalgebras of group invariant solutions can be derived. In this section, the optimal system is obtained by computing the adjoint representation of the vector fields [25]. We use the Lie series where is the commutator for the Lie algebra, is a parameter, and . The commutator table of the Lie point symmetries for Eq. (3) and the adjoint representation of the symmetry group on its Lie algebra are presented in Tables 1 and 2, respectively.

Given a nonzero vector our task is to simplify as many of the coefficients as possible through judicious applications of adjoint maps to .
Firstly, we suppose that . Scaling if necessary, we can assume that . Referring to Table 2, if we act on such a by , we can make the coefficient of vanish and the coefficients of cannot be eliminated further, so we can make the coefficient of either , , or . Thus any onedimensional subalgebra spanned by with is equivalent to one spanned by either , , or .
The remaining onedimensional subalgebras are spanned by vectors of the above form with . If , we scale to make , and then no coefficient vanishes by any action on , so that is equivalent to .
The remaining case, , is equivalent to . And it is impossible to make further simplification.
Until now, we have found the optimal system of onedimensional subalgebras spanned by
The list can be reduced slightly if we admit the discrete symmetry , which maps to , and thereby the number of inequivalent subalgebras is reduced to four.
2.3. Similarity Reductions
The HolmHone equation is expressed in the coordinates , so we try to reduce this equation in order to search for its form in specific coordinates which can be constructed by solving the characteristic equationBased on the optimal system presented before, we obtain the following four kinds of reductions:
Reduction 1. For , we get the reduction
Reduction 2. For , we get the traveling wave reduction , , where satisfies
Reduction 3. For , we get the reduction , where satisfies
Reduction 4. For , we get the reduction , where satisfies
3. Travelling Wave Solutions of the HolmHone Equation
The appearance of nonanalytic peakontype solutions has increased the menagerie of solutions appearing in nonlinear partial differential equations, both integrable and nonintegrable. The pulson solutions for Eq. (3) have a finite jump in second derivative of the solutions. In a series of papers, Wazwaz proposed some schemes to determine new sets of soliton solutions, in addition to the peakon solutions obtained before, for the family of CH equations [29, 30]. The method rests mainly on some ansatzes that use one hyperbolic function or combine two hyperbolic functions as follows:
(1) A sinhcosh Ansatz I
(2) A sinhcosh Ansatz II
(3) A tanh Ansatz or a coth Ansatz or
(4) The Exponential Peakon Ansatz where , and are parameters that will be determined. By this method, solitons, solitary patterns solutions, periodic solutions, compactons, and peakons solutions for a family of CH equations with distinct parameters are obtained. However, we should modify the first ansatz and combine it into the second one since and . In what follows, we try to describe some specific travelling wave solutions for the HolmHone equation via this modified method.
Firstly, we obtain the travelling wave solutions with or and are arbitrary constants. Moreover, it can be easily verified that there is no effective result for the third ansatz for the HolmHone equation. And in order to take the last exponential peakon ansatz, we are advised to rewrite Eq. (3) aswith
Supposing that , then Eq. (24) becomeswhereLet us find all possible nonconstant solutions satisfying and . Since , the corresponding characteristic equation is . Then the characteristic values are and , which yield It is easy to see that if and only if Nevertheless, when Then the solution can be reduced to Considering the continuity of solution at , we look for a solution of the following form:where , are arbitrary constants. Next we explore the relationship between the coefficients and . Substituting Eq. (32) into Eq. (27) and using the property we get the travelling wave solution of the original equation This solution is called the firstorder pulson solution which is different from the peakons of CH equation. Its first derivative is continuous and the second derivative of the cusp is not continuous.
4. Conservation Laws for the HolmHone Equation
Conservation laws are widely applied in the analysis of PDEs, particularly in the study of existence, uniqueness, and stability of solutions. The concept of conservation laws and the relationship between symmetries and conservation laws arise in a wide variety of applications and contexts [25, 26].
A conservation law for (3) is of the form where and denote the total derivatives as and the subscripts denote partial derivatives. The vector is a conserved vector for the partial differential equation. It was shown that Eq. (3) has the conservation law where and it does not own biHamiltonian structure [18]. In this section, we mainly solve the conservation laws of the HolmHone equation by multiplier method [31–33].
A multiplier has the property thatfor all solutions . Generally speaking, each multiplier is a function as , where denotes all th order derivatives of with respect to all independent variables . Here we only consider multipliers of the form , although multipliers which depend on the firstorder and higherorder partial derivatives of could also be considered, but the calculations become more complicated and we fail to find any result.
The righthand side of (38) is a divergence expression which leads to the determining equation for the multiplier aswhere is the standard Euler operator.
From the system (39), we can obtain the solutionwhere and are arbitrary constants. Therefore we get that any conserved vector of the HolmHone equation with multiplier of the form is a linear combination of the two conserved vectors
5. Conclusions
In summary, we have found the most general Lie point symmetries group for the nonintegrable HolmHone equation which is a fifthorder generalization of the CH equation. Meanwhile, we constructed the optimal system of onedimensional subalgebras. Afterwards, we created the preliminary classifications of similarity reductions. The Lie invariants and similarity reduced equations corresponding to infinitesimal symmetries have been obtained. In order to obtain the traveling wave solutions of the equation, we adopted the method of ansatz. We also found some conservation laws from the multiplier method.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work is supported by the 13th FiveYear National Key Research and Development Program of China with Grant No. 2016YFC0401407.
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Copyright © 2019 Guo Wang et al. 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.