Research Article | Open Access
Symmetry, Pulson Solution, and Conservation Laws of the Holm-Hone Equation
In this paper, we focus on the Holm-Hone equation which is a fifth-order generalization of the Camassa-Holm equation. It was shown that this equation is not integrable due to the nonexistence of a suitable Lagrangian or bi-Hamiltonian structure and negative results from Painlevé analysis and the Wahlquist-Estabrook method. We mainly study its symmetry properties, travelling wave solutions, and conservation laws. The symmetry group and its one-dimensional optimal system are given. Furthermore, preliminary classifications of its symmetry reductions are investigated. Also we derive some solitary pattern solutions and nonanalytic first-order pulson solution via the ansatz-based method. Finally, some conservation laws for the fifth-order equation are presented.
In the study of shallow water waves, Camassa and Holm derived a nonlinear dispersive shallow water wave equation which is called Camassa-Holm (CH) equation now. The here denotes the fluid velocity at time in the direction . Eq. (1) admits bi-Hamiltonian structure and it is completely integrable [3, 4]. What is more, the equation has infinite conservation laws  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 Bott-Virasoro group and owns the geometric interpretation . The CH equation possesses both global solutions and solutions developing singularities in finite time and the blow-up happens in a way which resembles wave breaking to some extent [9, 10]. The well-developed inverse scattering theory can also be used to integrate the CH flow . In , it was shown that the well-known CH equation is included in the negative order CH hierarchy and a class of new algebro-geometric solutions of the CH equation was presented. Moreover, the time evolution of traveling-wave solutions and the interaction of peaked and cusped waves were numerically studied .
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 higher-order CH equations are well considered. For example, Wazwaz studied the nonlinear fourth-order dispersive variants of the generalized CH equation by using sine-cosine method . The existence of global weak solutions was established for a higher-order CH equation describing exponential curves of the manifold of smooth orientation-preserving diffeomorphisms of the unit circle in the plane . Nonsmooth travelling wave solutions of a generalized fourth-order nonlinear CH equation were studied in . In , the CH model was extended to fifth order and some interesting solutions were obtained including explicit single pseudo-peakons, two-peakon, and -peakon solutions.
Especially, Holm and Hone discussed a fifth-order partial differential equation (PDE)which is a generalization of the integrable CH equation . We will call it the Holm-Hone equation in the following. This fifth-order 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 . These characteristics are reminiscent of solitons in integrable systems. However, after demonstrating the nonexistence of a suitable Lagrangian or bi-Hamiltonian structure  and obtaining negative results from Painlevé analysis  and the Wahlquist-Estabrook method , they asserted that the Holm-Hone 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  systematically unifies and extends well-known 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 lower-dimensional 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 blow-up. 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 Holm-Hone equation. Then we establish the optimal system of one-dimensional 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 Holm-Hone Equation
In this section, we investigate the Lie symmetries and similarity reductions of the Holm-Hone equation through the classical methods. The infinitesimal generators corresponding to the one-parameter transformation group are presented. Furthermore, the one-dimensional 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 , 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 first-order ordinary differential equations Exponentiating the infinitesimal symmetries we get the one-parameter groups generated by for as Accordingly, if is a solution of the Holm-Hone equation, so are the functions
2.2. One-Dimensional 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 1-dimensional 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 . 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 one-dimensional subalgebra spanned by with is equivalent to one spanned by either , , or .
The remaining one-dimensional 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 one-dimensional 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 Holm-Hone 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 Holm-Hone Equation
The appearance of nonanalytic peakon-type 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 sinh-cosh Ansatz I
(2) A sinh-cosh 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 Holm-Hone 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 Holm-Hone 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 first-order 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 Holm-Hone 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 bi-Hamiltonian structure . In this section, we mainly solve the conservation laws of the Holm-Hone 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 first-order and higher-order partial derivatives of could also be considered, but the calculations become more complicated and we fail to find any result.
The right-hand 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 Holm-Hone equation with multiplier of the form is a linear combination of the two conserved vectors
In summary, we have found the most general Lie point symmetries group for the nonintegrable Holm-Hone equation which is a fifth-order generalization of the CH equation. Meanwhile, we constructed the optimal system of one-dimensional 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.
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
This work is supported by the 13th Five-Year National Key Research and Development Program of China with Grant No. 2016YFC0401407.
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