From the nonlocal symmetries of the Whitham-Broer-Kaup system, an eight-dimensional Lie algebra is found and the corresponding one-dimensional optimal system is constructed to provide an inequivalent classification. Six types of inequivalent group invariant solutions are demonstrated, some of which reflect the interactions between soliton and other nonlinear waves.

1. Introduction

For a given nonlinear evolution equation, whether it is integrable or not, the Lie group theory is one of the most effective methods for seeking exact and analytic solutions. Due to the Lie point symmetries, one can obtain new solutions from the old one through the symmetry group and acquire the group invariant solutions by the similar reductions. With the development of symmetry theory, a variety of nonlocal symmetries have been intensely investigated in the literature, which may come from the Darboux transformation [15], Bäcklund transformation [6, 7], and Painlevé analysis [810]. Using these kinds of nonlocal symmetries, one can obtain new analytic solutions, especially the exact interaction solutions between soliton and cnoidal periodic waves.

Another interesting and important problem about the group invariant solutions is to find those inequivalent branches of solutions, which leads to the concept of the optimal system. A method of constructing optimal system was firstly established by Ovsiannikov [11], using a global matrix for the adjoint transformation. Hereafter, the method has received extensive development by authors in [1219] and others. Chou and Qu [2022] offer many numerical invariants to address the inequivalences among the elements in the optimal system. More recently, a direct and systematic algorithm that can guarantee both the comprehensiveness and the inequivalence of the optimal system is proposed in [23] to find one-dimensional optimal system.

The Whitham-Broer-Kaup (WBK) system [2426]is derived from the Hamiltonian theory of water waves and has been said to be “the oldest, simplest and most widely known set of equations…". Here is the amplitude of a surface wave, propagating along the -axis with the horizontal velocity . A good understanding of the WBK system is very helpful for coastal and civil engineers to apply the nonlinear water wave model in a harbor and coastal design. The multisoliton solutions and the scattering problem are presented by Kaup [26]. A large class of almost periodic solutions are found algebrogeometrically by Matveev and Yavor [27]. Some soliton excitations and periodic waves solutions without dispersion relation are obtained in [28, 29]. The classical symmetries and corresponding one-dimensional optimal system of (1)-(2) are performed in [30]. Recently, Zhou and Lu have obtained the nonlocal symmetries of (1)-(2) in [31]. So a question arises: can we construct the one-dimensional optimal system based on the nonlocal symmetries? The purpose of this paper is to answer this question. Thus, Section 2 is devoted to constructing the corresponding one-dimensional optimal system. To show the interactions between soliton and other nonlinear waves, six types of inequivalent group invariant solutions are demonstrated in Section 3. The last section contains a summary.

Here let us give a brief account of the nonlocal symmetries, which are somewhat different from [31], where six dependent variables were considered. We point out that five dependent variables are just enough to constitute a closed prolonged system. By direct calculations, we know that if the function satisfieswiththenis just the solution of the WBK system (1)-(2). Thus one can verify that the WBK system (1)-(2) possesses the following nonlocal symmetries:That is to say, if satisfy (1)-(2), so do . Because (6) is nonlocal for , we need to extend the original system by introducing two other new dependent variables:Furthermore, the compatibility conditions of and require

In the following, we investigate the prolonged system including five dependent variables and eight equations (i.e., (1), (2), (5), (7), (8), and (9)), instead of the two original equations, (1) and (2).

2. One-Dimensional Optimal System

For the original WBK system (1)-(2), its Lie point symmetries are provided in the following:with

For the whole prolonged system, we need to consider the following one-parameter Lie group of infinitesimal transformations with a small parameter ,and the vector field associated with the above group of transformations

Applying the standard Lie symmetry approach [1219] to the prolonged system, we have Thus we can obtain an eight-dimensional Lie algebra represented by the following eight generators:

Clearly seen from (15), all are point symmetries to the eight equations of the prolonged system, while is a nonlocal symmetry of the WBK system (1)-(2). The corresponding transformation groups of (15) are obtained directly by solving the initial value problems, sayingThe general Lie point symmetries of the prolonged system read

A meaningful and interesting job is to list those inequivalent group invariant solutions, which can be reduced to give a classification of the corresponding Lie algebra. We concentrate on constructing an optimal system of the eight-dimensional Lie algebra spanned by the basis in (15). Denote the symmetry algebra generated by as and the corresponding group as . The commutation relations between and are shown in Table 1, where the entry in row and column represents . Then the adjoint representation relations between and , that is,are directly provided in Table 2 with the help of Table 1.

A real function on the Lie algebra is called an invariant if for all and all . If two vectors and are equivalent to each other under the adjoint action, it is necessary that for any invariant . To compute the invariants of , applying any to it yieldswithFor any , it requiresExtracting the coefficients of all in (21) and setting them to be zeros, two basic invariants of are obtained, sayingMoreover, applying the separate adjoint actions of to , respectively, and with the help of Table 2, the corresponding adjoint matrices are obtained: Then the general adjoint transformation matrix is selected asThat is to say, after the adjoint action of is transformed into , shown byNow, by virtue of (26) and all the invariants, one can start to construct the optimal system of step by step. Since the degrees of and are both two, we just need to consider the following five cases:For each case, select a corresponding representative element in the simplest form named and solve (26). If (26) has the solution with respect to , it signifies that the selected representative element is right. If the chosen representative element makes (26) be incompatible, we need to adopt a new proper one. Repeat the process until all the cases in (27) are finished. Omitting the tedious but regular operational processes, the one-dimensional optimal system of eight-dimensional Lie algebra (15) is found to be

3. Group Invariant Solutions

For any given subgroup, an original nonlinear system can be reduced to a system with fewer independent variables, which may be easily solved to provide group invariant solutions. As Olver said, since there is almost always an infinite amount of such subgroups, it is usually not feasible to list all possible group invariant solutions to the system. In this section, some types of inequivalent group invariant solutions which correspond to the elements in the optimal system (28) are listed.

For with , we consider an equivalent case , where is an arbitrary constant. The group invariant solution isHere , , , , and are all group invariant functions of . Substituting the group invariant solutions (29) into the prolonged system leads to the symmetry reduction equations for , , , , and :withHere, and in the latter of the paper, the primes on the functions with only one independent variable denote derivatives. Thus by the transformation(31) becomesFor different values of , (33) is exactly solvable. For example, if we take , this equation has a solutionThen, by solving (30), one can directly obtain an exact solution of the WBK system (1)-(2):

One equivalent case of is and the group invariant solution in this case isHere , , , , and are all group invariant functions of , which should satisfyAndThus by means of the transformationwe can transform (38) intoOn differentiation, (40) becomes linear and of the fourth order:Thus, the general solution of (38) is a rational function of the constants of integration [32].

Remark 1. It is known that the solitary wave is a very important phenomenon in the nature and physics. In fact, these solitary waves must interact with other waves or spread on some nonconstant background. Here, the forms of solutions (29) and (36) directly embody this kind of interactions between the soliton and other nonlinear waves.

For and , it is not difficult to obtain the following group invariant solutions:

Instead of studying , consider an equivalent case: . The corresponding group invariant solution of the WBK system readswhereHere, and are the Airy wave functions, which are linearly independent solutions for .

For , it is equivalent to with being an arbitrary constant. The group invariant solution is presented aswith

By solving , one can easily give a simple exact solution:

4. Summary

In the beginning of [31], the authors also discussed the nonlocal symmetry (6) of the WBK system. To realize the localization of the nonlocal symmetry, they referred to six dependent variables. In fact, we point that five dependent variables are enough in this paper. Based on a Bäcklund transformation, [31] constructed some special exact solutions, which have no direct connection with the given nonlocal symmetry. The purpose of our article is to give a classification of the eight-dimensional Lie algebra (15) and then to find out some inequivalent group invariant solutions with respect to the nonlocal symmetry (6).

In this paper, we firstly localize the nonlocal symmetries of the (1+1)-dimensional WBK system (1)-(2) to Lie point symmetries by prolonging the original system to a larger one. Then, all the Lie point symmetries of the whole prolonged system are presented, which constitute an eight-dimensional Lie algebra (15). The one-dimensional optimal system (28) is constructed to give a classification of the elements in the Lie algebra. The adjoint transformation matrix and all the invariants of the Lie algebra are also displayed. Finally, we demonstrate six kinds of inequivalent group invariant solutions, some of which reflect the interactions between soliton and other nonlinear waves.

Data Availability

No data were used to support this study.

Conflicts of Interest

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


The project is supported by the Zhejiang Provincial Natural Science Foundation of China (Grants nos. LY14A010016, LY18A010033, and LY18A010034) and the National Natural Science Foundation of China (Grant no.11771395).