# Exact dynamical behavior for a dual Kaup–Boussinesq system by symmetry reduction and coupled trial equations method

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## Abstract

We propose a coupled trial equation method for a coupled differential equations system. Furthermore, according to the invariant property under the translation, we give the symmetry reduction of a dual Kaup–Boussinesq system, and then we use the proposed trial equation method to construct its exact solutions which describe its dynamical behavior. In particular, we get a cosine function solution with a constant propagation velocity, which shows an important periodic behavior of the system.

## Keywords

Trial equation method Symmetry The complete discrimination system for polynomial Exact solution Kaup–Boussinesq system## 1 Introduction

In general, it is difficult to study the exact dynamical behavior for nonlinear evolution problems. Therefore, some powerful methods, such as Ma and Lee’s transformed rational function method [7], Liu’s canonical-like transformation method [8] and trial equation method [9, 10, 11, 12, 13, 14], Ma and Zhu’s multiple exp-function method [15], and other direct expansion methods [16], and so forth, have been proposed to solve such problems. On the other hand, we can use the complete discrimination system for polynomial method to classify exact solutions for some nonlinear differential equations [17, 18, 19, 20, 21, 22, 23, 24, 25]. These methods have been extensively developed and applied to a lot of nonlinear problems [26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37]. Liu’s renormalization method and its applications can be found in [38, 39, 40, 41, 42, 43, 44, 45]. Some new methods and results on fractional differential equations can be seen in [46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57] and the references therein.

In the paper, we will study a dual Kaup–Boussinesq system. By proposing a coupled trial equation method and using symmetry reduction and a complete discrimination system for polynomial, we obtain its exact solutions which describe the dynamical behavior of the system. In particular, we find a cosine function solution which shows an important periodic motion.

This paper is organized as follows. In Sect. 2, we propose a coupled trial equation method. In Sect. 3, we give the reduction of the dual Kaup–Boussinesq system according to the symmetry property and the proposed trial equation method. In Sect. 4, we give the exact solutions by using the complete discrimination system for polynomial. In particular, we get an interesting periodic cosine solution. The last section is a short conclusion.

## 2 Trial equation method for a coupled system

*H*and

*G*are two unknown functions which need to be determined. Substituting these trial equations into the coupled system, we solve

*H*and

*G*, and then integrate the trial equation (5) or (7) to give the corresponding exact solutions such as

*H*is a polynomial, we will use the complete discrimination system for polynomial to classify the exact solutions. In the next section, we give the application of the proposed trial equation method to a dual Kaup–Boussinesq system.

## 3 Symmetry and reduction

*ξ*, so we can reduce the system. Substituting trial equations (5) and (6) into the above system and integrating them yield

## 4 Exact solutions

### Family 1

*α*,

*β*,

*γ*are real numbers, and \(\beta > \gamma \). When \(\gamma <\alpha <\beta \), we have

### Family 2

*α*,

*β*are real numbers. The solution is given by

### Family 3

### Family 4

*α*,

*β*, \(l_{1}\), and \(s_{1} \) are real numbers, and \(\alpha >\beta >\), \(s_{1}>0 \). The solutions can be represented in terms of the first and second kinds of elliptic integrals.

From Remark 1, we know that the above result gives the classification of all solutions of integral (20).

### Remark 2

Here we only write the expressions of *u*, by which *v* can be given from (4). For simplicity, we omit *v*.

From solution (32), we know that the dual KB system shows an important periodic dynamical behavior.

## 5 Conclusion

A dual Kaup–Boussinesq system is solved by symmetry reduction and a coupled trial equation method. The result includes four families of exact single traveling wave solutions for this system. Among those, if we consider *ξ* as the functions of *u* or *v* respectively, the solutions are given by the explicit functions, and reversely, the solutions are represented by implicit functions. In particular, when the wave propagation velocity is taken as a special constant, the KB system has a periodic cosine function solution. This solution shows an important periodic dynamical behavior. In summary, according to these exact solutions, a variety of evolution patterns for the coupled KB system are obtained.

## Notes

### Acknowledgements

The authors thank the reviewers for their helpful suggestions.

### Availability of data and materials

Not applicable.

### Authors’ contributions

WL completed the whole computation and wrote the draft, YW proposed the topic. Both authors read and approved the final manuscript.

### Funding

The first author is supported by the project of Youth Fund of Northeast Petroleum University (No. 2019QNL-38).

### Competing interests

The authors declare that they have no competing interests.

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