Abstract
In this study, by making use of the direct integral method and the complete discrimination system for the polynomial method, all the travelling wave solutions to the two-component Dullin–Gottwald–Holm (DGH2) system are obtained, including solitary wave solutions, singular periodic solutions and Jacobian elliptic function double periodic solutions. Some of them are initially given. Moreover, concrete examples are presented to make sure that several solutions can be realised, and the corresponding figures are also given to show their nature. This means every solution in the paper may reflect the corresponding natural phenomenon, such as tidal waves and tsunami waves.
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1 Introduction
The shallow water wave equation is a meaningful model that is used to describe the storm tide, tidal waves etc. [1,2,3]. Scientists found that many other real-world models could also be described by it. Thus, a growing academic interest has been drawn in the extension of this kind of equation [4,5,6].
Here, we study the two-component Dullin–Gottwald–Holm (DGH2) system
where u(x, t) is the fluid velocity in the x direction (or equivalently the height of the water’s free surface above a flat bottom), \(\rho (x,t)\) is related to the free surface elevation from equilibrium (or scalar density), the parameter \(A (A>0)\) characterises a linear underlying shear flow propagating in the positive direction of the x-coordinate (or the critical shallow-water speed) and the parameter \(\gamma \) is a constant determining the dispersion effect. The above system is an extension of the DGH equation developed by Dullin, Gottwald and Holm in 2001 [7]. Related results such as well-posedness and stability of this system can be seen in [8,9,10]. Furthermore, system (1) contains many famous models as specific examples. For example, if \(\gamma =0\) and \(\rho =0\), system (1) becomes the noted Camassa–Holm (CH) equation [11,12,13]. If \(\gamma =0\) and \(\rho \ne 0\), system (1) turns into the two-component CH system [14, 15].
System (1) can be used to describe shallow water waves with curl zero. It is applied in ocean exploitation, disaster prevention etc. [16,17,18,19,20,21,22,23]. Thus, constructing exact solutions to it would shed light on the related area. Zhu and Xu gave sufficient conditions for the existence of a strong global solution to system (1) in [24, 25]. Cheung [26] constructed some blow-up solutions of system (1) using the perturbation method.
The travelling wave solution mainly describes wave propagations with constant velocity, and so has wide applications in various areas. Different methods have been proposed to obtain such types of solutions [27,28,29], such as the F-expansion method [30], trial equation method [31,32,33,34] and the complete discrimination system for polynomial method (CDSPM) [35,36,37,38,39,40,41,42,43]. Among these, the complete discrimination system for the polynomials by Liu is more powerful, because it not only can construct all the travelling wave solutions if the original model is reduced to an integral form, but also can be applied to conduct qualitative analysis [44,45,46,47,48].
So in this paper, we use the CDSPM to system (1), and all the travelling wave solutions, i.e., the classification of travelling wave solutions are obtained. Some solutions, such as Jacobian elliptic function double periodic solutions are obtained, which is difficult to obtain by other methods. This also shows the effectiveness of the method adopted in this paper.
2 Simplify system
By taking the following travelling wave transformation
where \(k\ne 0\) is a real constant, and then substituting eq. (2) into system (1), we have
Integrating (3), once yields
where M and \(N\ne 0\) are integral constants. From system (4), we have
Thus, the following equation can be obtained:
whose general solution is shown as follows:
where
and \(\eta _{0}\), \(c_{0}\) are arbitrary constants.
For brevity, by using the transformation \(\psi =u-k\), (7) becomes
In the following, we shall construct exact solutions to the original equation according to (9).
3 Travelling wave solutions of the system
Case 1
If \(B_{0}=0\), eq. (9) turns into
According to the complete discrimination system of third order
four cases can be discussed.
Case 1.1
If \(\Delta =0\), \(D_{1}<0\), then we get \(F(\psi )=(\psi -\alpha )^{2}(\psi -\beta )\), where \(\alpha \ne \beta \). By the substitution
that is,
we can obtain
where
where
Case 1.2
If \(\Delta =0\), \(D_{1}=0\), then we get \(F(\psi )=(\psi -\alpha )^{3}\). Thus, the solution is as follows:
Case 1.3
If \(\Delta >0\), \(D_{1}<0\), then \(F(\psi )=(\psi -\alpha )(\psi -\beta )(\psi -\delta )\), where \(\alpha>\beta >\delta \). Using the transformation
that is,
we deduce that
where
Case 1.4
If \(\Delta <0\), we have \(F(\psi )=(\psi -\alpha )(\psi ^{2}+l_{1}\psi +s_{1})\), where \(l_{1}^{2}-4s_{1}<0\). Let
Then, eq. (9) becomes
where \(a_{0}=l_{1}\alpha +s_{1}+\alpha ^{2}\), \(b_{0}=-\gamma (2\alpha +l_{1})-l_{1}\alpha -2s_{1}\) and \(d_{0}=\gamma ^{2}+l_{1}\gamma +s_{1}\).
Additionally, based on the above analysis, the solutions of Cases 1.3 and 1.4 can be represented by the elliptic function of the first and third types, respectively.
Case 2. If \(B_{0}\ne 0\), let \(\psi _{1}=\psi +\frac{1}{4}B_{3}\). Then, eq. (9) becomes
where
We denote \(F(\psi _{1})=\psi _{1}^{4}+P_{2}\psi _{1}^{2}+P_{1}\psi _{1}+P_{0}\). Then the complete discrimination system of the fourth order is given as follows:
Case 2.1
If \(D_{4}=0\), \(D_{3}=0\), \(D_{2}=0\), we have \(F(\psi _{1})=\psi _{1}^{4}\). Then, eq. (17) turns into
By letting
when \(a<0\), we get
When \(a=0\), we have
and when \(a>0\), we get
Case 2.2
If \(D_{4}=0\), \(D_{3}=0\), \(D_{2}>0\), \(E_{2}>0\), we have
Suppose
Then, eq. (17) can be rewritten as
Then, we have
where
Case 2.3
If \(D_{4}=0\), \(D_{3}=0\), \(D_{2}<0\), \(E_{2}<0\), we have
Similarly, we can get
where
Case 2.4
If \(D_{4}=0\), \(D_{3}=0\), \(D_{2}>0\), \(E_{2}=0\), then we have \(F(\psi _{1})=(\psi _{1}-\alpha )^{3}(\psi _{1}+3\alpha )\), with the solution given by
where
In the same way, let
we obtain
where
and
Case 2.5
If \(D_{4}=0\), \(D_{3}>0\), \(D_{2}>0\), we have
Equation (9) can be rewritten as
where \(\alpha >\beta \), and \(\beta \ne -3\alpha \), \(\beta \ne -\frac{\alpha }{3}\).
Similar to the above case, let
then we infer that
Here,
Case 2.6
If \(D_{4}=0\), \(D_{3}<0\), we have \(F(\psi _{1})=(\psi _{1}-\alpha )^{2}[(\psi _{1}+\alpha )^{2}+\beta ^{2}]\). We get
where \(\beta \ne 0\). From eq. (30), we have
where
and
Case 2.7
If (\(D_{4}>0\), \(D_{3}>0\), \(D_{2}>0\)), or (\(D_{4}<0\), \(D_{2}>0\) \(\parallel \) \(D_{4}<0\), \(D_{2}<0\), \(D_{3}<0\) \(\parallel \) \(D_{4}<0\), \(D_{2}=0\), \(D_{3}\le 0\)), or (\(D_{4}>0\), \(D_{2}\le 0\) \(\parallel \) \(D_{4}>0\), \(D_{3}\le 0\), \(D_{2}>0\)), we have
where \(\varphi =\alpha +\beta +\delta \).
From Cases 2.4–2.7, these solutions can be represented by elliptic integral or elliptic functions. From all the cases we have discussed, the forms of travelling wave solutions of system (1) include solitary wave solutions, singular periodic solutions and double periodic solutions.
4 Physical representation
In this section, we show images of two types of solutions we obtained by adjusting the corresponding parameters. Other cases can be obtained in the same way.
Example 1
Take \(k=1\), \(\gamma =5\), \(\alpha =6\), \(\beta =2\), \(\eta _{0}=0\), then solution (12) becomes
Therefore, the graph of solution (12) can be seen in figure 1.
Example 2
Take \(k=1\), \(\gamma =3\), \(\alpha =2\), \(\eta _{0}=0\), then solution (14) becomes
Therefore, the graph of solution (14) can be seen in figure 2.
5 Conclusion
This study has shown all travelling wave solutions of the two-component DGH system. By the direct integral method and CDSPM, we attained solitary wave solutions, singular periodic solutions and double periodic solutions. In addition, double periodic solutions were initially presented. These travelling wave solutions will help us to better understand the propagation forms of shallow water waves.
Data availability
The data used to support the findings of this study are available from the corresponding author upon request.
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Acknowledgements
This work is financially supported by the National Natural Science Foundation of China (Grant No. 52174060).
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Guo, L., Xin, H. The classification of exact travelling wave solutions to two-component Dullin–Gottwald–Holm system. Pramana - J Phys 98, 86 (2024). https://doi.org/10.1007/s12043-024-02787-2
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DOI: https://doi.org/10.1007/s12043-024-02787-2
Keywords
- Two-component Dullin–Gottwald–Holm system
- complete discrimination system for polynomial
- direct integral method
- travelling wave solution