Propagation of lump-type waves in nonlinear shallow water wave

1.   Introduction

A lump wave is a rational soliton with a large amplitude, which is localized only in space and will not disappear due to time change. In 2015, an effective algebraic method to obtain lump solutions of integrable systems was proposed by Ma ^[1]. Subsequently, He ^[2] also provided theoretical support for this method and proved it, which made great progress in the solution of the lump wave and attracted the attention of a large number of researchers, such as Tian ^[3,4,5], Lü ^[6,7,8], He ^[9,10,11], Wen ^[12,13,14], Su ^[15,16,17], Lan ^[18,19,20], Chen ^[21,22,23], Li ^[24,25,26] and Ma ^[27,28] et al.

The shallow water wave equation has become a hot topic in recent years. The wave equation for shallow water is a model in which the depth of water is less than the wavelength of the free surface disturbance. Examples of shallow water wave equations are widely used in the field of oceanography and the atmosphere to simulate the dynamic behavior of water wave propagation. For example, the shallow water wave equation in a (2+1) dimension ^[29,30,31] is given below:

and

and the shallow water wave equation in (3+1) dimension:

Recently, a new extended shallow water wave equation in a (3+1) dimension was proposed by Wazwaz as follows ^[31]:

where $\Phi = \Phi(x, y, z, t)$, $\alpha$, $\beta$, $\gamma$ and $\delta$ are arbitrary constants. Wazwaz ^[31] obtained the multiple soliton and lump solutions. In addition, this equation has not been studied in other literatures.

Under the transformation

where $\xi = \xi(x, y, z, t)$, Eq (4) has the following Hirota bilinear form

The organization of this paper is as follows: Section two investigates the interaction between a lump wave and one soliton of Eq (4). Section three obtains the interaction solutions between a lump and two solitary waves for Eq (4). Section four studies the interaction between a lump and a periodic wave of Eq (4); Section five discusses the interaction among a lump, periodic and one solitary wave. Section six makes the conclusions.

2.   Lump-1-soliton solution

The lump solutions of Eq (4) have been obtained by Wazwaz. On this basis, we will further investigate the interaction between the lump wave and one soliton. For this reason, we assume

where $\mathcal{G}_i(i = 1, 2, \cdots, 12)$ and $k_1$ are undetermined constants. Substituting Eq (7) into (6), we obtain

All other parameters are free and unrestricted. By substituting the results of Eqs (8)–(10) into Eqs (5) and (7), we can get the corresponding interaction solution for Eq (4). In order to understand the dynamic properties of the interaction solutions between the lump wave and one soliton, we take Eq (8) as an example and select special values of parameters (see Figure 1) to obtain a special solution of the equation as follows:

Figure 1.  Solution (11). $\mathcal{G}_2 = \alpha = \beta = \delta = \gamma = 1$, $\mathcal{G}_1 = -4$, $\mathcal{G}_4 = \mathcal{G}_{12} = 3$, $k_1 = 6$, $y = 0$, $\mathcal{G}_{10} = -1$, (a) $z = -5$, (b) $z = 0$, (c) $z = 1$.

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The dynamic properties of Eq (11) are shown in Figure 1. In Figure 1(a), we can observe that a soliton and a lump wave propagate forward respectively. In Figure 1(b), the soliton and the lump wave slowly close together, and the amplitude of the lump wave begins to decrease. Until Figure 1(c), the soliton and the lump wave gradually converge, and the amplitude of the lump wave becomes smaller. It can be seen that the interaction between the soliton and the lump wave is an inelastic collision where the energy is consumed.

3.   Lump-2-soliton solution

Next, we want to further consider the interaction between the lump wave and double solitons. Therefore, we assume

Substituting Eq (12) into (6), we derive

All other parameters are free and unrestricted. By substituting the results of Eqs (13)–(15) into Eqs (5) and (12), the corresponding interaction solution of Eq (4) can be obtained. In order to understand the dynamic properties of the interaction solutions between the lump wave and two solitons, we take Eq (14) as an example and select special values of parameters (see Figure 2) to obtain a special solution of the equation as follows:

The dynamic properties of Eq (16) are described in Figure 2. In Figure 2(a), we can see that the lump wave interacts with one of the solitons. In Figure 2(b), the lump wave begins to move to the middle and interact with another soliton. At this time, the amplitude of the lump wave becomes larger. Until Figure 2(c), the lump wave shifts to another soliton and its amplitude decreases.

Figure 2.  Solution (16). $\mathcal{G}_2 = \alpha = \beta = \delta = \gamma = 1$, $\mathcal{G}_1 = 2$, $\mathcal{G}_4 = \mathcal{G}_{12} = 3$, $k_1 = -6$, $y = 0$, $\mathcal{G}_{10} = -1$, (a) $z = -4$, (b) $z = 0$, (c) $z = 4$.

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4.   Lump-periodic solution

In this section, we intend to investigate the interaction between the lump and periodic waves, so we suppose

Substituting Eq (17) into (6), we give

All other parameters are free and unrestricted. By substituting the results of Eqs (18) and (19) into Eqs (17) and (5), we can get the corresponding interaction solution between the lump and periodic waves of Eq (4). In order to understand the dynamic properties of the interaction solutions between lump and periodic waves, we take Eq (18) as an example and select special values of parameters (see Figure 3) to obtain a special solution of the equation as follows:

Figure 3.  Solution (20). $\mathcal{G}_2 = \alpha = \beta = \delta = \gamma = \mathcal{G}_{10} = 1$, $\mathcal{G}_1 = -3$, $k_1 = -1$, $\mathcal{G}_4 = \mathcal{G}_{12} = 3$, $y = 0$, (a) $z = -2$, (b) $z = 0$, (c) $z = 2$.

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The dynamic properties of Eq (20) are described in Figure 3. Different from the previous two sections, the lump wave and periodic wave have been entangled and propagated forward.

5.   Lump-periodic-1-soliton solutions

In order to investigate the interaction among the lump, periodic and one solitary wave, we suppose

where $\mathcal{G}_i(i = 14, 15, 16, 17)$ and $k_2$ are undetermined constants. Substituting Eq (21) into (6), we obtain

All other parameters are free and unrestricted. By substituting the results of Eqs (22) and (23) into Eqs (21) and (5), we can derive the corresponding interaction solution among the lump, periodic and one solitary wave. In order to understand the dynamic properties of the interaction solutions among lump, periodic and one solitary wave, we take Eq (22) as an example and select special values of parameters (see Figure 4) to obtain a special solution of the equation as follows:

Figure 4.  Solution (24). $\mathcal{G}_2 = \alpha = \beta = \delta = \gamma = \mathcal{G}_{10} = 1$, $\mathcal{G}_1 = \mathcal{G}_{16} = -2$, $k_1 = -1$, $y = 0$, $\mathcal{G}_4 = \mathcal{G}_{12} = 3$, $k_2 = 4$, (a) $x = -5$, (b) $x = 0$, (c) $x = 5$.

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The dynamic properties of Eq (24) are described in Figure 4.

6.   Conclusions

In this article, we investigated a new extended shallow water wave equation in (3+1) dimensions based on the Hirota bilinear form and symbolic computation ^{[32,33,34,35,36,37,38,39,40,41,42,43,44]}. The interaction between the lump wave and single solitary wave is studied. The interaction between the lump wave and two solitary waves as well as the interaction between lump wave and periodic wave was also discussed. Finally, we obtained the interaction solutions among the lump, periodic and one solitary wave and discribed the dynamic properties of the obtained results in Figures 1–4. Since its discovery, the Hirota bilinear method has been widely used in solving lump wave solutions of nonlinear integrable systems, and this method is simple, direct, and effective. In addition to the lump wave solution, this method can also be used to obtain hybrid rogue wave and breather solutions and has been promoted by many famous scholars in China ^{[45,46,47,48]}.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

Project supported by National Natural Science Foundation of China (Grant No. 12161048), Doctoral Research Foundation of Jiangxi University of Chinese Medicine (Grant No. 2021WBZR007) and Jiangxi University of Chinese Medicine Science and Technology Innovation Team Development Program (Grant No. CXTD22015).

Conflict of interest

The authors declare there is no conflict of interest.