Abstract
The soliton molecules, as bound states of solitons, have attracted considerable attention in several areas. In this paper, the -dimensional higher-order Boussinesq equation is constructed by introducing two high-order Hirota operators in the usual -dimensional Boussinesq equation. By the velocity resonance mechanism, the soliton molecule and the asymmetric soliton of the higher-order Boussinesq equation are constructed. The soliton molecule does not exist for the usual -dimensional Boussinesq equation. As a special kind of rational solution, the lump wave is localized in all directions and decays algebraically. The lump solution of the higher-order Boussinesq equation is obtained by using a quadratic function. This lump wave is just the bright form by some detail analysis. The graphics in this study are carried out by selecting appropriate parameters. The results in this work may enrich the variety of the dynamics of the high-dimensional nonlinear wave field.
1. Introduction
The -dimensional Boussinesq equation can describe the propagation of small-amplitude long waves in shallow water. The physical and dynamical structures of the -dimensional Boussinesq equation are investigated by using various methods [1–4]. The -dimensional Boussinesq equation reads where , , and are arbitrary constants. It can be transformed into the Hirota form: with the dependent variable transformation:
The -dimensional Boussinesq equation reduces the -dimensional Boussinesq form with . The -dimensional Boussinesq equation includes the “good” Boussinesq form and “bad” Boussinesq form with and , respectively [5]. Investigating deeper into properties of this model (1), the extended -dimensional Boussinesq equations are introduced based on the usual Boussinesq equation (1) [6, 7]. The topological kink-type soliton solutions of the extended -dimensional Boussinesq equation are obtained by the sine-Gordon expansion method [6]. The modified exponential expansion method is applied to the coupled Boussinesq equation [7]. The multisoliton solutions, breather solutions, and rogue waves of the generalized Boussinesq equation are obtained via the symbolic computation method [8] and the polynomial functions in the bilinear form [9]. Generally, seeking exact solutions to nonlinear evolution equations is a vital task in soliton theory. Many methods have been proved effective in finding the exact solutions of the soliton equation [10–12]. By using the extended auxiliary equation method and the extended direct algebraic method, the solitary traveling wave solutions and the stability of these solutions are analyzed [10–12]. In this work, we shall study the soliton molecule and lump wave of the higher-order Boussinesq equation by solving the bilinear form of the higher-order Boussinesq equation.
The soliton molecule which is formed by the balance of repulsive and attractive forces between solitons is treated as a boundary state [13]. It was first predicted within the framework of the nonlinear Schrödinger-Ginzburg-Landau equation [14]. Many effects including nonlinear and dispersive effects are a key role in the soliton molecule. The soliton molecule has become a focus of intense research in both experiment and simulation [13–17]. The theoretical frameworks to address the soliton molecule have been introduced [18, 19]. Recently, Lou proposed the velocity resonance mechanism to construct the soliton molecules of the -dimensional nonlinear systems [20]. The velocity resonance mechanism is one of the useful methods to form the soliton molecule [20]. To balance the nonlinear effects, the high-order dispersive terms may play a key role in the velocity resonance mechanism [21]. The soliton molecule of a variety of integrable systems has been verified with the velocity resonance mechanism: the fifth-order Korteweg-de Vries (KdV) equation [22, 23], the modified KdV equation [24, 25], the -dimensional Boiti-Leon-Manna-Pempinelli equation [26], and so on [27]. The dynamics between soliton molecules and breather solutions and between soliton molecules and dromions are presented by the velocity resonance mechanism, the Darboux transformation, and the variable separation approach [25–28].
In this paper, we try to construct the -dimensional higher-order Boussinesq equation which possesses the soliton molecule. The soliton molecule is absent in the usual -dimensional Boussinesq equation. This paper is organized as follows. In Section 2, the soliton molecule and the asymmetric soliton of the -dimensional higher-order Boussinesq equation are constructed by the velocity resonance condition. In Section 3, the lump solution of the higher-order Boussinesq equation is obtained by solving the corresponding Hirota bilinear form. Finally, the conclusions of this paper follow in the last section.
2. Soliton Molecule for the -Dimensional Higher-Order Boussinesq Equation
Based on the bilinear form of the -dimensional Boussinesq equation, we can construct the higher-order form by introducing the high-order Hirota operators ( and ): where is the bilinear derivative operator [29]:
Two-soliton solution of the higher-order Boussinesq equation can be calculated as where . By substituting (6) into (4), the phase shift and the dispersion relation are written as
The soliton molecule can be constructed with the velocity resonance condition [30]. The velocity resonance condition reads
By solving condition (8), the velocity resonant condition becomes
Above velocity resonant condition (9) cannot be obtained while equation (4) is absent in the high-order Hirota operators and . A soliton molecule and an asymmetric soliton can be constructed by selecting appropriate parameters in (8) or (9). These phenomena are shown in Figures 1 and 2. We select the same parameters and different phases for Figures 1 and 2. The parameters are
(a)
(b)
(a)
(b)
The phases of Figures 1 and 2 are and , respectively. The soliton molecule and the asymmetric soliton are described in Figures 1 and 2. The soliton molecule and the asymmetric soliton can be transformed with each other by selecting different parameters. Two solitons in the molecule have different amplitudes, while two solitons in the molecule possess the same velocity.
3. Lump Solution of the -Dimensional Higher-Order Boussinesq Equation
Lump solutions, which can be considered a kind of rational function solutions, decay polynomially in all directions of space [31–36]. One can construct lump solutions by the Hirota bilinear method and the Darboux transformation [37–45]. Lump waves of the high-dimensional nonlinear systems are constructed by solving the Hirota bilinear method [46–49]. A symbolic computation approach is one of the useful methods to search the lump wave [31]. The interaction between the lump waves and other complicated waves is presented by the symbolic computation approach [38–43]. In this section, we shall study the dynamics of lump waves by using the symbolic computation approach.
To obtain the lump solution of the -dimensional higher-order Boussinesq equation, a quadratic function of is shown as where are arbitrary constants. By substituting (11) into the Hirota bilinear form (4) and balancing the different powers of , , and , the parameters are constrained as the following three cases.
Case 1.
The solution of can be localized in the -plane with the parameters satisfying
Case 2.
Case 3.
In order to localize the solution of in the -plane for Cases 2 and 3, the parameters should be satisfied:
Take Case 1 as an example to describe the dynamics of lump waves. By substituting (11) into (3), the lump wave of the -dimensional higher-order Boussinesq equation in Case 1 is generated:
To describe the lump wave of the -dimensional higher-order Boussinesq equation, the parameters are selected as
The spatiotemporal structure and the density of a lump wave are described in Figures 3(a) and 3(b), respectively. The critical points of the lump wave are solved:
(a)
(b)
By solving above condition (19), we find that the function reaches the maximum value at the point and the minimum values at two points . By substituting above three points values into (17), the maximum and minimum values of the function are and , respectively. The value of the maximum point is bigger than zero due to . The ratio between the maximum and minimum amplitudes is . The lump wave of the higher-order Boussinesq equation is just the bright form by the above detail analysis.
4. Conclusion
In summary, the soliton molecule and lump solution of the -dimensional higher-order Boussinesq equation are studied by solving the Hirota bilinear form (4). The soliton molecule and the asymmetric soliton are obtained by the velocity resonance mechanism. The lump solution can be derived by using a positive quadratic function. The lump wave of the higher-order Boussinesq equation is just the bright form after some detail analysis. Figures 1–3 show the dynamics of the soliton molecule and lump wave by putting suitable parameters. The soliton molecule and the asymmetric soliton can be transformed with each other by selecting different phases. The soliton molecule and the asymmetric soliton cannot be derived in the -dimensional Boussinesq equation (1).
In this paper, the -dimensional higher-order Boussinesq equation is constructed by introducing the high-order Hirota bilinear operators and based on the usual -dimensional Boussinesq equation. Similar to introducing the high-order Hirota bilinear operator procedure, we propose one equation with and being arbitrary constants. The soliton molecule and lump wave of (20) are worthy of study by the velocity resonance mechanism and the symbolic computation approach. Rogue waves are unexpectedly high-amplitude single waves that have been reported by using the Hirota bilinear method [50, 51]. These nonlinear excitations of (20) are valuable to increase understanding of the phenomena between different nonlinear waves.
Data Availability
The datasets supporting the conclusions of this article are included in the article.
Conflicts of Interest
The authors declare that they have no conflict of interest.
Acknowledgments
This work is supported by the National Natural Science Foundation of China (No. 11775146).