# Characteristics of the Soliton Molecule and Lump Solution in the2+1-Dimensional Higher-Order Boussinesq Equation

Authors: Bo Ren
Journal: Advances in Mathematical Physics
 (2021)

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## Abstract

The soliton molecules, as bound states of solitons, have attracted considerable attention in several areas. In this paper, the2+1-dimensional higher-order Boussinesq equation is constructed by introducing two high-order Hirota operators in the usual 2+1-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 2+1-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.


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## Body

## 1. Introduction

The2+1-dimensional Boussinesq equation can describe the propagation of small-amplitude long waves in shallow water. The physical and dynamical structures of the 2+1-dimensional Boussinesq equation are investigated by using various methods [1–4]. The 2+1-dimensional Boussinesq equation reads
(1)utt+γuxx+3γu2xx−αuxxxx+μuyy=0,where α, γ, and μ are arbitrary constants. It can be transformed into the Hirota form:
(2)Dt2+γDx2−αDx4+μDy2f·f=0,with the dependent variable transformation:
(3)u=2lnfxx.The2+1-dimensional Boussinesq equation reduces the 1+1-dimensional Boussinesq form with μ=0. The 1+1-dimensional Boussinesq equation includes the “good” Boussinesq form and “bad” Boussinesq form with α<0 and α>0, respectively [5]. Investigating deeper into properties of this model (1), the extended 2+1-dimensional Boussinesq equations are introduced based on the usual Boussinesq equation (1) [6, 7]. The topological kink-type soliton solutions of the extended 2+1-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 1+1-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 3+1-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 the2+1-dimensional higher-order Boussinesq equation which possesses the soliton molecule. The soliton molecule is absent in the usual 2+1-dimensional Boussinesq equation. This paper is organized as follows. In Section 2, the soliton molecule and the asymmetric soliton of the 2+1-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 the2+1-Dimensional Higher-Order Boussinesq Equation

Based on the bilinear form of the2+1-dimensional Boussinesq equation, we can construct the higher-order form by introducing the high-order Hirota operators (Dx6 and Dy4):
(4)Dt2+γDx2−αDx4−βDx6+μDy2+νDy4f·f=0,where D is the bilinear derivative operator [29]:
(5)DxlDynDtmf·g=∂∂x−∂∂x′l∂∂y−∂∂y′n∂∂t−∂∂t′mfx,y,t·gx′,y′,t′x′=x,y=y′,t′=t.Two-soliton solution of the higher-order Boussinesq equation can be calculated as(6)f=1+expη1+expη2+a12expη1+η2,where ηi=kix+liy+ωit+cii=1,2. By substituting (6) into (4), the phase shift a12 and the dispersion relation are written as
(7)a12=2γk1k2+2μl1l2+2νl1l22L−3l1l2−2αk1k22K−3k1k2−k1k26K2−15k1k2K+8k12k22+2ω1ω22γk1k2+2μl1l2+2νl1l22L+3l1l2−2αk1k22K+3k1k2−k1k26K2+15k1k2K+8k12k22+2ω1ω2,K=k12+k22,L=l12+l22,ωi2+γki2−αki4−βki6+μli2+νli4=0.The soliton molecule can be constructed with the velocity resonance condition [30]. The velocity resonance condition ki≠kj reads
(8)kikj=lilj=ωiωj=αki4+βki6−γki2−μli2−νli4αkj4+βkj6−γkj2−μlj2−νlj4.By solving condition (8), the velocity resonant condition becomes
(9)kj=±βνli4−αki4−βki6ki2,lj=±liβνli4−αki4−βki6ki3.Above velocity resonant condition (9) cannot be obtained while equation (4) is absent in the high-order Hirota operators Dx6 and Dy4. 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
(10)k1=12,k2=28,l1=14,l2=216,α=−14,β=1,γ=−1,μ=1,ν=12.Figure 1
(a) Soliton molecule of the2+1-dimensional higher-order Boussinesq equation. (b) Density plot of the corresponding soliton molecule.
(a)(b)Figure 2
(a) Asymmetric soliton of the2+1-dimensional higher-order Boussinesq equation. (b) The wave propagation pattern along the x-axis by selecting different times t=60, t=0, and t=−60 (from left to right).
(a)(b)The phases of Figures1 and 2 are c1=0,c2=10 and c1=0,c2=1, 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 the2+1-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 the2+1-dimensional higher-order Boussinesq equation, a quadratic function of f is shown as
(11)f=a1x+a2y+a3t2+a4x+a5y+a6t2+a7,where aii=1,2,⋯,7 are arbitrary constants. By substituting (11) into the Hirota bilinear form (4) and balancing the different powers of x, y, and t, the parameters are constrained as the following three cases.Case 1.
(12)a1=μa52+a62γ,a2=a3a5a6,a4=a3a6μa52+a62γ,a7=−3νa54a32+a62a62μa52+a62+3αa32+a62μa52+a62γ2a62.The solution ofu can be localized in the x,y-plane with the parameters satisfying
(13)μν>0,a7>0.Case 2.
(14)a1=−μa52+a62γ,a2=−a5,a4=−μa52+a62γ,a7=6αμa52−a622γ2a62−6νa54a62.Case 3.
(15)a1=−μa52+a62γ,a3=−a6,a4=−μa52+a62γ,a7=6αμa52−a622γ2a62−6νa54a62.In order to localize the solution ofu in the x,y-plane for Cases 2 and 3, the parameters should be satisfied:
(16)αμa52−a622−νγ2a54>0.Take Case1 as an example to describe the dynamics of lump waves. By substituting (11) into (3), the lump wave of the 2+1-dimensional higher-order Boussinesq equation in Case 1 is generated:
(17)u=4a32+a62μa52+a62γa62f−8a32+a622μa52+a622x2γ2a64f2.To describe the lump wave of the2+1-dimensional higher-order Boussinesq equation, the parameters are selected as
(18)α=1,γ=1,a3=1,a5=3,a6=2,μ=1,ν=12.The spatiotemporal structure and the density of a lump wave are described in Figures3(a) and 3(b), respectively. The critical points of the lump wave are solved:
(19)∂ux,y,t∂x=0,∂ux,y,t∂y=0.Figure 3
(a) The three-dimensional plot of a lump wave of the2+1-dimensional higher-order Boussinesq equation with the parameters in (18). (b) The corresponding density plot.
(a)(b)By solving above condition (19), we find that the function u reaches the maximum value at the point 0,−a6/a5t and the minimum values at two points ±3αμa52+a622−νγ2a52/γμa52+a62,−a6/a5t. By substituting above three points values into (17), the maximum and minimum values of the function u are 4γμa52+a622/3αμa52+a622−νγ2a52 and −γμa52+a622/6αμa52+a622−νγ2a52, respectively. The value of the maximum point is bigger than zero due to a7>0. The ratio between the maximum and minimum amplitudes is 8. 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 the2+1-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 2+1-dimensional Boussinesq equation (1).In this paper, the2+1-dimensional higher-order Boussinesq equation is constructed by introducing the high-order Hirota bilinear operators Dx6 and Dy4 based on the usual 2+1-dimensional Boussinesq equation. Similar to introducing the high-order Hirota bilinear operator procedure, we propose one equation
(20)Dt2+γDx2−∑i=1nαiDx2+2i+∑j=1mβjDy2jf·f=0,with αi and βj 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.



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