Abstract
In this paper, the bilinear method is employed to investigate the rogue wave solutions and the rogue type multiple lump wave solutions of the (2+1)-dimensional Benjamin-Ono equation. Two theorems for constructing rogue wave solutions are proposed with the aid of a variable transformation. Four kinds of rogue wave solutions are obtained by means of Theorem 1. In Theorem 2, three polynomial functions are used to derive multiple lump wave solutions. The 3-lump solutions, 6-lump solutions, and 8-lump solutions are presented, respectively. The 3-lump wave has a “triangular” structure. The centers of the 6-lump wave form a pentagram around a single lump wave. The 8-lump wave consists of a set of seven first order rogue waves and one second order rogue wave as the center. The multiple lump wave develops into low order rogue wave as parameters decline to zero. The method presented in this paper provides a uniform method for investigating high order rational solutions. All the results are useful in explaining high dimensional dynamical phenomena of the (2+1)-dimensional Benjamin-Ono equation.
1. Introduction
Rogue wave is an isolated huge wave, which plays an important role in analyzing many science problems, such as ocean’s waves [1–3], optical fibers [4], Bose-Einstein condensates [5, 6], and financial markets [7, 8]. The Darboux transformation method [9, 10] is one of the most effective methods to construct the rogue wave solutions of the integrable systems. Rogue wave solutions are an interesting class of lump-type solutions [11]. It is interesting that lump functions can provide approximate prototypes to model rogue waves [12]. Based on the bilinear method, the bilinear forms can be used to construct the lump solutions by choosing appropriate multivariate positive substitution quadratic functions. In 2015, Ma investigated the lump solutions of the (2+1)-dimensional Kadomtsev-Petviashvili equation by means of the bilinear forms and positive quadratic functions [13]. Inspired by this work, many researchers studied the lump solutions of the integrable systems, such as the (2+1)-dimensional Sawada-Kotera equation [14], the (2+1)-dimensional asymmetrical Nizhnik-Novikov-Veselov equation [15], the (3+1)-dimensional B-type KP equation [16], and the generalized Calogero-Bogoyavlenskii-Schiff equation [17]. Furthermore, the generalized bilinear forms can also be used to derive the lump solutions [18].
Searching for the interaction solutions between the lump wave and other types of wave is one of the hot issues. There are many methods to investigate the lump solutions and the interaction solutions among solitons, such as the inverse scattering transformation [19], the Painlevé analysis [20–22], the symmetry analysis [23–27], and the Darboux transformation [28, 29]. Fokas et al. studied the completely elastic collision between a lump wave and a line soliton of the DSII equation [30]. Tang et al. investigated the nonelastic collision between a lump wave and a stripe soliton of the (2+1)-dimensional Ito equation [31]. Then the interaction lump-soliton solutions [32, 33], lump-twin-stripe solutions [34], and lump-periodic wave solutions [35] were investigated.
Compared with the construction of single lump wave solutions, multiple wave solutions are more difficult to obtain. Zhang et al. obtained the multiple lump solutions of a (3+1)-dimensional nonlinear evolution equation by taking a “long wave” limit for the corresponding N-solitons [36]. The dynamical behaviors of the interactions of multilumps within the Kadomtsev-Petviashvili-1 equation were analyzed by numerical simulation [37]. The multiple lump solutions of a (3+1)-dimensional Boiti-Leon-Manna-Pempinelli equation in incompressible fluid were derived [38]. The high-order lumps of the (3+1)-dimensional KP-Boussinesq equation were presented by using Hirota’s bilinear method [39]. Zhaqilao et al. proposed a new method to construct the multiple lump solutions with a controllable center [40, 41]. We constructed the multiple lump solutions of the (3+1)-dimensional potential Yu-Toda-Sasa-Fukuyama equation [42].
In this paper, we consider the (2+1)-dimensional Benjamin-Ono equationwhere . Eq.(1) is the reduction of a generalized (2+1)-dimensional Boussinesq equation [43, 44]. The (2+1)-dimensional Benjamin-Ono equation arises in the study of internal waves in deep stratified fluids [45].
The organization of the paper is as follows. In Section 2, multiple rogue wave solutions of (1) are constructed with the aid of bilinear method. A theorem is proposed to establish a unified approach to study multiple rogue wave solutions. In Section 3, three polynomial functions are introduced to construct the rogue type multiple lump wave solutions. The 3-lump, 6-lump, and 8-lump solutions are investigated in Sections 4, 5, and 6. Finally some conclusions are given in the last section.
2. Multiple Rogue Wave Solutions
In this section, we shall investigate rogue wave solutions of (1). Eq.(1) can be reduced to the following equation:by setting . With the aid of variable transformation (2) can be reduced to the bilinear equationThis bilinear form can be used to construct multiple rogue wave solutions of (1).
Theorem 1. (2+1)-Dimensional Benjamin-Ono equation (1) has the multiple rogue wave solutionswhere, is the solution of bilinear equation (4) and the coefficients are the parameters to be determined.
Theorem 1 provides a uniform method for constructing the multiple rogue wave solutions of (1). Using this method we obtain the following polynomials: which corresponded to the rogue wave solutions , and .
Figures 1, 2, 3, and 4 show the dynamical behaviors of the rogue wave solutions , and , respectively.
(a)
(b)
(c)
(a)
(b)
(c)
(a)
(b)
(c)
(a)
(b)
(c)
Figures 1(a) and 1(b) show that the rogue wave has one peak which is higher than the water level. This kind of rogue wave is called lump wave that maintains the localization property in the planes . The lump wave reaches the higher peak at the point . The lump wave Figures 1(a) and 1(b) have two lower peaks. The coordinates of the lower peaks are and . The center of the lump wave in the plane is . The one order rogue wave develops into one-soliton solution in Figure 1(c).
If we take in rogue wave , we obtain the second order rogue wave described in Figures 2(a) and 2(b). The rogue wave has the structure of two-line solitons in the plane . Figures 3 and 4 plot the rogue waves and . These show that the number of the higher peaks of the multiple rogue wave solutions is equivalent to the subscript of . From Figures 1(a), 2(a), 3(a), and 4(a), we can conclude that the maxima of the solutions all lie on the line . Figures 3(c) and 4(c) exhibit the fact that the rogue wave develops into the multiple soliton solutions in the plane .
3. Rogue Type Multiple Lump Wave Solutions
Theorem 2. (2+1)-Dimensional Benjamin-Ono equation (1) has the rogue type multiple lump wave solutionswhere, is the solution of bilinear equation (4), the coefficients , and are the parameters to be determined, and and are arbitrary constants.
Remark 3. Theorem 2 gives a uniform method for constructing the rogue type multiple lump wave solutions of (1). One can obtain the multiple wave solutions more than six lump waves by selecting . The parameters and determine the shape of the multiple lump wave. The multiple lump wave tends to a low order rogue wave as declines to . All peaks of the lump wave tend to the same height when is sufficiently large. In what follows, we shall construct the multiple lump solutions of (1).
4. 3-Lump Solutions
Based on Theorem 2, we construct the 3-lump solutions with the aid of following polynomial functions:Substituting (11) into (4) and setting all the coefficients of the different polynomials of and to zero yield a set of determining equations. Solving these equations, one hasThen we obtain the 3-lump solutionswhere , the coefficients of are determined by (12), and , , , , and are arbitrary real constants. Figure 5 plots the 3-lump wave (13) when , , . The 3-lump wave is the arrangement of three first order rogue waves in the planes and , which forms a structure of triangular. From Figures 5(c) and 5(f), we observe that 3-lump wave solution (13) develops into two-soliton solution in the plane .
(a)
(b)
(c)
(d)
(e)
(f)
5. 6-Lump Solutions
In order to obtain the 6-lump solutions of (1), we choosewhereSubstituting (14) into (4) and setting all the coefficients of the different polynomials of and to zero yield a set of determining equations. Solving these equations, one hasThen we obtain the 6-lump solutionswhere and the coefficients of are determined by (16), , , , , and are arbitrary real constants.
Figures 6(a) and 6(e) plot 6-lump wave (17) when , . This case illustrates that as tends to the six lump wave becomes a third order rogue wave in Section 2. Figures 6(b), 6(c), 6(f), and 6(g) exhibit the fact that three more peaks appear as and increase. As and become more large, six lump waves appear and all the centers of the single lump waves form a structure of pentagram. Figure 6(d) shows that all the peaks of the 6-lump wave tend to the same hight when is sufficiently large. Figure 7 plots the 6-lump wave (17) when , . The corresponding 6-lump wave is composed of six single lump waves, which array a pentagon. Figure 7(c) shows that the 6-lump wave solutions have the structure of three solitons in the plane .
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(a)
(b)
(c)
(d)
(e)
(f)
6. 8-Lump Solutions
To find the 8-lump solutions of (1), we investigate polynomial function solutions to (4) with the assumptionwhere Submitting (18) into (4), we obtain a set of constraining equations for the parameters This set leads to a 8-lump solution to (1)where , the coefficients of are determined by the above constraining equations, and , , , , and are arbitrary real constants. When setting , , , we find that solution (21) (Figures 8(a), 8(b), 8(d), and 8(e)) possesses eight lump waves, namely, seven single lump waves and one second order rogue wave (see Figures 2(a) and 2(b)). The seven single lump waves array a heptagon around the second order rogue wave. If we set , the 8-lump solution develops into four-soliton solution, which is plotted in Figures 8(c) and 8(f). Figures 9(a) and 9(e) show that the 8-lump wave develops into fourth order rogue wave of Section 2 when declines to . From Figures 9(b)–9(d) and 9(f)–9(h), we can observe that the fouth order rogue wave splits into a set of seven first order rogue waves and one second order rogue as the center with parameters and gradually become large enough.
(a)
(b)
(c)
(d)
(e)
(f)
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
7. Conclusions
In this paper, we focus on investigating the rogue wave solutions and the rogue type multiple lump solutions of the (2+1)-dimensional Benjamin-Ono equation by means of the bilinear method. Based on the reduced bilinear equations, four kinds of rogue wave solutions are obtained. The number of the higher peaks of the multiple rogue wave solutions is equivalent to the subscript of . The maxima of the solutions all lie on the line . The multiple rogue waves develop into the multiple stripe solitons in the plane . A theorem for constructing multiple lump solutions has been presented by introducing three polynomial functions , , and (10) of degree . The multiple lump waves tend to the low order rogue waves as the parameter is near . Numerical experiments show that the multiple lump solutions will produce new peaks with the increase of parameters. The number of the lumps will be fixed and all peaks will tend to the same height when is sufficiently large. The method presented in this paper shows that Hirota’s bilinear method is a powerful approch to construct the high order rational solutions to nonlinear integrable equations. Based on the bilinear forms, one can construct different types of rational solutions by choosing different polynomial functions. The results might be helpful in explaining nonlinear wave phenomena in fluid mechanics.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This research is supported by Scientific and Technologial Innovation Programs of Higher Education Institutions in Shanxi, Initial Scientific Research Fund of the High-level Talents in 2019 in North University of China (No. 11012411), and the Fund for Shanxi “1331KIRT”.