Abstract
Through the method of -KP hierarchy, we propose a new ()-dimensional weakly coupled B-KP equation. Based on the bilinear form, we obtain the lump and rational solutions to the dimensionally reduced cases by constructing a symmetric positive semidefinite matrix. Then, we do numerical analysis on the rational solutions and fit the trajectory equation of the crest. Furthermore, we verify the accuracy of the trajectory equation by numerical analysis. This method of solving the lump and rational solutions can also be applied to other nonlinear evolution equations.
1. Introduction
During the recent years, in the field of nonlinear evolution equations, the exact solution is also widely concerned, such as the rational solutions, lump solutions, hybrid solutions, and rogue wave solutions [1–5]. And it has been applied to many fields: solid-state physics, fluid dynamics, plasma physics, and other nonlinear engineering problems. Furthermore, constructing exact solutions for nonlinear evolution equations is important in many scientific or engineering applications. Therefore, a series of systematic approaches, the Lie group method [6], Bäcklund transformation method, Darboux transformation method [7–10], Hirota bilinear method [11–13], inverse scattering transform method [14], and other methods [15], have been rapidly developed. It is meaningful and interesting to search for solutions to nonlinear evolution equations.
In this paper, the ()-dimensional B-KP equation is expressed as
The B-KP equation describes the evolution of quasi-one-dimensional shallow water waves, when the effects of surface tension and viscosity are negligible and these are widely used in various physics fields such as the surface and internal oceanic waves, nonlinear optics, plasma physics, ferromagnetics, Bose-Einstein condensation, and string theory. Because of its importance, many properties have been studied. Wazwaz [16] established one and two soliton solutions of this equation by using the simplified Hereman-Nuseir form. Huang et al. [17] derived the bilinear Bäcklund transformations and soliton solutions to Equation (1); Bright-Dark lump wave solutions for a new form of the ()-dimensional B-KP-Boussinesq equation have been explored in [18]. Lump, breather, and solitary wave solutions to the new reduced form of the generalized B-KP equation have been found in [19], and the dynamical analysis of lump solutions is exhibited in [20].
Based on bilinear derivatives, the B-KP equation can be transformed into the following bilinear equation:under a dependent variable transformation
The operator is the Hirota bilinear differential operator, which is defined by
By taking the -KP hierarchy method [21], a new ()-dimensional weakly coupled B-KP equation can be derived in the following form:
Equation (5) will be reduced to the following ()-dimensional weakly coupled B-KP equation under :
In this work, we mainly explore the lump and rational solutions to weakly coupled B-KP Equation (6). The rest of the paper is arranged as follows. In Section 2, the bilinear formalism of ()-dimensional weakly coupled B-KP equation is derived. In Section 3, we obtain the lump and rational solutions to this equation by constructing symmetric positive semidefinite matrices, and some properties of the lump solutions are analyzed and discussed. Furthermore, we get the trajectory equation of the crest by using numerical analysis methods. Finally, our conclusion is presented in Section 4.
2. Bilinear Formalism
We start from a potential field to construct bilinear transformation
By substituting and into Equation (6), the result is presented as
Equation (6) becomes the Hirota bilinear equation:where and are real functions with respect to variables , , and and the operator is the Hirota bilinear operate.
3. Lump Solutions and Rational Solutions
3.1. Lump Solutions
In order to obtain rational solutions to Equation (6), we setwithwhere , , and are the real parameters that need to be determined; the matrices and are symmetric matrices.
After directly utilizing Maple symbolic calculation with , a series of constraint equations for parameters are as follows:
By combining Equations (10)–(12), the is presented as follows:
To obtain lump solutions to Equation (6), the matrix and parameter should satisfy the following conditions: (1) the matrix must be a positive semidefinite matrix; (2) the real parameter must be greater than ; (3) the elements of matrix must also satisfy the following constraints and . Among the above conditions, (1) and (2) are to ensure that is always greater than 0. Condition (3) is to ensure that has only one minimum at any time.
Under the above conditions, we found that we can take the following special parameter values:
Substituting Equation (14) into in Equation (7), we find
Therefore, we obtain a lump solution of the -dimensional B-KP Equation (6):
Figure 1 shows the contour plot, -curve, and -curve of the lump solution. And the lump solution has a maxium point and a minimum point . Besides, the lump solution reaches the maximum value of and the minimum value of .
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We find a class of lump solutions to Equation (6). As mentioned above, since is a lump solution to Equation (6), then we make , so is also a lump solution to Equation (6). The correlative solutions are presented in the following form:
As can be seen in Figure 2, when , the lump solution has a maximum point and two minimum points. The maximum value is obtained at , and the maximum value is ; the minimum value is obtained at , , and the minimum value is .
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3.2. Rational Solutions
In order to get the value of and make the result clear, we substitute Equation (14) directly into Equation (8). And the related constraint equations are presented in the following form:
Equation (18) contains three arbitrary parameters , , and ; we assign values to parameters as follows:
The rational solution can be presented aswith
Therefore, it can be seen from the above that is the rational solution to Equation (6), and Equations (16) and (20) are a set of solutions to weakly coupled Equation (6).
After a detailed analysis, we find that rational solution has some fascinating properties. This property is that is an odd function, which means . Therefore, when we know the trajectory of the crest at at a certain period of time, we can know the corresponding trajectory of the crest at . We find that the solutions of and have extremely complex algebraic structures, so we do not know what curve of the crest is over time. Therefore, we investigate the motion trajectory of the rational solution by taking the numerical analysis method.
When , the position of the crest at different times can be seen in Table 1. Here, is the height of the crest and the and are used to measure the error of numerical calculation. When and are closer to 0, the position of the crest will be more accurate.
From the above description, we construct a nonlinear fit to the data in Table 1. During a certain period of time , the motion trajectory of this crest can be fitted to the following equation:
We find from Equation (22) that as the time increases gradually, the trajectory of the crest will fit more linearly. And, an interesting phenomenon that can be seen from Figure 2 is that the strange wave of is very similar to the lump wave at every moment, and its amplitude and time are proportional. Since is an odd function, we obtain the trajectory equation of at by combining Equation (23). So the result is presented in the following form:
Thus, Equations (22) and (23) can accurately express the trajectory of at every moment. Next, we do some numerical tests on the equations in Equations (22) and (23). The results are presented in Table 2.
As can be seen from Table 2, when , and are very close to 0, so it can be explained that Equations (22) and (23) can accurately represent the position of the crest.
4. Conclusion
In this paper, by using the -KP hierarchy, a new ()-dimensional weakly coupled B-KP equation has been constructed. And its dimensionally reduced form has been investigated. We obtain the lump solutions to Equation (6) by taking the Hirota bilinear method and constructing the symmetric positive semidefinite matrix technique. The lump solutions are displayed in three-dimensional graphics (see Figures 1 and 2). We also found an interesting set of rational solutions to Equation (6) by using the same method. And the height of will also increase proportionally over time. Time-varying rational solutions are presented in graphics (see Figure 3). By taking a numerical analysis of the rational solution , the motion trajectory of is approximately linear motion over time (see Table 1). And we also test the equations of the moving trajectory by using numerical analysis methods. The position and fitting results of the trajectory equation are presented (see Table 2). In this paper, the method of finding the lump and rational solutions to weakly coupled B-KP equations by constructing a symmetric positive semidefinite matrix can also be applied to other nonlinear evolution equations. We hope that our results can provide valuable help in the field of nonlinear science.
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Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work is supported by the National Natural Science Foundation of China under Grant Nos. 11805106, 11775121, and 11435005 and the K.C.Wong Magna Fund in Ningbo University.