Abstract
The N-soliton solution of the (2+1)-dimensional Sawada-Kotera equation is given by using the Hirota bilinear method, and then, the conjugate parameter method and the long-wave limit method are used to get the breather solution and the lump solution, as well as the interaction solution of the elastic collision properties between them. In addition, according to the expression of the lump-type soliton solution and the striped soliton solution, the completely inelastic collision, rebound, absorption, splitting, and other particle characteristics of the two solitons in the interaction process are directly studied with the simulation method, which reveals the laws of physics reflected behind the phenomenon.
1. Introduction
As one of the most important research fields in nonlinear science [1], the phenomenon of their interaction is commonly found in the fields of fluid mechanics [2], electromagnetism [3], optics [4, 5], plasma physics [6], nuclear physics [7], and so on. Feng et al. have explored the interaction solution between the soliton and elliptic cosine wave of (2+1)-dimensional DLWE by CRE method [8]. Zhang and Yan have used the backscattering method to analyze the dynamic behavior and interaction of nonreflective potential solitons in the nonlocal mKdV equation [9]. Guo et al. have used the Darboux transform method to study the three soliton interaction solutions of the Davey-Stewartson I equation [10], and the results are conducive to improving the understanding of the mechanism of rogue waves. Lü et al. have used the Wronskian formal expansion method to find rich interaction solutions to the mKdV-sine-Gordon equation [11]. At the same time, it has been noted that the lump solution, as a rational function solution [12] that is local in all directions of space, has recently attracted more and more attention from researchers [13–21]. Zhang and Chen have explored the interaction between the lump solution and the hyperbolic function solution through the Hirota bilinear method [22–24] and scientifically explained the formation of a new type of strange wave and named this new rogue wave as a resonance rogue wave [25].
As a supplement and development, this paper takes the (2+1)-dimensional Sawada-Kotera equation [26–28] as an example, first solves its N-soliton solution by Hirota bilinear method, and then gets its breather solution and lump solution by conjugate parameter method and long-wave limit method, respectively. In addition, the motion law of the interaction solution between striped soliton, breather, and lump-type soliton is also explored.
2. Nonlinear Wave Solution
Nonlinear waves are a common phenomenon in nonlinear physics. Its types include solitons, respirators, and lump solitons [29, 30]. The study of nonlinear waves is helpful to clarify the motion change law of physical systems under nonlinear action and reasonably explain related natural phenomena.
For the (2+1)-dimensional Sawada-Kotera equation,
It is widely used in various branches of physics, such as the field of two-dimensional quantum gravimeters, conformal field theory, and nonlinear science Liouville flow conservation equations.
First, transform Equation (1)
Thus, the bilinear form of the (2+1)-dimensional SK equation is obtained [31]. where is the bilinear operator and .
2.1. N-Soliton Solution
To get the soliton solution of the (2+1)-dimensional Sawada-Kotera equation, assume
Then, substitute Equation (4) into Equation (3) to get its dispersion relation as
After substituting the above two equations into Equation (2), we get a single soliton solution of the (2+1)-dimensional SK equation
Similarly, in order to get the 2-soliton solution of the (2+1)-dimensional SK equation, hypothesize
Then, substitute Equation (7) into Equation (3) and calculate it.
Then from Equation (2) and Equations (7) and (8), we can get the double soliton solution of the (2+1)-dimensional SK equation.
Finally, on the basis of Equations (5) and (8), the expression for the N-soliton solution of the (2+1)-dimensional SK equation can be written from literature [32]. where and interaction coefficient are, respectively,
For expressions (9)–(11) of N-soliton solutions, expressions of 3-soliton solution and 4-soliton solution can be written as
Then, according to the above discussion, by setting the relevant parameters, the images of 1-soliton, 2-soliton, 3-soliton, and 4-soliton of the (2+1)-dimensional SK equation can be expressed, as shown in Figure 1.
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It can be seen from Figure 1(a) that the soliton solution of this equation is a bell-shaped soliton, and it can be observed from the images (Figures 1(b) and 1(c)) of two and three solitons that the amplitude height of the soliton at the interaction is obviously slightly higher than that around it. In addition, by observing the four-soliton solution images (Figures 1(d)–1(f)) at different times, it can be seen that the shape and moving speed of the interaction between solitons remain unchanged before and after the interaction, so the interaction process belongs to elastic collision.
2.2. Breather Solution
As a spatially localized nonlinear wave with periodic vibration, the breather can be obtained by conjugate assignment transformation of the coefficients in the 2 N-soliton solutions. First, let and then, make
The expression N-breather can be expressed.
For 2-soliton solution (7), there is
That is, where and are real constants. Then, substitute the above equation into Equation (7) to get
Substituting Equations (20)–(23) into Equation (2) gives a single-breather solution of the (2+1)-dimensional SK equation. Finally, set the relevant parameters to draw its soliton image.
From Figure 2, it can be seen that the image of the single breather propagates periodically along the -axis, while the -axis direction is local. And its amplitude oscillates periodically with time like breathing. In addition, the breather as a whole moves at a uniform speed along the positive -axis.
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Similarly, the following constraints are imposed on the parameters in the 4-soliton solution expression (13):
That is,
The 4-soliton solution can be converted to a 2-breather solution, and the image looks like this:
From Figure 3, it can be observed that the 2-breather is a periodic image of two rows parallel to the -axis direction, and both move in the positive direction of the -axis; in addition, the rear breather moves faster than the front breather, resulting in catching up and overtaking the front breather.
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2.3. Lump Solution
The lump solution, as a special class of rational functions in nonlinear partial differential equations, is local in all directions of space. And it can be converted by the soliton solution by the long-wave limit method. For the N-soliton solution (9)–(11) of the (2+1)-dimensional SK equation, first let to get
Then, let , considering thathas the same progressive order, which is presented as
After ignoring the remainder in Equation (27), consider that factor in can be eliminated by transformation (2), so in the case of ,
Then, let to get the M-lump solution of the (2+1)-dimensional SK equation.
For 2-soliton solution (7), let ; then, there is where . Then, make , and then, there is in the case of and .
After the transformation of Equation (2), in the above equation can be eliminated, so in the above equation is ignored, and get
Finally, under the condition of , pass Equation (2) to get the 1-lump solution of the (2+1)-dimensional SK equation. Select the appropriate parameters below to plot the image of the 1-lump solution.
From Figure 4, it is easy to see that the 1-lump solution image of the (2+1)-dimensional SK equation consists of a towering peak and two shallow troughs. That is, it has one global maximum value over two global minimums.
(a) 3D structure diagram
(b) Density map
(c) Contour diagram
(d) image
In order to explore the dynamic characteristics of the 1-lump solution, the following parameter transformation is performed in the Equations (34)–(36) setting to get
Then, substitute the above equation into Equation (2)
In Equation (39), find the first partial derivatives of and , respectively, and let the first partial derivative equal to 0 to obtain the coordinates of the 3 stations, i.e.,
The Hessian matrix [33] and at the station are calculated
At the stations and , Hessian matrices and are
Therefore, takes a maximum value at , which is also the maximum value. That is, the crest height is
takes the minimum value at and and is also the minimum value; that is, the trough depth is
Then, in Equations (40)–(45), the time is derived, and the coordinate expression is eliminated from the time ; the motion velocity of the lump-type soliton and the equation of the trajectory of the crest and trough are
Select the same parameters as in Figure 4; that is, substitution (50)–(53) is presented as
Then, according to Equation (57), the trajectory image of the lump-type soliton is drawn.
From Figure 5, you can see that the 1-lump soliton moves at a uniform speed in the direction of .
At the same time, high-order lump solitons can also be obtained through (28)–(30), such as set
Then, under the constraint conditions of and , the 2-lump solution and 3-lump solution can be obtained after the transformation of Equation (2). Finally, appropriate parameters were selected to draw 2-lump solution and 3-lump solution images.
As can be seen from Figure 6, the image of the multilump solution is similar to the 1-lump solution, which consists of multiple towering peaks and shallow troughs.
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3. Interaction Solution
The interaction between solitons is similar to the collision between classical particles, namely, elastic collision [34] and inelastic collision [35]. If the shape and velocity of the soliton do not change before and after the interaction, it is an elastic collision; otherwise, it is an inelastic collision.
3.1. Soliton and Breather
Under the constraints of Equations (17)–(19), parameters in three-soliton solution (12) can be converted into the interaction solution of 1-soliton and 1-breather, and the image is shown below.
It can be observed from Figure 7 that the shape and velocity of 1-soliton and 1-breather do not change before and after the interaction, so the interaction belongs to elastic collision.
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Similarly, for the 4-soliton solution (13), the constraints of Equations (17)–(19) on the parameters can be converted into the interaction solution between 1-breather and 2-solitons, and the image is shown below.
It can be seen from Figure 8 that the two solitons intersect each other, while the breather moves along the positive -axis, and it moves much faster than the solitons.
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3.2. Soliton and Lump
With respect to the three-soliton solution (12) and four-soliton solution (13), 1-lump solution process (31)–(36) can be transformed into
Then, under the restriction of , the solution of 1-lump interaction with 1-soliton and the solution of 1-lump interaction with 2-solitons of the (2+1)-dimensional SK equation are obtained by Equation (2).
It can be found from Figures 9 and 10 that lump-type soliton did not get absorbed and disappeared by the striped soliton after contact with the striped soliton as described in the literature [19]. On the contrary, after passing through the striped soliton, the original shape and velocity are restored to continue to move, so the interaction process is also an elastic collision.
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Fitting solution for 1-lump [13],
It can be superimposed with single soliton solution (4) and (5).
Then, the above equation is substituted into Equation (3), and the coefficients of and are collected and set as 0 to obtain a set of algebraic equations. Then, solve the system and get two sets of solutions:
Group 1:
Group 2:
Finally, Equation (64) or Equation (65) was substituted into Equation (63), and the interaction solution of lump-type soliton and single soliton was presented by Equation (2).
In Equation (64), select parameters
The images of lump-type soliton and single soliton interaction at different time are drawn.
It can be found from Figure 11 that in addition to the phenomenon of lump being swallowed by single soliton mentioned in literature [17–19], it can also be found that after the lump is collided with and absorbed by the single soliton, the single soliton go backwards in the opposite direction of the original motion, just like the process of particle rebound after particle collision.
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In fact, with all other parameters constant, adjust the parameter to equal -0.6 and then draw the moving image of the place at different times.
It can be observed from Figure 12 that the striped soliton did not regress after absorbing lump but continued to advance. This reflects that when the value is increased, the kinetic energy of the striped soliton becomes larger, which exceeds the kinetic energy of the lump-type soliton, so the striped soliton can continue to advance after the collision with the lump-type soliton.
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At the same time, command when other parameters remain unchanged and then draw the moving images at different times.
As can be seen from Figure 13, the lump-type soliton with slow forward motion increased significantly after absorbing the lump-type soliton that caught up quickly from behind. The explanation for this phenomenon can be given as follows: when two objects moving in the same direction but with different speeds merge together, according to the law of conservation of momentum, the overall speed will be greater than the original slow object but smaller than the original fast object.
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Then, adjust parameter to equal 0.8 and 0.6. And draw the images at different times.
From Figures 14 and 15, it can be seen that the single soliton spit lump forward or backward in the process of motion and then lump away from the single soliton at a greater speed, while the single soliton also continued to advance in the original direction. This is equivalent to a particle splitting into two parts of different sizes and continuing to move at different speeds, which is an important law of momentum conservation in physics (partial momentum changes, but the total momentum is conserved before and after changes).
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3.3. Soliton, Lump, and Breather
It can be seen from Sections 2.2 and 2.3 that the soliton solution can be presented by using the long-wave limit method and the conjugate parameter method, respectively, and the interaction solution between the soliton, lump, and breather can be presented by using these two methods successively for the N-soliton solution.
For the 5-soliton solution of the (2+1)-dimensional Sawada-Kotera equation, it can be simplified by following the same process of lump solution obtained in Section 2.2, that is, by setting and then
The definitions of and are the same as above. Constraints are then applied to the parameters in the above equation
The interaction solution between 1-soliton, 1-lump, and 1-breather can be presented.
As can be seen from Figure 16, 1-soliton, 1-lump, and 1-breather did not disappear after interaction but continued to advance with the original speed according to the original direction of motion, so the interaction process belonged to elastic collision.
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4. Conclusion
Taking the (2+1)-dimensional Sawada-Kotera equation as an example, the soliton solutions, breather solutions, and lump-type soliton solutions of the equation and their interactions with each other were presented by the Hirota bilinear method. For the interaction solution obtained from the expression of N-soliton solution which has been determined by the relation between parameters through the long-wave limit method and the conjugate parameter method, the collisions between different types of solitons are elastic collisions. But when the interaction solution of lump-type soliton whose parameters are not determined is directly superposition with the single soliton solution whose expressions have been determined, it can be found that the collision between the two is completely inelastic, and the particle properties (collision, rebound, and split) of the interaction solution are also shown, which presents that as a quasiparticle, the lump-type soliton is a kind of lump particle with energy, momentum, and other particle characteristic quantity. In the previous literature, lump-type soliton solution was simply added to or , in which and were all arbitrary constants that had not been determined, but the result was lump-type soliton swallowed by striped soliton.
It is worth pointing out that the particle properties presented by the interaction between soliton of lump-type and single soliton have not been carefully and comprehensively explained in this paper, and it is expected that there will be further research in this field.
Data Availability
All data are included in the article.
Conflicts of Interest
The author declares that he has no conflicts of interest to report regarding the present study.
Acknowledgments
The author would like to thank Jiao Yan for her support and help.