Advances in Mathematical Physics
Volume 2017 (2017), Article ID 1743789, 6 pages
https://doi.org/10.1155/2017/1743789

Research Article

Lump Solutions and Resonance Stripe Solitons to the (2+1)-Dimensional Sawada-Kotera Equation

Xian Li, Yao Wang, Meidan Chen, and Biao Li

Ningbo Collaborative Innovation Center of Nonlinear Hazard System of Ocean and Atmosphere and Department of Mathematics, Ningbo University, Ningbo 315211, China

Correspondence should be addressed to Biao Li; libiao@nbu.edu.cn

Received 1 June 2017; Accepted 3 July 2017; Published 11 September 2017

Academic Editor: Ming Mei

Copyright © 2017 Xian Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Based on the symbolic computation, a class of lump solutions to the (2+1)-dimensional Sawada-Kotera (2DSK) equation is obtained through making use of its Hirota bilinear form and one positive quadratic function. These solutions contain six parameters, four of which satisfy two determinant conditions to guarantee the analyticity and rational localization of the solutions, while the others are free. Then by adding an exponential function into the original positive quadratic function, the interaction solutions between lump solutions and one stripe soliton are derived. Furthermore, by extending this method to a general combination of positive quadratic function and hyperbolic function, the interaction solutions between lump solutions and a pair of resonance stripe solitons are provided. Some figures are given to demonstrate the dynamical properties of the lump solutions, interaction solutions between lump solutions, and stripe solitons by choosing some special parameters.

1. Introduction

In soliton theories [1–8], as a special kind of rational solution, rogue wave has been published in different fields since Solli et al. first reported the existence of optical rogue wave in 2007 [9]. Its lethality is very strong and can lead to devastated impact on the navigation. Compared with the rogue wave, lump solution is a special kind of solution, rationally localized in all directions in the space. So the lump solution has also attracted more and more attention [10–14], and it can be studied through Hirota bilinear equation. One equation can be transformed into a new equation with Hirota bilinear method [15–17]; the new equation is called the Hirota equation. Some special examples of lump solutions have been found, such as KPI equation [12], p-gBKP equation [11], KdV equation [18], and Davey-Stewartson II equation [19, 20]. More importantly, Zhang and Chen showed lump solution and its interaction phenomenon with a pair of stripe (line) solitons of a reduced (3+1)-dimensional Jimbo-Miwa equation [10]. The general Sawada-Kotera (SK) equation

where is an arbitrary nonzero and real parameter, is first produced by Sawada and Kotera [21]. It is an important unidirectional nonlinear evolution equation and it has been studied extensively over the last three decades and its mathematical properties are well-documented in the literatures [22–28]. For instance, the multisoliton solutions, conserved quantities, Bäcklund transformation, and Darboux transformation of the equation have been discussed in [22–25]. In [29], a (2+1)-dimensional integrable generalization of the Sawada-Kotera (2DSK) equation has the following form:

The equation is widely used in many physical branches, such as conformal field theory, two-dimensional quantum gravity, and conserved current of Liouville equation [22, 30]. It is interesting to study the 2DSK equation. So the main purpose of this paper is to investigate the lump solutions and the interaction solutions between lump solutions and resonance stripe solitons of 2DSK equation.

The outline of the paper is organized as follows. In Section 2, based on the bilinear method and one positive quadratic function which can guarantee the solutions to be nonsingular, the lump solutions of the 2DSK equation are obtained. In Section 3, by adding an exponential function into the original positive quadratic function, the interaction solutions between lump solutions and one stripe (line) soliton are provided. In Section 4, we extend this method to investigate the interaction solutions between the lump solutions and a pair of stripe solitons through combining the positive quadratic function and hyperbolic cosine function. The last section contains a short summary and discussion.

2. Lump Solutions to (2+1)-Dimensional Sawada-Kotera (2DSK) Equation

In this part, we consider a dependent variable transformation of 2DSK equation

where is positive; with this transformation, we obtain the following Hirota bilinear form of 2DSK equation:

and here is a real function with respect to variables , , and , and the derivatives , , , and are the Hirota bilinear operators.

Therefore, if solves bilinear 2DSK equation (4), then will solve the 2DSK equation. In order to get lump solutions, we make the following assumption:

where () are real parameters to be determined. Substituting (5) into (4), equating all the coefficients of different polynomials of to zero, we obtain a set of algebraic equations in ; solving the set of algebraic equations, we can find the following relations of these parameters:

in order to guarantee that is positive, it needs ; then the parameters need to satisfy these conditions

These sets lead to guarantee of the well-defined function and a class of positive quadratic function solutions to the bilinear 2DSK equation in (4):

which in turn generates a class of lump solutions to the 2DSK equation through transformation (3):

where the quadratic function is defined by (8), and the functions of and are given as follows:

In this class of lump solutions, , , , , , and are arbitrary so that the solutions are well defined. That is to say, if determinants (7) are satisfied, these conditions precisely imply that two directions and are not parallel in the -plane.

Note that solutions in (9) are analytic in the -plane if and only if the parameter . Conditions (7) guarantee the analyticity of the solutions in (9); they also lead to , and so . It is readily observed that, at any given time , all the above lump solutions if and only if the corresponding sum of squares , or equivalently, due to conditions (7). Therefore, conditions (7) guarantee both analyticity and localization of the solutions in (9). Actually, based on the above observation, we can see that two determinant conditions (7), the analyticity, and the localization of the solutions in (9) are equivalent to each other.

The plots are shown in Figure 1 when and , respectively.

(a)

(b)

Figure 1: Profiles of (9) with , , , , , .

3. The Interaction Solutions between Lump Solutions and One Stripe Soliton

In Section 2, the lump solutions of 2DSK equation are presented through the quadratic function. In order to get the interaction solutions between lump solutions and one stripe soliton, we make as a combination of positive quadratic function and one exponential function in this part; that is,

where

through substituting (11) into (4) and symbolic calculation, these parameters can be expressed:

which should satisfy

Under the transformation , the solutions of 2DSK equation will be obtained again:

where

By taking special choices of these parameters, the dynamic plots of collision between lump and one stripe soliton are depicted in Figure 2.

(a)

(b)

(c)

Figure 2: Evolution plot of (15) with , , , : (a) indicates one stripe soliton and lump solution, (b) denotes that lump soliton and stripe soliton begin to impact, and (c) denotes that the lump solution is swallowed by the stripe soliton.

4. The Interaction Solutions between Lump Solutions and a Pair of Resonance Stripe Solitons

We study the collision between lump and one stripe soliton; on that basis, we begin to discuss the collision between lump and a pair of stripe solitons. In this section, we redefine as the following formula:

where

Through substituting solution (17) into (4), these parameters have the following relations:

which should satisfy

Once again, by substituting (17) into (4), with transformation , the solutions of 2DSK equation are obtained

where

In Figure 3, we can see the dynamic plots when changes.

(a)

(b)

(c)

(d)

(e)

Figure 3: Evolution plot of (21) with , , , , .

Figure 3(a) shows when , a pair of resonance solitons appear, while lump is hidden in one of the stripe solitons; Figure 3(b) shows when , lump propagates and tangles with one of the resonance solitons. Furthermore, the shapes of these two resonance solitons change at the same time and in the same location. Figure 3(c) shows when , lump’s energy reaches up to the maximum, whereafter, its energy transfers into the other stripe soliton until it disappears, and we can see the lump tangles with the other stripe soliton successfully and then vanishes.

5. Summary and Discussion

Through Hirota bilinear form and symbolic calculation, we investigate the (2+1)-dimensional Sawada-Kotera equation. Its lump solutions are provided first, and the analyticity and localization of the resulting solutions are guaranteed by two determinant conditions. And then the interaction solutions between lump solutions and one stripe soliton are obtained and the results show that lump will be drowned or swallowed by the stripe soliton. Furthermore, we study the interaction solutions between lump solutions and a pair of solitons. In the beginning, there exist a pair of resonance stripe solitons; lump is hidden in one of the solitons. As time goes on, lump propagates gradually and it tangles with one of the resonance stripe solitons. When is close to , lump’s energy reaches the maximum, whereafter its energy transfers into the other stripe soliton until it disappears, and we can see the lump tangles with the other stripe solitons successfully and then blends into the soliton.

In future work, we will be devoted to investigating the interaction solutions between lump solutions and other solutions to some equations. These problems will be worth discussing.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work is supported by National Natural Science Foundation of China under Grant nos. 11271211 and 11435005 and K. C. Wong Magna Fund in Ningbo University.

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