Advances in Mathematical Physics

Volume 2017 (2017), Article ID 1743789, 6 pages

https://doi.org/10.1155/2017/1743789

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

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; nc.ude.ubn@oaibil

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) equationwhere 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 equationwhere 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.