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

Authors: Xian Li; Yao Wang; Meidan Chen; Biao Li
Journal: Advances in Mathematical Physics
 (2017)

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## 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.


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## Body

## 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(1)ut+uxxxxx+cuxuxx+cuuxxx+c25u2ux=0,where c 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:(2)ut=uxxxxx-5uxuxx-uuxxx+u2ux+uxxy+uuy+ux∂x-1uy-∂x-1uyy.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 Section2, 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(3)u=6ln⁡fxx,where f is positive; with this transformation, we obtain the following Hirota bilinear form of 2DSK equation:(4)DxDt-Dx6-5Dx3Dy+5Dy2f·f=2ffxt-fxft-ffxxxxxx-6fxfxxxxx+15fxxfxxxx-10fxxx2-53fxxfxy-3fxfxxy+ffxxxy-fxxxfy+5ffyy-fy2,and here f is a real function with respect to variables x, y, and t, and the derivatives DtDy, Dx3Dy, Dx6, and Dy2 are the Hirota bilinear operators.Therefore, iff solves bilinear 2DSK equation (4), then u=6(ln⁡f)xx will solve the 2DSK equation. In order to get lump solutions, we make the following assumption:(5)f=g2+h2+a9,g=a1x+a2y+a3t+a4,h=a5x+a6y+a7t+a8,where ai (1≤i≤9) are real parameters to be determined. Substituting (5) into (4), equating all the coefficients of different polynomials of x,y,t to zero, we obtain a set of algebraic equations in ai(1≤i≤9); solving the set of algebraic equations, we can find the following relations of these parameters:(6)a3=-5a1a22-a1a62+2a2a5a6a12+a52;a7=-52a1a2a6-a22a5+a5a62a12+a52,a9=a12+a522a1a2+a5a6a1a6-a2a52,in order to guarantee that f is positive, it needs a9>0; then the parameters need to satisfy these conditions(7)Δ1≔a1a6-a2a5=a1a2a5a6≠0,Δ2≔a1a2+a5a6=a1a6-a5a2>0.These sets lead to guarantee of the well-defined functionf and a class of positive quadratic function solutions to the bilinear 2DSK equation in (4):(8)f=a1x+a2y-5a1a22-a1a62+2a2a5a6a12+a52t+a42+a5x+a6y-52a1a2a6-a22a5+a5a62a12+a52t+a82+a12+a522a1a2+a5a6a1a6-a2a52,which in turn generates a class of lump solutions to the 2DSK equation through transformation (3):(9)u=12a12+a52f-24a1g+a5h2f2,where the quadratic functionf is defined by (8), and the functions of g and h are given as follows:(10)g=a1x+a2y-5a1a22-a1a62+2a2a5a6a12+a52t+a4,h=a5x+a6y-52a1a2a6-a22a5+a5a62a12+a52t+a8.In this class of lump solutions,a1, a2, a4, a5, a6, and a8 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 (a1,a2) and (a5,a6) are not parallel in the (x,y)-plane.Note that solutions in (9) are analytic in the (x,y)-plane if and only if the parameter a9>0. Conditions (7) guarantee the analyticity of the solutions in (9); they also lead to a12+a52≠0, and so a9>0. It is readily observed that, at any given time t, all the above lump solutions u→0 if and only if the corresponding sum of squares g2+h2→∞, or equivalently, x2+y2→∞ 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 Figure1 when t=0 and y=0, respectively.Figure 1
Profiles of (9) with a1=2, a2=1, a5=1, a6=-1, a4=0, a8=0.
(a)
t
=
0
(b)
y
=
0


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

In Section2, 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 f as a combination of positive quadratic function and one exponential function in this part; that is,(11)f1=m12+n12+l1+a9,where(12)m1=a1x+a2y+a3t+a4;n1=a5x+a6y+a7t+a8,l1=kexp⁡k1x+k2y+k3t,through substituting (11) into (4) and symbolic calculation, these parameters can be expressed:(13)a2=-9a12k14-4a6212a1k12,a3=-581a14k18-216a12a62k14+16a64144a13k14,a4=0,a5=0,a7=5a69a12k14-4a626a12k12,a8=0,a9=-a129a12k14-a624k12a62,k2=-3a12k14-4a6212k1a12,k3=81a14k18-360a12a62k14+80a64144a14k13,which should satisfy(14)a1a6k1≠0,k>0.Under the transformationu=6(ln⁡f1)xx, the solutions of 2DSK equation will be obtained again:(15)u=62a12+2a52+k12f1-62m1a1+2a5n1+k1l12f12,where(16)f1=a1x-9a12k14-4a6212a1k12y-581a14k18-216a12a62k14+16a64144a13k14t2+a6y+5a69a12k14-4a626a12k12t2+kl1-a129a12k14-a624k12a62,m1=a1x-9a12k14-4a6212a1k12y-581a14k18-216a12a62k14+16a64144a13k14t,n1=a6y+5a69a12k14-4a626a12k12t,l1=exp⁡k1x-3a12k14-4a6212k1a12y+81a14k18-360a12a62k14+80a64144a14k13t.By taking special choices of these parameters, the dynamic plots of collision between lump and one stripe soliton are depicted in Figure2.Figure 2
Evolution plot of (15) with a1=2, a6=-4, k=1, k1=-1: (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.
(a)
t
=
-
1
(b)
t
=
0.25
(c)
t
=
3


## 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 redefinef as the following formula:(17)f2=m22+n22+l2+a9,where(18)m2=a1x+a2y+a3t+a4,n2=a5x+a6y+a7t+a8,l2=kcosh⁡k1x+k2y+k3t.Through substituting solution (17) into (4), these parameters have the following relations:(19)a1=2a63k12,a2=-3k12a52,a3=15k12a62,a4=0,a7=45a5k144,a9=9k2k1629a52k14+4a62,k2=k132,k3=9k154,which should satisfy(20)a5a6k1≠0,k>0.Once again, by substituting (17) into (4), with transformation u=6(ln⁡f2)xx, the solutions of 2DSK equation are obtained(21)u=62a12+2a52+kk12cosh⁡k1x+k2y+k3tf2-62m2a1+2n2a5+kk1sinh⁡k1x+k2y+k3t2f22,where(22)f2=2a63k12x-3k12a52y+15k12a62t+a5x+a6y+45a5k144t+a8+9k2k1629a52k14+4a62+kcosh⁡k1x+k132y+9k154t.In Figure3, we can see the dynamic plots when t changes.Figure 3
Evolution plot of (21) with a5=1, a6=1, a8=0, k=1, k1=1.
(a)
t
=
-
12
(b)
t
=
-
2
(c)
t
=
0
(d)
t
=
2
(e)
t
=
12Figure3(a) shows when t=-12, a pair of resonance solitons appear, while lump is hidden in one of the stripe solitons; Figure 3(b) shows when t=-2, 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 t=0, 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. Whent is close to 0, 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.



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*Source: 1743789-2017-09-11.xml*