Lump-type solutions and interaction phenomenon to the bidirectional Sawada–Kotera equation

(1)

MS received 9 June 2018; revised 24 July 2018; accepted 27 July 2018; published online 1 February 2019 Abstract. In this paper, we use the Hirota bilinear method. With the help of symbolic calculation and applying this method, we solve the(2+1)-dimensional bidirectional Sawada–Kotera (bSK) equation to obtain some new lump-kink, lump-solitons, periodic kink-wave, periodic soliton and periodic wave solutions.

Keywords. Bidirectional Sawada–Kotera equation; Hirota bilinear method; symbolic calculation; lump-kink;

lump-solitons; periodic kink-wave; periodic soliton and periodic wave solutions.

PACS Nos 02.60.Lj; 02.70.Wz; 02.90.+p; 04.30.Nk

1. Introduction

The(1+1)-dimensional Sawada–Kotera (SK) equation [1] and the (2 +1)-dimensional SK equation [2] are given, respectively, as

ut +ux x x x x+5uux x x +5uxux x x +5u²ux x =0, (1) 5uxu²+5uuy+5uux x x−ut +5uxux x+5ux x y

+ux x x x x−5

uyydx+5ux

uydx =0. (2) To further understand the nonlinear phenomena, namely, the (1+1)-dimensional SK equation using symbolic computation, many powerful works are presented [3–

6]. Modifications of eq. (1) have also been investigated in [7–9]. Many other interesting findings such as Darboux–Levi-type transformations [10], the ∂- dressing method [11], the truncated Painle´v expansion [12], the Bäcklund transformations and Lax pairs [13], the simplified Hirota bilinear method [14], the Hirota bilinear method [15], the travelling wave solutions [16] and the multisoliton solutions [17] were also obtained.

Also, a bidirectional generalisation of the SK equation is given as

ut +45u²ux−15uxux x−15uux x x+ux x x x x =0. (3)

In this paper, we consider the following (2+1)- dimensional dynamical model called the (2+1)- dimensional bidirectional Sawada–Kotera (bSK) equation, first introduced by Lv and Bilige [18]:

5(∂x)⁻¹(uyy)−5ux xt+15uut −15ux(∂x)⁻¹(uy)

−45u²u_x−15u_xu_{x x}−15uu_{x x x}−u_{x x x x x}=0, (4) in which we have(∂x)⁻ⁿ =(d/dx)⁻ⁿ.

The (2 + 1)-dimensional bSK equation has been studied well by many scholars. For example, exact solutions and other equations, which contain the exact solitary wave and periodic solutions, the non-travelling wave solutions, the new periodic soliton solutions, the conservation laws, soliton-type solutions and rational solutions, the periodic cross-kink-wave solutions and breather-type solutions, the lump and its interaction solutions, the multiple wave solutions, etc. were obtained in [18–21]. One can easily understand that the lump solutions are a kind of rational function localised in all directions of the space with applications for integrable equations. To know about the important mathematical merits of the lump solutions, the reader is referred to refs [22–33]. Lump solutions have been sys- tematically analysed by Ma and Zhou [34] and Chen and Ma [35], and the interaction solutions have been dis- cussed for a few integrable equations in(2+1)dimen- sions by Zhang and Ma [36] for constructing lump-kink solutions to the bilinear Kadomtsev–Petviashvili (BKP)

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the Kadomtsev–Petviashvili-based system [42], utilis- ing homoclinic test approach and Hirota bilinear method for the (2+ 1)-dimensional generalised Kadomtsev–

Petviashvili equation [43], employing Hirota bilinear method to study phase shifts and trajectories during the overtaking collision of multisolitons [44], to study the two coupled Kadomtsev–Petviashvili (KP) equations by using the simplified Hereman form of Hirota bilinear method [44] and finally to investigate the (4 +1)-dimensional Fokas equation with the help of Hirota bilinear method in which single-soliton solution, double-soliton solution and three-soliton solution are obtained [45]. The authors of [46,47] used the numeri- cal Galerkin method for solving the nanofluid flow and non-Newtonian flow.

Our most important objective here is to find exact solutions of the bSK equation by considering the Hirota bilinear method for obtaining the five classes of interaction solutions in terms of a new merge of positive quadratic functions, trigonometric functions and hyperbolic functions. Discussion about the lump-type solutions and their interaction solutions including lump-kink solution, first and second types of lump-soliton solutions and kinky breather soliton solution is given. The stripe soliton solution is investigated and the exact solutions are derived. In continuation, we shall present the graphi- cal illustrations of some solutions of the aforementioned model. After that, we shall deal with the investigation of solutions and we shall end by a conclusion. As a result, we obtain some new periodic kink-wave, periodic soliton and periodic wave solutions. To the best of our knowledge, they have not been reported. At the same time, this paper can also be regarded as a supplement to other associated works.

2. The lump-type solutions and their interaction solutions

In this section, the Hirota bilinear method will be used to handle the bSK equation for acquiring new lump

where the Hirota bilinear operatorDand its derivatives are defined as

D_x^λD^δ_yD_t^θ(f ·g)

= ∂

∂x − ∂

∂x1

_λ

∂

∂y − ∂

∂y1

_δ

∂

∂t − ∂

∂t1

_θ

×f(x,y,t)g(x1,y1,t1)|x1=x,y1=y,t1=t. (7) 2.1 Lump-kink solutions of the bSK equation

To present the lump-kink solutions, we select the following test functions:

f=g²+h²+l1+a13, g =a1x+a2y+a3t+a4, h =a5x+a6y+a7t+a8,

l1 =exp(a9x+a10y+a11t+a12), (8) wherea_i,i =1, . . . ,13,are free constants. By inserting (8) into (6), we obtain the following results.

SetI:

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

a1 =a1, a2= −5 9

(a⁴₁−16a₁²a₅²−16a⁴₅)a₃²

a₁⁵ ,

a₃ =a₃, a₄=a₄, a₅ =a₅, a6 = 5

9

(7a₁⁴−8a₁²a₅²−16a₅⁴)a5a²₃

a₁⁶ ,

a7 = a5a3(3a₁²+4a²₅) a₁³ , a8 =a8, a9= 4a₅

a1

a₃ 3a1, a10 = − 4

27

a³₃a5(5a₁⁴−40a²₁a²₅+16a₅⁴) a₁⁸

3a₁ a3 , a11 = 4

9

a₃²a5(3a₁²−4a₅²) a⁵₁

3a1

a3 , a12 =a12, a13 = 3(a₁⁴−3a₁²a₅²−4a⁴₅)a₁

16a₃a²₅

(9)

(3)

(d) (c)

Figure 1. Plot of eq. (11) by taking the parametersa1=a5=2,a3=a8=1,a4=7,a7=a12 =5,t = −1. (a) 3D plot, (b) density plot, (c) contour plot and (d) 2D plot (y=1 – red,y=2 – blue andy=3 – green).

and

a1,a3,a5 =0, a1a3 >0, (10) where a1,a3,a4,a5,a8 and a12 are free constants.

Putting (9) into (8), we can acquire the solution f, and then the solution is

u= −2

2a₁²+2a₅²+a₉²l1

f +2(2a1g+2a5h+a9l1)²

f² ,

f =g²+h²+l1+a13, (11) where

g=a1x−5 9

(a⁴₁−16a₁²a₅²−16a⁴₅)a₃²

a₁⁵ y+a3t+a4, h=a5x+5

9

(7a₁⁴−8a₁²a₅²−16a⁴₅)a₅a₃²

a₁⁶ y

+a5a3(3a²₁+4a₅²) a³₁ t+a8,

l1=exp

4a5

a₁ a3

3a₁x− 4 27

a₃³a₅(5a⁴₁−40a₁²a₅²+16a₅⁴) a₁⁸

×

3a1

a3

y+4 9

a²₃a5(3a₁²−4a²₅) a⁵₁

3a1

a3

t+a₁₂

. (12)

The 3D, 2D plots, density plot and contour plot of u(x,y,t) for the lump-kink solution are presented in figure1. The following condition is called the collision phenomenon that is given as

a₃²+

a5a3(3a₁²+4a²₅) a₁³

2

+

4 9

a₃²a5(3a₁²−4a₅²) a⁵₁

3a1

a3

2

=0.

(4)

(d) (c)

Figure 2. Plot of eq. (14) by taking the parametersa1=a8=a9=2,a4=a12 =3,a5=1,a13 =5,y= −1.(a)3D plot, (b)density plot,(c)contour plot and(d)2D plot (x=1 – red,x=2 – blue andx =3 – green).

Set II:

⎧⎪

⎪⎨

⎪⎪

⎩

a₁=a₁, a₂ =a₃ =a₆ =a₇=0, a4=a4 a5=a5,

a8=a8, a9 =a9, a10 = −a⁵₉, a11 = −a₉³, a12 =a12, a13 =a13,

(13)

wherea1,a4,a5,a8,a9,a12 anda13 are free constants.

Inserting (13) into (8), we can acquire the solutionuas u= −2(2a₁²+2a₅²+a₉²l1)

f +2(2a₁g+2a₅h+a₉l₁)²

f² ,

f =g²+h²+l1+a13, (14) where

g=a1x+a4, h =a5x+a8,

l₁=exp(a₉x−a₉⁵y−a₉³t+a₁₂). (15) The 3D, 2D plots, density plot and contour plot of u(x,y,t) for the lump–kink solution are presented in figure2.

SetIII:

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩

a1 =0, a2= − 5 54

(4a₃²−9a₅²a₉⁴)a3

a²₅a²₉ , a3 =a3, a4 =a4, a5 =a5,

a6 = − 5 1296

81a₅⁴a₉⁸−216a²₅a⁴₉a²₃+16a₃⁴ a₅³a₉⁴ , a7 = 4a₃²−9a²₅a⁴₉

12a5a₉² , a8 =a8, a9=a9, a₁₀=− 1

1296

81a⁴₅a⁸₉−360a₅²a₉⁴a₃²+80a⁴₃ a₅⁴a₉³ , a₁₁ = 4a²₃−3a₅²a₉⁴

12a9a₅² ,

a₁₂ =a₁₂, a₁₃ = a₅²(4a²₃−9a₅²a₉⁴) 4a₉²a₃²

(16)

and

a₃,a₅,a₉=0, (17)

(5)

(d) (c)

Figure 3. Plot of eq. (18) by taking the parametersa1=a8=a9=2,a4=a12=3,a5=1,a13 =5,y= −1. (a) 3D plot, (b) density plot, (c) contour plot and (d) 2D plot (x=1 – red,x=2 – blue andx=3 – green).

where a₃, a₄, a₅, a₈,a₉ and a₁₂ are free constants.

Putting (16) into (8), we can acquire the solution f, and then the solution is

u= − 2

2a²₁+2a²₅+a₉²l1

f +2(2a1g+2a5h+a9l1)²

f² ,

f =g²+h²+l1+a²₅(4a₃²−9a₅²a₉⁴)

4a₉²a₃² , (18) where

g= − 5 54

(4a₃²−9a₅²a₉⁴)a3

a₅²a²₉ y+a3t+a4, h=a5x − 5

1296

81a₅⁴a₉⁸−216a₅²a₉⁴a²₃+16a₃⁴ a₅³a₉⁴ y +4a²₃−9a₅²a₉⁴

12a5a²₉ t+a₈, l1=exp

a9x − 1 1296

81a₅⁴a₉⁸−360a²₅a⁴₉a²₃+80a₃⁴ a₅⁴a₉³ y +4a₃²−3a₅²a₉⁴

12a9a₅² t+a12

. (19)

The 3D, 2D plots, density plot and contour plot of u(x,y,t) for the lump-kink solution are presented in figure3.

2.2 First type of lump-soliton solution of the bSK equation

In this subsection, to get the lump-soliton solutions, suppose f is a combination of the hyperbolic cosine function and positive quadratic function:

f =g²+h²+cosh(l₁)+a₁₃, g =a₁x+a₂y+a₃t+a₄, h =a5x+a6y+a7t+a8,

l1 =a9x+a10y+a11t+a12, (20) where a_i,i = 1, . . . ,13, are free constants. Insert- ing (20) into eq. (6), we shall acquire the following results:

(6)

(d) (c)

Figure 4. Plot of eq. (23) by taking the parametersa3=0.9,a5 =1.2,a8 =1.3,a4 =a12 =1,a9 =2,t = −1. (a) 3D plot, (b) density plot, (c) contour plot and (d) 2D plot (x= −1 – red,x=0 – blue andx=1 – green).

Set I:

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩

a₁=0, a₂ = − 5 54

(4a₃²−9a₅²a₉⁴)a3

a₅²a₉² , a3=a3, a4 =a4, a5 =a5,

a6= − 5 1296

81a₅⁴a₉⁸−216a₅²a₉⁴a₃²+16a⁴₃ a₅³a₉⁴ , a7= 4a₃²−9a₅²a₉⁴

12a5a₉² , a8 =a8, a9 =a9, a10 = − 1

1296

81a⁴₅a⁸₉−360a²₅a⁴₉a²₃+80a₃⁴ a₅⁴a₉³ , a11 = 4a₃²−3a²₅a⁴₉

12a₉a₅² , a12 =a12,

a13 = 4a₃²a₉⁴+9a⁸₉a₅²+16a₃²a₅⁴−36a⁶₅a⁴₉ 16a₉²a₅²a₃²

(21)

and

a₃,a₅,a₉ =0, (22)

wherea3,a4,a5,a8,a9 anda12 are free constants. By usingu= −2(ln(f))x x, the solution will be

u = −2[2a²₁+2a₅²+a₉²cosh(l1)]

f

+2[2a1g+2a5h+a9sinh(l1)]²

f² , (23)

where

f =g²+h²+cosh(l1)

+4a₃²a₉⁴+9a⁸₉a²₅+16a₃²a₅⁴−36a₅⁶a₉⁴ 16a₉²a₅²a₃² , g= − 5

54

(4a²₃−9a₅²a₉⁴)a3

a₅²a₉² y+a₃t+a₄, h =a₅x− 5

1296

81a₅⁴a⁸₉−216a₅²a₉⁴a₃²+16a₃⁴ a₅³a₉⁴ y +4a₃²−9a²₅a⁴₉

12a5a₉² t+a8,

(7)

(d) (c)

Figure 5. Plot of eq. (26) by taking the parametersa3=a4=a13 =1,a8=1.2,a9 =a12 =2,t = −1. (a) 3D plot, (b) density plot, (c) contour plot and (d) 2D plot (x= −1 – red,x=0 – blue andx=1 – green).

l1=a9x − 1 1296

81a₅⁴a₉⁸−360a₅²a₉⁴a₃²+80a⁴₃ a⁴₅a³₉ y +4a²₃−3a₅²a₉⁴

12a₉a²₅ t+a₁₂. (24)

One can see that (24) is a merge of the hyperbolic cosine function and positive quadratic function. The 2D plot, 3D plot, density plot and contour plot ofu(x,y,t)for the lump-soliton solution are presented in figure4.

Set II:

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

a1 = −4a3

3a₉², a2= 5

12a₃a₉², a3 =a3, a4=a4, a5 =

18a₉⁶−64a²₃

6a₉² , a6 = − 5 96

18a₉⁶−64a₃²a₉², a₇ = −1

8

18a₉⁶−64a₃², a₈ =a₈, a₉=a₉, a₁₀ = − 1

16a₉⁵, a₁₁=−1

4a³₉, a₁₂ =a₁₂, a₁₃ =a₁₃, (25)

where a3, a8, a9(=0),a10, a12 anda13 are free constants. Then solution of eq. (4) is given as

u= −2[2a²₁+2a₅²+a₉²cosh(l1)]

f

+2[2a1g+2a5h+a9sinh(l1)]²

f² , (26)

where

f =g²+h²+cosh(l1)+a13, g= −4a3

3a₉²x + 5

12a₃a₉²y+a₃t+a₄, h =

18a₉⁶−64a₃² 6a²₉ x− 5

96

18a₉⁶−64a²₃a²₉y

−1 8

18a₉⁶−64a²₃t+a₈, l1=a9x− 1

16a₉⁵y−1

4a₉³t+a12. (27)

(8)

(d) (c)

Figure 6. Plot of eq. (31) by taking the parametersa3=4,a4=a8=a12 =1,a5=1.2,a9=1.4,t =1.(a)3D plot,(b) density plot,(c)contour plot and(d)2D plot (x= −1 – red,x=0 – blue andx=1 – green).

Moreover, one can see that (26) is a lump-kink solution, which is presented in figure5.

2.3 Second type of lump-soliton solution of the bSK equation

In this subsection, to get the second type of lump-soliton solutions, let

f =g²+h²+sinh(l1)+a13, g=a1x+a2y+a3t+a4, h =a5x+a6y+a7t+a8,

l1=a9x +a10y+a11t+a12, (28) where a_i,i = 1, . . . ,13 are free constants. Putting (28) into eq. (6), we shall acquire the following results.

SetI:

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩

a1 =0, a2= − 5 54

(4a₃²−9a₅²a⁴₉)a3

a₅²a₉² , a₃ =a₃, a₄=a₄, a₅ =a₅,

a6 = − 5 1296

81a₅⁴a₉⁸−216a₅²a₉⁴a₃²+16a₃⁴ a³₅a₉⁴ , a7 = 4a²₃−9a₅²a₉⁴

12a₅a²₉ , a8=a8, a9 =a9, a10 = − 1

1296

81a₅⁴a₉⁸−360a₅²a₉⁴a₃²+80a₃⁴ a₅⁴a₉³ , a11 = 4a₃²−3a₅²a₉⁴

12a9a²₅ , a12 =a12,

a13 = −4a²₃a⁴₉−9a₉⁸a₅²+16a²₃a⁴₅−36a₅⁶a₉⁴ 16a₉²a₅²a₃²

(29)

and

a3,a5,a9=0, (30)

(9)

(c) (d)

Figure 7. Plot of eq. (34) by taking the parametersa3=a4=a13 =1,a8=1.2,a9 =a12 =2,t = −1. (a) 3D plot, (b) density plot, (c) contour plot and (d) 2D plot (x= −1 – red,x=0 – blue andx=1 – green).

where a₃,a₄,a₅,a₈,a₉ anda₁₂ are free constants. By usingu = −2(ln(f))x x, then the solution will be u = −2[2a₁²+2a₅²+a₉²sinh(l₁)]

f

+2[2a1g+2a5h+a9cosh(l1)]²

f² , (31)

where

f = g²+h²+sinh(l1)

+−4a₃²a₉⁴−9a⁸₉a²₅+16a₃²a₅⁴−36a₅⁶a⁴₉ 16a²₉a²₅a²₃ , g= − 5

54

(4a²₃−9a₅²a₉⁴)a3

a₅²a₉² y+a3t+a4, h= a5x − 5

1296

81a₅⁴a₉⁸−216a²₅a⁴₉a²₃+16a₃⁴ a₅³a₉⁴ y +4a²₃−9a₅²a₉⁴

12a5a₉² t+a8,

l1= a9x− 1 1296

81a⁴₅a⁸₉−360a₅²a₉⁴a₃²+80a₃⁴ a₅⁴a₉³ y +4a₃²−3a²₅a⁴₉

12a9a₅² t+a12. (32)

One can see that (32) is a merge of the hyperbolic sine function and positive quadratic function. The 2D plot, 3D plot, density plot and contour plot ofu(x,y,t)for the lump-soliton solution are presented in figure6.

SetII:

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

a1 = −4a₃

3a₉², a2 = 5

12a3a²₉, a3=a3, a4 =a4, a5= b

3a₉², a6= − 5 48ba₉², a₇ = −1

4b, a₈=a₈, a₉ =a₉, a₁₀ = − 1

16a⁵₉, a₁₁= −1 4a³₉, a12 =a12, a13 =a13,

(33)

(10)

(d) (c)

Figure 8. Plot of eq. (38) by taking the parametersa1=b1=2,a4=1.2,a5=1.3,a8=1.5,a9=1,p =0.8,p1=1.5, y= −1.(a)3D plot,(b)density plot,(c)contour plot and(d)2D plot (t =1 – red,t =2 – blue andt =3 – green).

wherebsolves 81a₉¹²+1024a⁴₃+128b²a₃²+4b⁴ =0, anda3,a8,a9(=0),a10,a12anda13are free constants.

Then solution of eq. (4) is given by u = −2[2a₁²+2a₅²+a₉²sinh(l₁)]

f

+2[2a1g+2a5h+a9cosh(l1)]²

f² , (34)

where

f =g²+h²+sinh(l1)+a13, g= −4a3

3a²₉x+ 5

12a₃a₉²y+a₃t+a₄, h= b

3a₉²x− 5

48ba²₉y−1

4bt+a8, l1=a9x − 1

16a₉⁵y−1

4a³₉t+a12. (35)

Likewise, one can understand that (34) is a mix of the hyperbolic sine function and positive quadratic function, which is presented in figure7.

2.4 Kinky breather-soliton solution

To achieve kinky breather-soliton solution, we utilise the following function:

f =e⁻^pg+b1e^pg+b0cos(p1h)+a9, g=a1x +a2y+a3t+a4,

h =a5x +a6y+a7t+a8, (36) where a_i,i = 1, . . . ,9,b₀ and b₁ are free constants.

Inserting (36) into eq. (6), we shall obtain the following results.

(11)

(d) (c)

Figure 9. Plot of eq. (41) by taking the parametersa1 =1.2,a4 = a8 =1,a5 =1.3,a8 =1,a9 =1.4,p =1,p1 =2, y= −1. (a) 3D plot, (b) density plot, (c) contour plot and (d) 2D plot (t =0.1 – red,t =0.2 – blue andt=0.3 – green).

Set I:

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

a₁=a₁, a₂ = −p⁴a⁵₁, a₃= −p²a₁³, a4=a4, a5 =a5,

a6= −p⁴₁a₅⁵, a7 = p₁²a₅³,

a8=a8, a9 =a9, b0 =b0, b1=b1,

(37)

wherea1,a4,a5,a8anda9are free parameters. By using u = −2(ln(f))x x, then the solution of eq. (4) is presented as

u = −2(p²a₁²e⁻^pg+b1p²a²₁e^pg−b0p²₁a²₅cos(p1h)) f

+2

−pa₁e⁻^pg+b₁pa₁e^pg−b₀p₁a₅sin(p₁h)2

f² ,

(38) where

f =e⁻^pg+b₁e^pg+b₀cos(p₁h)+a₉, g=a₁x −p⁴a₁⁵y−p²a₁³t+a₄,

h=a5x −p₁⁴a₅⁵y+p₁²a₅³t+a8. (39)

SetII:

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩

a₁ =a₁, a₂ = − 1

16a₁(p⁴a₁⁴−10p²a₁²a²₅p₁²+5a⁴₅p₁⁴), a3 = −1

4a1(p²a₁²−3a₅²p₁²), a4 =a4, a5=a5,

a6 = − 1

16a5(5p⁴a₁⁴−10p²a²₁a²₅p²₁+a₅⁴p₁⁴), a7 = 1

4a5(p₁²a₅²−3a²₁p²), a8 =a8, a9 =a9, b0 =b0, b1 = a₅⁴p⁴₁b²₀

4p⁴a⁴₁ ,

(40)

wherea₁,a₄,a₅,a₈anda₉are free parameters. By using u = −2(ln(f))x x, then the solution of eq. (4) is presented as

(12)

Figure 10. Plot of eq. (45) by taking the parametersa1=1.5,a2 =a4=1,a5 =2,y = −1. (a) 3D plot and (b) 2D plot (t =1–red,t =2–blue andt =3–green).

u = −2(p²a₁²e⁻^pg+^a₄⁵⁴_p^p2¹⁴a^b₁²²⁰e^pg−b0p₁²a₅²cos(p1h)) f

+2(−pa1e⁻^pg+^a₄⁴⁵_p^p3⁴¹a^b³₁²⁰e^pg−b0p1a5sin(p1h))²

f² ,

(41) where

f =e⁻^pg+b1e^pg+b0cos(p1h)+a9, g=a₁x − 1

16a₁(p⁴a₁⁴−10p²a²₁a²₅p²₁+5a₅⁴p⁴₁)y

−1

4a₁(p²a²₁−3a₅²p²₁)t+a₄, h=a5x − 1

16a5(5p⁴a⁴₁−10p²a₁²a₅²p²₁+a₅⁴p⁴₁)y +1

4a5(p²₁a²₅−3a₁²p²)t+a8. (42) The analytical solutions (38) and (41) are really a kinky breather-soliton solutions. Figures8and9indicate the plots of the breather-soliton solutions.

3. Stripe soliton solution

As the last test, we consider the stripe soliton solution for the bSK equation and choose the following function:

f =1+a5eâ¹^x⁺â²^y⁺â³^t⁺â⁴, (43) whereai,i =1, . . . ,5, are free constants. By inserting (43) into (6), we shall acquire the following result:

a₁ =a₁, a₂ =a₂, a₃ = 1

2a₁³+ 3 10

5a₁⁶−20a₁a₂,

a4 =a4, a5=a5, (44)

and by using u = −2(ln(f))x x, then the solution of eq. (4), one can be stated as

u = − 2a²₁a₅e^a¹^x⁺^a²^y⁺

1 2a³₁+₁₀³

5a₁⁶−20a₁a₂

t+a₄

⎛

⎝1+a5e^a¹^x+a²^y+

1 2a₁³+₁₀³

5a⁶₁−20a1a2

t+a4

⎞

⎠² ,

(45) wherea₁,a₂,a₄anda₅ are free constants. The asymp- totic behaviour of the stripe soliton solution (45) can be acquired whenu → −∞, asa1 > 0,t → +∞or a1 < 0,t → −∞;u → 0, asa1 > 0,t → −∞ or a1 < 0,t → −∞. The kink-type solitary wave solution, and its corresponding 3D and 2D plots can be seen in figure10.

4. Conclusion

With the help of the symbolic calculation and applying the Hirota bilinear method, we have found some new exact solutions for the(2+1)-dimensional bSK equation. As a result, some new solutions, which include the lump-kink, lump-solitons, periodic kink-wave, solitary wave and periodic wave solutions, were obtained.

We then think that these results will help in conduct- ing future research in various areas of physics such as mathematical physics, nonlinear mechanics and other applied fields.