Over the past few decades, the research of the nonlinear evolution equations (NLEEs) is flourishing because of the rich findings of these equations. NLEEs display many interesting nonlinear dynamic behaviors, such as plasma physics, optical systems, ocean, superfluids, hydrodynamics and other nonlinear fields [1] [2] [3] [4]. The solutions of NLEEs can provide much physical information and more insight into the physical aspects and then lead to further applications, so seeking exact solutions is an important problem in nonlinear science. In fact, there are a variety of powerful methods have been used to solve the NLEEs. For instance, the Darboux transformation [5], the Lie group method [6], the Hirota bilinear method [7], the inverse scattering method [8] and other methods. Recent study for constructing lump solution by taking the function f in its bilinear form as a positive quadratic function based on Hirota bilinear method was obtained in [9]. Then, by using this method, more and more the lump solutions and the interaction solutions have been perfectly obtained in NLEEs equations [10] - [26].

In this paper, we focus on the following (2 + 1)-dimensional KdV-Calogero-Bogoyavlenkskii-Schiff (KdV-CBS) equation [27] [28] [29].

4 u t − α ( 4 u u y + 2 u x ∂ x − 1 u y + 2 u x x y ) − β ( 6 u u x + u x x x ) = 0 , (1)

which with two arbitrary constant α and β , Equation (1) can be reduced to some other equations with physical meanings. If we setting α ≠ 0 , β = 0 , Equation (1) is reduced to Calogero-Bogoyavlenskii-Schiff equation [30].

4 u t − α ( 4 u u y + 2 u x ∂ x − 1 u y + u x x y ) = 0. (2)

If setting { α = 0 , β ≠ 0 } , Equation (1) is famous KdV equation

4 u t − β ( u x x x + 6 u u x ) = 0. (3)

Let v x = u y , Equation (1) is transformed into the following system

u t − α ( 4 u u y + 2 u x v + u x x y ) − β ( 6 u u x + u x x x ) = 0 , (4a)

v x − u y = 0. (4b)

By applying Painlevé truncated extension, some types of explicit solutions (1) were studied in [31]. A direct bilinear Bäcklund transformation on the basis of quadrilinear representation was constructed in [32] [33]. Ref. [34] obtained soliton solutions, quasiperiodic wave solutions of the Equation (1) based on the Riemann-Bäcklunk method. Interaction solutions of Equation (1) were obtained under consistent Riccati expansion in [29].

The outline of this paper is as follows. In Section 2, we convert the original KdV-CBS equation to a trilinear equation by using the truncated Painlevé expansion. In Section 3, we derived exact solutions for the KdV-CBS equation by taking function f as a positive quadratic function; another kind of exact solutions were got by taking function f as a positive quadratic and an exponential function; by taking function f as two exponential terms and the quadratic function, we obtained three kinds of exact solutions respectively via different parameters. A short conclusion is included in the last section.

Based on the Painlevé analysis proposed in Ref. [4], Equation (4a) and Equation (4b) possess a truncated Painlevé expansion as follows:

u = u 0 + u 1 f + u 2 f 2 , v = v 0 + v 1 f + v 2 f 2 , (5)

with u 0 , u 1 , v 0 , v 1 , v 2 , f being the function of x , y and t. We have

v 1 = f x y , v 2 = − 2 f x f y , u 2 = − 2 f x 2 , u 0 = λ − f x x x 2 f x + f x x 2 4 f x 2 , u 1 = 2 f x x , v 0 = − 3 β λ α + 2 f t α f x − 2 λ f y f x + 3 β 4 α f x x 2 f x 2 + f x x f x y f x 2 − f x x y f x − β 2 α f x x x f x , (6)

the Equation (6) is from Ref. [35]. The function f satisfies the following trilinear equation,

α f x x x y f x 2 + 4 α λ f x 2 f x y + β f x x x x f x 2 − α f f x f x x x f x y − 4 β f x x x f x f x x − 3 α f x f x x f x y   − 4 λ α f x x f x f y + 3 α f x y f x x 2 + 3 β f x x 3 − 4 f x 2 f x t + 4 α 4 f x 2 f x f x = 0. (7)

By solving the Equation (5), we can get

u = λ − 1 2 f x x x f x + f x x 2 4 f x 2 , (8a)

v = − 3 β λ α + 2 f t α f x − 2 λ f y f x + 3 β 4 α f x x 2 f x 2 + f x x f x y f x 2 − f x x y f x − β 2 α f x x x f x . (8b)

Based on the idea in Ref. [9], the (2 + 1)-dimensional KdV-Calogero-Bogoy-avlenkskii-Schiff equation has a set of rational solutions determined by

f = ( a 1 x + a 2 y + a 3 t + a 4 ) 2 + ( a 5 x + a 6 y + a 7 t + a 8 ) 2 + a 9             + m exp ( k 1 x + k 2 y + k 3 t + k 4 ) + n exp ( − ( k 1 x + k 2 y + k 3 t + k 4 ) ) , (9)

where a i , k , k j ,1 ≤ i ≤ 9,1 ≤ j ≤ 4 are real parameters to be determined. Substituting function (9) into trilinear Equation (7), we give five kinds of exact solutions of Equation (4a) and Equation (4b) via function (9).

Case I

Substitution function (9) into trilinear Equation (7), a set of constraining equations for the parameters can be obtained as following

a 1 = 0 , a 2 = a 2 , a 3 = a 2 λ α , a 4 = a 4 , a 5 = a 5 , a 6 = − a 5 β λ , a 7 = a 7 , a 8 = a 8 , a 9 = a 9 , m = 0 , n = 0 , k 1 = k 1 , k 2 = k 2 , k 3 = k 3 , k 4 = k 4 , β = β , λ = λ (10)

where

λ ≠ 0 , a 9 > 0 (11)

to guarantee that the corresponding f is positive. By substituting Equation (10) into function (9), then we get the function f reads

f = ( a 2 y + a 2 λ α t + a 4 ) 2 + ( a 5 x − a 5 β λ y + a 7 t + a 8 ) 2 + a 9 . (12)

By means of Equation (8a) and Equation (8b), the solutions of (4a) and (4b) can be obtained

u = λ + a 5 2 2 A 1 2 , (13a)

v = − 3 β λ α + 2 B 1 A 1 − 2 λ C 1 A 1 − β a 5 2 α A 1 2 , (13b)

where

f = g 2 + h 2 + a 9 , g = a 2 y + a 2 λ α t + a 4 , h = a 5 x − a 5 β λ y + a 7 t + a 8 , A 1 = 2 a 5 f , B 1 = 2 a 2 α λ f + 2 a 7 g , C 1 = 2 a 2 f − 2 a 5 β λ g . (14)

The solutions of u via Equation (13a) which can be seen in Figure 1(a); Figure 1(b) is the corresponding density plot. The solutions of v via Equation (13b) can be seen in Figure 1(c); Figure 1(d) is the corresponding density plot.

Case II

a 1 = a 1 , a 2 = − a 1 β α , a 3 = a 3 , a 4 = a 4 , a 5 = 0 , a 6 = a 6 , a 7 = a 6 α λ , a 8 = a 8 , a 9 = a 9 , m = m , k 1 = 0 , k 2 = k 2 , k 3 = k 2 α λ , k 4 = k 4 , (15)

which should satisfy the constraint conditions

α ≠ 0 , a 9 > 0 , m > 0 , (16)

to ensure that the corresponding f is positive and well defined. By substituting Equation (15) into function (9), then the function f reads

f = ( a 1 x − a 1 β α y + a 3 t + a 4 ) 2 + ( a 6 y + a 6 α λ t + a 8 ) 2             + m exp ( k 2 y + k 2 α λ t + k 4 ) + a 9 . (17)

Using Equation (8a) and Equation (8b), the solutions for (4a) and (4b) can be obtained

u = λ + a 1 2 4 g 2 , (18a)

v = − 3 β λ α + A 2 α λ B 2 α a 1 g + 3 β a 1 2 16 λ g 2 − a 1 3 β α g 2 , (18b)

where

f = g 2 + h 2 + m exp ( ξ ) + a 9 , g = a 1 x − a 1 β α y + a 3 t + a 4 , h = a 6 y + a 6 α λ t + a 8 , A 2 = 2 a 3 g + 2 a 6 α λ h + k 2 m α λ exp ( ξ ) , B 2 = − 2 a 1 β g α + 2 a 6 h + k 2 m exp ( ξ ) , ξ = k 2 y + k 2 α λ t + k 4 , (19)

The solutions of u via Equation (18a) can be seen in Figure 2(a); Figure 2(b) is the corresponding density plot. The solutions of v via Equation (18b) can be seen in Figure 2(c); Figure 2(d) is the corresponding density plot.

Case III

a 1 = a 1 , a 2 = a 2 , a 3 = a 3 , a 4 = a 4 , a 5 = 0 , a 6 = a 6 , a 7 = a 6 α λ , a 8 = a 8 , a 9 = a 9 , α = α , β = a 2 α a 1 , k 1 = 0 , k 2 = k 2 , k 3 = k 2 α λ , k 4 = k 4 , λ = λ , m = m , n = n , (20)

where

a 1 ≠ 0 , a 9 > 0 , m > 0 , n > 0. (21)

to ensure that the corresponding f is positive and well defined. By substituting Equation (20) into function (9),

f = ( a 1 x + a 2 y + a 3 t + a 4 ) 2 + ( a 6 x + a 6 α λ t + a 8 ) 2 + a 9     + m exp ( k 2 y + k 2 α λ t + k 4 ) + n exp ( − ( k 2 y + k 2 α λ t + k 4 ) ) , (22)

where a i , k j ,1 ≤ i ≤ 9,1 ≤ j ≤ 4 are all real parameters to be determined. Then we get the solutions of the Equation (4a) and Equation (4b)

u = λ + a 1 2 4 g 2 , (23a)

v = − 3 a 2 λ a 1 + 2 B 3 − λ α C 3 α A 3 + a 1 3 a 2 A 3 2 , (23b)

with

f = h 2 + g 2 + a 9 + m exp ( ξ ) + n exp ( − ξ ) , g = a 1 x + a 2 y + a 3 t + a 4 , h = a 6 x + a 6 α λ t + a 8 , ξ = k 2 y + k 2 α λ t + k 4 , A 3 = 2 a 1 g , C 3 = 2 a 2 g + 2 a 6 h + m k 2 exp ( ξ ) − n k 2 exp ( − ξ ) B 3 = 2 a 3 g + 2 a 6 α λ h + m k 2 α λ exp ( ξ ) − n k 2 α λ exp ( ξ ) (24)

The exact solutions for Equation (23a) and Equation (23b) can be seen in Figure 3(a) and Figure 3(c), Figure 3(b) and Figure 3(d) are the corresponding density plots.

Case IV

a 1 = a 1 , a 2 = − a 1 β α , a 3 = − a 1 a 7 α a 6 , a 4 = − a 1 a 8 a 6 α , a 5 = − α a 6 β , a 6 = a 6 , a 7 = a 7 , a 8 = a 8 , a 9 = a 9 , m = m , n = n , α = α , β = β , k 1 = 0 , k 2 = k 2 , k 3 = k 2 α λ , k 4 = k 4 , λ = λ (25)

where

α β a 6 ≠ 0 , a 9 > 0 , m > 0 , n > 0. (26)

to guarantee that the corresponding f is positive and well defined. We substitute Equation (25) into Equation (9), hence, reinstall function f as the following formula:

f = ( a 1 x − a 1 β α y − a 1 a 7 α a 6 t − a 1 a 8 a 6 α ) 2 + ( a 5 x + a 6 y + a 7 t + a 8 ) 2 + a 9             + m exp ( k 2 y + k 2 α λ t + k 4 ) + n exp ( − ( k 2 y + k 2 α λ t + k 4 ) ) , (27)

where a i , k j ,1 ≤ i ≤ 9,1 ≤ j ≤ 4 are all real parameters to be determined. Then, we get

u = λ + D 4 2 A 4 2 , (28a)

v = − β λ α + 2 C 4 α A 4 + 2 λ B 4 A 4 + D 4 ( 3 β D 4 + 4 α E 4 ) 4 α A 4 2 , (28b)

where

f = h 2 + g 2 + α exp ( ξ ) + β exp ( − ξ ) + a 9 , h = a 1 x − a 1 β α y − a 1 a 7 α a 6 t − a 1 a 8 a 6 α , g = a 5 x + a 6 y + a 7 t + a 8 , ξ = k 2 y + k 2 α λ t + k 4 ,

A 4 = 2 ( a 1 h + a 5 g ) , B 4 = − 2 a 1 β α h + 2 a 6 g + k 2 ( m exp ( ξ ) − n exp ( − ξ ) ) , C 4 = − 2 a 1 a 7 α a 6 h + 2 a 7 g + k 2 α λ ( m exp ( ξ ) − n exp ( − ξ ) ) , D 4 = 2 ( a 1 2 + a 5 2 ) , E 4 = 2 ( a 5 a 6 − a 1 2 β α ) . (29)

The exact solution for Equation (28b) can be seen in Figure 4(a), Figure 4(b) Figure 4(c) the corresponding density plots with the different time.

Case V.

a 1 = 0 , a 2 = a 2 , a 3 = α λ a 2 , a 4 = a 4 , a 5 = a 6 k 1 k 2 a 6 = a 6 , a 7 = a 7 , a 8 = a 8 , a 9 = a 9 , α = − β k 1 k 2 , β = β , k 1 = k 1 , k 2 = k 1 , k 3 = a 7 k 2 a 6 , k 4 = k 4 , m = m , n = n . (30)

with

k 2 a 6 ≠ 0 , a 9 > 0 , m > 0 , n > 0. (31)

to guarantee that the corresponding f is positive and well defined. We substitute Equation (30) into Equation (9), then function f reads

f = ( a 2 y + α λ a 2 t + a 4 ) 2 + ( a 6 k 1 k 2 x + a 6 y + a 7 t + a 8 ) 2     + a 9 + m exp ( k 1 x + k 2 y + a 7 k 2 a 6 t + k 4 )     + n exp ( − ( k 1 x + k 2 y + a 7 k 2 a 6 t + k 4 ) ) , (32)

where a i , k j ,1 ≤ i ≤ 9,1 ≤ j ≤ 4 are all real parameters to be determined. Then, the rational solution of system (4a) and (4a) can be got again.

u = λ − k 1 3 ( m exp ( ξ ) − n exp ( − ξ ) ) 2 A 5 + D 5 4 A 5 2 , v = − 3 β λ α + 2 B 5 A 5 + 2 λ C 5 A 5 , + 3 β D 5 2 4 α A 5 2 + D 5 E 5 A 5 2 − F 5 A 5 − β G 5 2 α A 5 , (33)

and where

f = g 2 + h 2 + m exp ( ξ ) + n exp ( − ξ ) + a 9 , g = a 2 y + α λ a 2 t + a 4 , h = a 6 k 1 k 2 x + a 6 y + a 7 t + a 8 , ξ = k 1 x + k 2 y + a 7 k 2 a 6 t + k 4 , A 5 = 2 a 6 k 1 k 2 + k 1 h + m k 1 exp ( ξ ) − n k 1 exp ( − ξ ) ,

B 5 = 2 α λ a 2 g + 2 a 7 h + a 7 k 2 a 6 ( m exp ( ξ ) + n exp ( − ξ ) ) , C 5 = 2 a 2 g + 2 a 6 h + k 2 ( m exp ( ξ ) + n exp ( − ξ ) ) , D 5 = 2 a 6 2 k 1 2 k 2 2 + k 1 2 ( m exp ( ξ ) + n exp ( − ξ ) ) , E 5 = k 1 k 2 ( m exp ( ξ ) + n exp ( − ξ ) ) + 2 a 6 a 6 k 1 k 2 , F 5 = k 1 2 k 2 ( m exp ( ξ ) + n exp ( − ξ ) ) . (34)

The exact solutions for Equation (33) can be seen in Figure 5(a) and Figure 5(c), Figure 5(b) and Figure 5(d) are the corresponding density plots.

In this paper, by using the truncated Painlev expansion, the (2 + 1)-dimensional KdV-CBS equation can be changed to a trilinear Equation (7), and by means of symbolic computations, we presented five types of exact solutions ((13a), (13b), (18a), (18b), (23a), (23b), (28a), (28b), (33)) to the (2 + 1)-dimensional KdV-CBS equation. Three-dimensional plots and the corresponding density plots of these exact solutions are given respectively in Figure 1 and Figure 5. This work shows the power of the methods, the exact solutions of some nonlinear equations can be obtained by using this means. It is expected that our results can enrich the exact solutions of the nonlinear equations.

This work is supported by the National Natural Science Foundation of China under grant number 11571008.

The authors declare no conflicts of interest regarding the publication of this paper.

Li, Y. and Chaolu, T. (2020) Exact Solutions for (2 + 1)-Dimensional KdV-Calogero-Bogoyavlenkskii-Schiff Equation via Symbolic Computation. Journal of Applied Mathematics and Physics, 8, 197-209. https://doi.org/10.4236/jamp.2020.82015