Exact Solutions for (2 + 1)-Dimensional KdV-Calogero-Bogoyavlenkskii-Schiff Equation via Symbolic Computation

DOI: 10.4236/jamp.2020.82015   PDF   HTML   XML   167 Downloads   295 Views  

Abstract

This paper constructs exact solutions for the (2 + 1)-dimensional KdV-Calogero-Bogoyavlenkskii-Schiff equation with the help of symbolic computation. By means of the truncated Painlev expansion, the (2 + 1)-dimensional KdV-Calogero-Bogoyavlenkskii-Schiff equation can be written as a trilinear equation, through the trilinear-linear equation, we can obtain the explicit representation of exact solutions for the (2 + 1)-dimensional KdV-Calogero-Bogoyavlenkskii-Schiff equation. We have depicted the profiles of the exact solutions by presenting their three-dimensional plots and the corresponding density plots.

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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. doi: 10.4236/jamp.2020.82015.

1. Introduction

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.

2. Trilinear Equation

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).

3. Lumps and Interaction Solutions

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)

Figure 1. Profiles of the solution u via Equation (13a) and profiles of the solution v via Equation (13b) with a 2 = 1 , a 4 = 1 , a 5 = 1 , a 7 = 1 , a 8 = 0 , a 9 = 0.5 , β = 1 , α = 1 , λ = 2 , t = 0 . (a) and (c) are the corresponding 3-dimensional plots; (b) and (d) are the corresponding density plots.

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)

Figure 2. Profiles of the solution u via Equation (18a) and the solution v via Equation (18b) with a 1 = 0.5 , a 3 = 2 , a 4 = 1 , a 5 = 0 , a 6 = 0 , a 9 = 0.5 , a 8 = 1 , β = 0 , α = 1 , λ = 1 , m = 1 , k 1 = 0 , k 2 = 1 , k 4 = 1 , t = 1 . (a) and (c) are the corresponding 3-dimensional plots; (b) and (d) are the corresponding density plots.

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

Figure 3. Profiles of the solution u via Equation (23a) and v via Equation (23b) with a 1 = 1 , a 2 = 1 , a 3 = 2 , a 4 = 2 , a 5 = 0 , a 6 = 1 , a 8 = 1 , a 9 = 1 , k 1 = 0 , k 2 = 1 , k 4 = 1 , m = 1 , n = 1 , α = 1 , λ = 1 , t = 0 . (a) and (c) are the corresponding 3-dimensional plots; (b) and (d) are the corresponding density plots.

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.

Figure 4. Profiles of the solution v via Equation (28b) with a 1 = 1 , a 4 = 0 , a 5 = 2 , a 6 = 4 , a 8 = 0 , a 9 = 1 , α = 1 , β = 1 , k 1 = 1 , k 2 = 1.5 , k 3 = 2 , k 4 = 0 . (a) 3-dimensional plot with the time t = 0; (b) (c) (b) the corresponding density plot 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)

Figure 5. Profiles of the solution u and v via Equation (33) with a 2 = 1 , a 4 = 1 , a 6 = 1 , a 7 = 1 , a 8 = 2 , a 9 = 1 , k 1 = 1 , k 2 = 1 , k 4 = 1 , m = 1 , n = 1 , λ = 1 , β = 1 , t = 0 . (a) and (c) are the 3-dimensional plots; (b) and (d) are the corresponding density plots.

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.

4. Conclusion

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.

Acknowledgements

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

Conflicts of Interest

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

References

[1] Gear, J.A. and Grimshaw, R. (1984) Weak and Strong Interactions between Internal Solitary Waves. Studies in Applied Mathematics, 70, 235-258.
https://doi.org/10.1002/sapm1984703235
[2] Ruggieri, M. and Speciale, M.P. (2013) Similarity Reduction and Closed Form Solutions for a Model Derived from Two-Layer Fluids. Advances in Difference Equations, 1, 355-363.
https://doi.org/10.1186/1687-1847-2013-355
[3] Christodoulides, R.G.P. (2010) Steady Gap Solitons in a Coupled Korteweg-de Vries System: A Dynamical Systems Approach. Physica D: Nonlinear Phenomena, 239, 635-639.
https://doi.org/10.1016/j.physd.2010.01.016
[4] Weoss, J., Tabor, M. and Carnevale, G. (1983) Painlevé Property for Partial Differential Equations. Journal of Mathematical Physics, 24, 522-526.
https://doi.org/10.1063/1.525721
[5] Matveev, V.B. and Salle, M.A. (1991) Darboux Transformmations and Solitons. Springer, Berlin.
https://doi.org/10.1007/978-3-662-00922-2
[6] Bluman, G.W., Checiakov, A.F. and Anco, S.C. (2010) Applications of Symmetry Methods to Partial Differential Equations. Springer, New York.
https://doi.org/10.1007/978-0-387-68028-6
[7] Hirota, R. (2004) The Direct Method in Soliton Theory. Spring, Berlin.
https://doi.org/10.1017/CBO9780511543043
[8] Ablowitz, M.J. and Clarkkson, P.A. (1991) Solitons, Nonlinear Evolution Equations and Inverse Scattering. Cambridge University Press, Cambrige.
https://doi.org/10.1017/CBO9780511623998
[9] Ma, W.X. (2015) Lump Solutions to the Kadomtsev-Petviashvili Equation. Physics Letters A, 379, 1975-1978.
https://doi.org/10.1016/j.physleta.2015.06.061
[10] Ma, W.X., Zhou, Y. and Dougherty, R. (2016) Lump-Type Solutions to Nonlinear Differential Equantions Derived from Generalized Bilinear Equations. International Journal of Modern Physics B, 30, Artical ID: 1640018.
https://doi.org/10.1142/S021797921640018X
[11] Zhang, H.Q. and Ma, W.X. (2016) Lump Solution to the (2 + 1)-Dimensional Sawada-Kotera Equation. Nonlinear Dynamics, 87, 2305-2310.
https://doi.org/10.1007/s11071-016-3190-6
[12] Hua, Y.F., Guo, B.L. and Ma, W.X. (2016) Interaction Behavior Associated with a Generalized (2 + 1)-Dimensional Hirota Bilinear Equation for Nonlinear Waves. Applied Mathematical Modelling, 74, 184-198.
https://doi.org/10.1016/j.apm.2019.04.044
[13] Zhang, H.Q. and Ma, W.X. (2017) Mixed Lump-Kink Solutions to the KP Equation. Computers and Mathematics with Applications, 74, 1399-1405.
https://doi.org/10.1016/j.camwa.2017.06.034
[14] Yang, J.Y. and Ma, W.X. (2017) Abundant Lump-Type Solutions of the Jimbo Miwa Equation in (2 + 1)-Dimensions. Computers and Mathematics with Applications, 73, 220-225.
https://doi.org/10.1016/j.camwa.2016.11.007
[15] Wang, Y.H., Wang, H., Zhang, H.S. and ChaoLu, T.M. (2017) Exact Interaction Solutions of an Extended (2 + 1)-Dimensinal Shallow Water Wave Equation. Commnications in Theoretical Physics, 68, 165.
https://doi.org/10.1088/0253-6102/68/2/165
[16] Yang, J.Y., Ma, W.X. and Qin, Z.Y. (2017) Lump and Lump-Soliton Solutions to the (2 + 1) Dimensional Ito Equation. Analysis and Mathematical Physics, 8, 427-436.
[17] Wang, H., Wang, Y.H. and Ma, W.X. (2018) Lump Solutions of a New Extended (2 + 1)-Dimensional Boussinesq Equation. Modern Physics Letters B, 32, Article ID: 1850376.
https://doi.org/10.1142/S0217984918503761
[18] Ma, W.X. (2018) Abundant Lumps and Their Interaction Solutions of (2 + 1)-Dimensional Linear PDEs. Journal of Geometry and Physics, 133, 10-16.
https://doi.org/10.1016/j.geomphys.2018.07.003
[19] Sumayah, B. and Ma, W.X. (2018) A Study of Lump-Type and Interaction Solutions to a (2 + 1)-Dimensional Jimbo-Miwa-Like Equation. Computers and Mathematics with Applications, 76, 1576-1582.
https://doi.org/10.1016/j.camwa.2018.07.008
[20] Ma, W.X. and Zhou, Y. (2018) Lump Solutions to Nonlinear Partial Differential Equations via Hirota Bilinear Forms. Journal of Differential Equations, 264, 2633-2659.
https://doi.org/10.1016/j.jde.2017.10.033
[21] Ma, W.X., Yong, X. and Zhang, H.Q. (2018) Diversity of Interaction Solutions to the (2 + 1)-Dimensinal Ito Equation. Computers and Mathematics with Applications, 75, 289-295.
https://doi.org/10.1016/j.camwa.2017.09.013
[22] Huang, L.L., Yue, Y.F. and Chen, Y. (2018) Localized Waves and Interaction Solutions to a (2 + 1)-Dimensional Generalized KP Equation. Computers and Mathematics with Applications, 76, 831-844.
https://doi.org/10.1016/j.camwa.2018.05.023
[23] Ma, W.X. (2019) Lump and Interaction Solutions to Linear (2+1)-Dimensional PDEs. Acta Mathematica Scientia, 39, 498-508.
[24] Ma, W.X. (2019) Lump and Interaction Solutions of Linear PDEs in (2 + 1)-Dimensions. Aast Asian Journal on Applied Mathematics, 9, 185-194.
https://doi.org/10.4208/eajam.100218.300318
[25] Ren, B., Ma, W.X. and Yu, J. (2019) Characteristics and Interactions of Solitary and Lump Waves of a (2 + 1)-Dimensional Coupled Nonlinear Partial Differential Equation. Nonlinear Dynamics, 96, 717-727.
https://doi.org/10.1007/s11071-019-04816-x
[26] Zhou, Y., Manukure, S. and Ma,W.X. (2019) Lump and Lump-Soliton Solutions to the Hirota-Satsuma-Ito Equation. Communications in Nonlinear Science and Numerical Simulation, 68, 56-62.
https://doi.org/10.1016/j.cnsns.2018.07.038
[27] Wang, Y.H. and Chen, Y. (2013) Binary Bell Polynomial Manipulations on the Integrability of a Generalized (2 + 1)-Dimensional Korteweg-de Vries Equation. Journal of Mathematical Analysis and Applications, 400, 624-634.
https://doi.org/10.1016/j.jmaa.2012.11.028
[28] Peng, Y.Z. (2010) A New (2 + 1)-Dimensional KdV Equation and Its Localized Structures. Communications in Theoretical Physics, 54, 863-865.
https://doi.org/10.1088/0253-6102/54/5/17
[29] Chen, J.C. and Ma, Z.Y. (2017) Consistent Riccati Expansion Solvability and Soliton-Cnoidal Wave Interaction Solution of a (2 + 1)-Dimensional Korteweg-de Vries Equation. Applied Mathematics Letters, 64, 87-93.
https://doi.org/10.1016/j.aml.2016.08.016
[30] Toda, K. and Yu, S.J. (2000) The Investigation into the Schwarz-Korteweg-de Vries Equation and the Schwarz Derivative in (2 + 1)-Dimensional. Journal of Mathematical Physics, 41, 4747-4751.
https://doi.org/10.1063/1.533374
[31] Zhang, Y. and Xu, G.Q. (2014) Intergrability and Exact Solutions for a (2 + 1)-Dimensional Variable-Coefficient KdV Equation. Applications of Mathematics, 9, 646-658.
[32] Lü, X., Lin, F.H. and Qi, F.H. (2015) Analytical Study on a Two Dimensional Korteweg-de Vries Model with Bilinear Representation, Bäcklund Transformaation and Soliton Solutions. Applications of Mathematical Models, 39, 3221-3226.
https://doi.org/10.1016/j.apm.2014.10.046
[33] Lü, X., Ma, W.X. and Khalique, C.M. (2015) A Direct Bilinear Bäcklund Transformaation of a (2 + 1)-Dimensional Korteweg-de Vries Equation. Applied Mathematics Letters, 50, 37-42.
https://doi.org/10.1016/j.aml.2015.06.003
[34] Zhao, Z.L. and Han, B. (2017) The Riemann-Bäcklund Method to a Quasiperiodic Wave Solvable Generalized Variable-Coefficient (2 + 1)-Dimensional KdV Equation. Nonlinear Dynamics, 87, 2661-2676.
https://doi.org/10.1007/s11071-016-3219-x
[35] Huang, L.L. and Chen, Y. (2018) Nonlocal Symmetry and Similarity Reductions for a (2 + 1)-Dimensional Korteweg-de Vries Equation. Nonlinear Dynamics, 92, 221-234.
https://doi.org/10.1007/s11071-018-4051-2

  
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