LIU Dong , JU Xiaodong ILHAN Onur Alp, MANAFIAN Jalil , and ISMAEL Hajar Farhan

1) State Key Laboratory of Petroleum Resources and Prospecting, China University of Petroleum, Beijing 102249, China

2) China Petroleum Materials Company Limited, Beijing 100029, China

3) Department of Mathematics, Erciyes University, Melikgazi-Kayseri 38039, Turkey

4) Department of Applied Mathematics, Faculty of Mathematical Sciences, University of Tabriz, Tabriz 5166616471, Iran

5) Department of Mathematics, Faculty of Science, University of Zakho, Zakho 42002, Iraq

Abstract The present article deals with multi-waves and breathers solution of the (2+1)-dimensional variable-coefficient Caudrey-Dodd-Gibbon-Kotera-Sawada equation under the Hirota bilinear operator method. The obtained solutions for solving the current equation represent some localized waves including soliton, solitary wave solutions, periodic and cross-kink solutions in which have been investigated by the approach of the bilinear method. Mainly, by choosing specific parameter constraints in the multi-waves and breathers, all cases the periodic and cross-kink solutions can be captured from the 1- and 2-soliton. The obtained solutions are extended with numerical simulation to analyze graphically, which results in 1- and 2-soliton solutions and also periodic and cross-kink solutions profiles. That will be extensively used to report many attractive physical phenomena in the fields of acoustics, heat transfer, fluid dynamics, classical mechanics, and so on. We have shown that the assigned method is further general, efficient, straightforward, and powerful and can be exerted to establish exact solutions of diverse kinds of fractional equations originated in mathematical physics and engineering. We have depicted the figures of the evaluated solutions in order to interpret the physical phenomena.

Key words variable-coefficient Caudrey-Dodd-Gibbon-Kotera-Sawada equation; Hirota bilinear operator method; soliton; multiwaves and breathers; periodic and cross-kink; solitray wave solutions

1 Introduction

Partial differential equations (PDEs) play important roles in the numerous areas such as biology, physics, chemistry,fluid mechanics and many engineering and sciences applications among others (Daiet al., 2008; Dehghanet al., 2011;Ma and Zhu, 2012; Manafian and Lakestani, 2016a; Foroutanet al., 2018). Furthermore, the approaches to solving these types of equations alongside nonlinear PDEs ranging from analytical to numerical methods are very important in many engineering and sciences applications. Some of these methods include finding the exact solutions by using the special techniques in which can be manifested to new works with the vigorous references (Dehghan and Manafian, 2009; Wanget al., 2010; Manafian, 2015; Baskonus and Bulut, 2016; Manafian and Lakestani, 2016b; Tanget al., 2016; Zhouet al., 2016; Gao, 2017; Wang and Liu,2018; Chenet al., 2019a).

The nonlinear (2+1)-dimensional Caudrey-Dodd-Gibbon-Kotera-Sawada equation is given as in whichu=u(x,y,t), andposed by Konpelchenko and Dubrovsky by the help of the inverse scattering transform method (Konopelchenko and Dubrovsky, 1984). The nonlinear (2+1)-dimensional Caudrey-Dodd-Gibbon-Kotera-Sawada equation has been investigated for finding the exact solutions in which can be pointed to vigorus works containing the algebraic method with symbolic computation (Yang, 2006), the solitary waves and lump waves with interaction phenomena by the way of vector notations (Penget al., 2018), the quasi- periodic solutions by the Riemann theta functions (Caoet al.,1999; Genget al., 2019), some novel group invariant solutions by utilizing the classical symmetry reduction method(Chenget al., 2019), and in continue we take the (2+1)-dimensional variable coefficient Caudrey-Dodd-Gibbon-Kotera-Sawada (VC CDGKS) equation which reads

in whichbj=bj(t) (j= 1, 2, ···, 9) are functions with respect tot. Chenget al.(2014) obtained the bilinear form, bilinear BT, Lax pair, and infinite conservation law for Eq. (2).Moreover, the VC CDGKS equation with essential applications in the incompressible fluid has been investigated by Wanget al.(2019a), and new non-traveling lump solutions, their interaction solutions, and mixed lump-kink solutions for the considered equation have been achieved to explain relative physics or expect some new physical phenomena.

For getting to the lump solutions and their interactions authors have conjugated sufficient time to search the exact rational soliton solutions, for example, the Kadomtsev-Petviashvili (KP) equation (Ma, 2015), the B-Kadomtsev-Petviashvili equation (Yang and Ma, 2016), the reduced p-gKP and p-gbKP equations (Maet al., 2016), the (2+1)-dimensional KdV equation (Wang, 2016), the (2+1)-dimensional generalized fifth-order KdV equation (Lüet al.,2018), the (2+1)-dimensional Burger equation (Wanget al.,2016), the nonlinear evolution equations (Tanget al., 2016),the generalized (3+1)-dimensional Shallow water-like equation (Zhanget al., 2017), (2+1)-dimensional Sawada-Kotera equation (Huang and Chen, 2017), and (2+1)-dimensional bSK equation (Lu and Bilige, 2017; Manafian and Lakestani, 2019). Various types of work for finding the periodic solitary wave solutions on the (2+1)- dimensional extended Jimbo-Miwa equations (Manafian, 2018),the interaction between lump and other kinds of solitary,periodic and kink solitons for the (2+1)- dimensional Breaking Soliton equation (Manafianet al., 2019), the lump and interaction between different types of those on the variable-coefficient Kadomtsev-Petviashvili equation(Ilhanet al., 2019), and periodic type and periodic crosskink wave solutions (Ilhan and Manafian, 2019) are achieved through the Hirota bilinear operator.

Due to simplifications obtained by Chenget al.(2019)as the below form, we have

in whichλ0andλ1are the free constants. Through the relation betweenuandf, one can get to the following conversion as

Then, by using Eq. (4) in Eq. (2), the following bilinear model concludes

Suppose the Hirota derivatives in terms of the functionsfandgcan be written as

where the vectorsj= (j1,j2,j3,j4) = (x,y,z,t),j'= (j'1,j'2,j'3,j'4) = (x',y',z',t') andβ1,β2,β3,β4are the arbitrary nonnegative integers, and its corresponding bilinear formalism equals as below form

The soliton solutions to a few (3+1)-dimensional generalized nonlinear integrable equations have been constructed. Recently, a special kind of reductions of soliton solutions to rational functions that are actively studying is lump solutions to nonlinear partial differential equations by Ma and Zhou (2018), their interactions with solitons to Hirota-Satsuma-Ito equation in (2+1)-dimensions by Ma(2019a), and even for linear PDEs by the same author (Ma,2019b). Also, Ma (2020) presented the inverse scattering transforms and soliton solutions for nonlocal reverse-time nonlinear Schrodinger equations. The same author offered an application of the nonlinear steepest descent method to a three-component coupled mKdV system associated with a 4×4 matrix spectral problem (Ma, 2019b). Nowadays NLPDEs have been creating a significant opportunity for the researchers to explain the tangible incidents. Therefore,mathematicians and scientists are working tirelessly to bring out different kinds of soliton solutions. As a result,in the past few years several effective, rising and realistic methods have been initiated and dilated to extract closedform solutions to the NLPDEs, videlicet, observational/experimental consideration on certain (2+1)-dimensional waves in the cosmic/laboratory dusty plasmas (Gao, 2019),a vector nonlinear Schrodinger equation in a birefringent optical fiber (Yinet al., 2020), dark-bright semi-rational solitons and breathers for a higher-order coupled nonlinear Schrodinger system (Duet al., 2020), conservation laws of a (2+1)-dimensional nonlinear Schrodinger equation(Duet al., 2019), rogue waves, and modulation instability for the coherently coupled nonlinear Schrodinger equations (Chenet al., 2019b), the higher-order Boussinesq-Burgers system, auto- and non-auto-B?cklund transformations (Gaoet al., 2020), lump wave- soliton interactions for a (3+1)-dimensional generalized Kadomtsev- Petviashvili equation (Huet al., 2019), rogue waves for a (2+1)-dimensional reduced Yu-Toda-Sasa- Fukuyama equation(Wanget al., 2019b), and interactions of the couple Fokas-Lenells system (Zhanget al., 2020). We clearly confirm that others’ published papers do not cover theirs and made work is really new.

Our purpose here is to discover exact solutions of the VC CDGKS equation under consideration of the Hirota bilinear method for getting the multi-waves, breathers solution, periodic solution, cross-kink solution, and new solitary wave solutions in which can be captured from the 1- and 2-soliton. Discussion about the nonlinear VC CDGKS equation and the Hirota bilinear method is given.In the continuation, we will offer the graphical illustrations of some solutions of the considered model. After that, we will deal with the probe of solutions and we will finish by a conclusion.

2 New Multi-Waves Solutions for VC CDGKS Equation

Here, we will compose multi-waves solutions of the Eq.(2), we choose the three waves hypothesis which can be discovered through employing Hirota operator (Geng and Ma, 2007). The solution can be expressed in the below form as:

whereai,bi,ci,qj,i= 1, ···, 4,j= 1, 2, 3 are the free parameters in which are to find later. Plugging relations(8) into the Eq. (7) and then collecting the coefficients,we get to system of the nonlinear algebraic equations.

Solving the obtained equations we achieve to obtained cases:

Inserting Eq. (9) into Eq. (8), we get a multi-wave solu- tion of the Eq. (2) as follows:

whereH1′=q1sinh(…),H3′=q1sinh(…),a1,a4,b1,b4,c1,c4,q1,q2andq3are arbitrary values.

Moreover, we obtained five sets of solutions as mentioned above, we neglect to bring those categories of solutions (see Fig.1).

3 New Breather Solutions for VC CDGKS Equation

Here, we will compose breather wave solutions of theEq. (2), we choose the following function that can be expressed in the below form as:

Fig.1 Diagram of multi-waves Eq. (10) using values a1 = 0.6, a4 = 1, b1 = 0.5, b4 = 1, c1 = 1.2, c4 = 1, q1 = 2, q2 = 0.5, q3 = 2, δ1 =0.5, λ0 = ?1, λ1 = 1, b9(t) = cos(t), y = ?10, and (a) 3D plot, (b) density plot, and (c) 2D plot with (red x = ?1, blue x = 0, and green x = 1).

whereai,bi,qj,i= 1, ···, 4,j= 1, 2, 3 are the free parameters in which are to find later. Plugging Eq. (11) into the Eq. (7) and then collecting the coefficients, we get to system of the nonlinear algebraic equations.

Solving the obtained equations we achieve to obtained cases:

Inserting Eq. (12) into Eq. (11), we get a breather wave solution of the Eq. (2) as follows:

4 New Instanton Wave Solution for VC CDGKS Equation

Here, we will compose a special rogue-wave that is generated by cutting the lump wave through a pair of resonance stripe soliton waves of the Eq. (2), we choose the following function that can be expressed in the below form as:

whereai,bi,ci(i= 1, ···, 4) andq1, are the free parameters in which are to find later. Plugging Eq. (14) into the Eq.(7) and then collecting the coefficients, we get to the follow- ing results:

Inserting Eq. (15) into Eq. (14), we get a instanton wave solution of the Eq. (2) as follows:

whereH3′=sinh(…),a1,a4,b1,b4, andc4are the arbitrary values (see Fig.3).

Fig.3 Diagram of instanton wave Eq. (16) using values b1 = 1.5, b4 = 1, c1 = 0.5, c2 = 0.2, c4 = 1, δ1 = 0.5, λ0 = ?1, λ1 = 1,b9(t) = cos(t), q1 = 2, y = ?4, and (a) 3D plot, (b) density plot, and (c) 2D plot with (red x = ?1, blue x = 0, and green x = 1).

5 Novel Periodic Wave Solutions of the VC CDGKS Equation

To get for the periodic wave solutions of the VC CDGKS equation, we would like to commence from a function as below form

whereai,bi,ci,kj(i= 1, ···, 4;j= 1, 2, 3) are the free parameters in which are to find later. Plugging Eq. (17)into the Eq. (7) and then collecting the coefficients, we get to system of the nonlinear algebraic equations.

Solving the obtained equations we achieve to obtained cases:

Substituting Eq. (18) into Eq. (17), we obtain a periodic solution of the Eq. (2) as follows:

wherea1,a4,c1,c4,k1,k3, andk4are the arbitrary values.

Moreover, we obtained twelve sets of solutions as mentioned above, we neglect to bring those categories of solutions (see Fig.4).

Fig.4 Diagram of periodic wave (19) using values a1 = 0.5, a4 = 1, c1 = 1.5, c4 = 1, δ1 = 0.5, λ0 = ?1, λ1 = 1, b9(t) = cos(t),k1 = 1, k3 = 1.5, k4 = 2, y = ?4, and (a) 3D plot, (b) density plot, and (c) 2D plot with (red x = ?10, blue x = 0, and green x= 10).

6 Novel Cross-Kink Wave Solutions of the VC CDGKS Equation

To get for the cross-kink wave solutions of the VC CDGKS equation, we would like to commence from a function as below form

whereai,bi,ci,kj(i= 1, ···, 4;j= 1, 2, 3) are the free parameters in which are to find later. Plugging Eq. (20)into the Eq. (7) and then collecting the coefficients, we get to system of the nonlinear algebraic equations.

Solving the obtained equations we achieve to obtained cases:

Plugging Eq. (21) into relations (20), we get a cross- kink wave solution of the Eq. (2) as follows:

wherea1,a4,b1,b4,c1,c4,k1,k2,k3andk4are the arbitrary values.

7 Novel Solitary Wave Solutions of the VC CDGKS Equation

To get for the new solitary wave solutions of the VC CDGKS equation, we would like to commence from a function as below form

whereai,bi,ci,kj(i= 1, ···, 4;j= 1, 2, 3) are the free parameters in which are to find later. Plugging Eq. (23)into the Eq. (7) and then collecting the coefficients, we get to the following results case:

Appending Eq. (24) into relations (23), we get a crosskink wave solution of the Eq. (2) as follows:

wherea1,a2,a3,a4,b3,b4,c3,c4,k2andk3are the arbitrary values.

We obtained twelve sets of solutions as mentioned above, we neglect to bring those categories of solutions.The three-dimensional dynamic graphs of the wave and corresponding density plots, contour plots, and two-dimensional plots were successfully depicted in Figs.1–4 with the help of the Maple. We can see that the exponential function, the sine function, and the hyperbolic sine function react with each other and move forward. Due to analyzing the dynamics properties briefly, we would like to discuss the evolution characteristic.

8 Conclusions

Through the symbolic calculation and employing the Hirota bilinear operator, we have discovered some novel analytic solutions for the VC CDGKS equation. As a consequence, some new solutions, which include the new multi-wave, breathers, periodic, cross-kink wave solutions were catched. Through of Maple, the evolution phenomenon of these waves is seen in Figs.1–4, respectively. The obtained solutions for solving the VC CDGKS equation shown some localized waves such as soliton, periodic and cross-kink solutions in which have been investigated by the approach of the bilinear method. Mainly, by choosing specific parameter constraints in all cases the two-dimension, and three-dimension in solitons can be captured from the multi-wave, breathers, periodic, cross-kink wave solutions. The obtained solutions are extended with numerical simulation to analyze graphically, which results in multiwave, breather wave, periodic, cross-kink wave solutions.The attained solutions are in broad-ranging form and the definite values of the included parameters of the attained solutions yield the soliton solutions and help to analyze the quantum mechanics, the signal processing waves, the meteorology, and biomedical engineering,etc.That will be extensively used to report many attractive physical phenomena in the fields of acoustics, heat transfer, fluid dynamics, classical mechanics, and so on. Moreover, the established results have shown that the Hirota bilinear method is further general, straightforward, and more powerful and helped to examine traveling wave solutions of NLPDEs.

Acknowledgements

This work is supported by the National Science and Technology Major Project (Nos. 2017ZX05019001 and 2017ZX05019006), the PetroChina Innovation Foundation (No. 2016D-5007-0303), and the Science Foundation of China University of Petroleum, Beijing (No. 2462016 YJRC020).