Romanian Reports in Physics 72, 117 (2020)

CHAOTIC AND SOLITONIC SOLUTIONS FOR A NEW TIME-FRACTIONALTWO-MODE KORTEWEG-DE VRIES EQUATION

MARWAN ALQURAN1,a, IMAD JARADAT1,b, SHAHER MOMANI2,3,c, DUMITRUBALEANU4,5,d

1Department of Mathematics and Statistics, Jordan University of Science and Technology,P.O. Box: 3030, Irbid, 22110, Jordan

Email: amarwan04@just.edu.jo, biajaradat@just.edu.jo2Department of Mathematics and Sciences, College of Humanities and Sciences,

Ajman University, Ajman, UAE3Department of Mathematics, Faculty of Science, The University of Jordan,

Amman 11942, JordanEmail: cs.momani@ju.edu.jo

4Department of Mathematics, Cankaya University, Ankara, TurkeyEmail: ddumitru@cankaya.edu.tr

5Institute of Space Sciences, Magurele, Bucharest, Romania

Received April 6, 2020

Abstract. The two-mode Korteweg-de Vries (TMKdV) equation is a nonlin-ear dispersive wave model that describes the motion of two different directional wavemodes with the same dispersion relations but with various phase velocities, nonlin-earity, and dispersion parameters. In this work, we study the dynamics of the modelanalytically in a time-fractional sense to ensure the stability of the extracted waves ofthe TMKdV equation. We use the fractional power series technique to conduct ouranalysis. We show that there is a homotopy mapping of the solution as the Caputotime-fractional derivative order varies over (0,1] and that both waves have the samephysical shapes but with reflexive relation.

Key words: Caputo time-fractional derivative, Two-mode Korteweg-de Vriesequation, Fractional power series.

1. INTRODUCTION

There are many recent research studies regarding the mathematical physics dy-namical models involving fractional-order derivatives instead of integer-order deriva-tives. For example, the dynamics of dissipative solitons in the framework of a one-dimensional complex Ginzburg-Landau equation of a fractional order has been ex-plored in [1]. Also, in a recent work [2], in the framework of the nonlinear fractionalSchrodinger equation, the asymmetric, symmetric, and antisymmetric soliton solu-tions have been found and their stability features have been studied numerically. Theparity-time symmetric optical modes and the phenomenon of spontaneous symmetrybreaking in the space-fractional Schrodinger equation have been investigated by ade-(c) 2020 RRP 72(0) 117 - v.2.0*2020.7.20 —ATG

Article no. 117 Marwan Alquran et al. 2

quate numerical methods [3]. The dynamics of a nonlinear fractional model respon-sible for the transition of turbulence phenomena and cellular instabilities to chaoshas been studied in [4]. For other interesting recent works in the area of fractionalmodels in many physical contexts, see, for example, Refs. [5–9].

Four decades ago, Hirota and Satsuma [10] investigated the interaction of twolong waves having different dispersion parameters. They reported that if there is noeffect of one wave on the other one, then these two waves will coincide and act as asingle wave with a single dispersive parameter; on other words, these waves obey theKorteweg-de Vries (KdV) equation. As a result, the two-mode KdV equation wasfirst established in [11] to reflect the dynamics of moving two different wave modespropagating in the same direction. Later in [12], the two-mode KdV equation hasbeen reformulated in a scaled form to have the following form

�D2

t �s2D2x

�w+(Dt�↵sDx)wwx+(Dt��sDx)wxxx = 0, (1)

where Dt =@@t , D

2t =

@2

@t2 , w = w(x,t) is a field function, s is the phase velocity, ↵is the nonlinearity parameter, and � is the dispersive parameter with s � 0, |↵| 1,|�| 1. In [13], by means of the simplified Hirota’s method, tanh/coth method, andthe tan/cot method, different set of solutions to (1) with distinct physical structuresare obtained. The conservation laws are used in [14] and a quasi-soliton behavioris reported to the two-mode KdV equation. In [15] and [16], the (G0/G)-expansionmethod and the Jacobi elliptic function method are implemented to extract more newsolitary wave solutions of Eq. (1).

The aim of the current work is to revisit the two-mode KdV equation (1) andto investigate the effect of replacing the integer-order derivative of the time coordi-nate with fractional-order derivative. The new model under investigation is of thefollowing form

�D2�

t �s2D2x

�w+(D�

t �↵sDx)wwx+(D�t ��sDx)wxxx = 0, � < 1, (2)

where now D�t is defined as

D�t w(x,t) =

@�w(x,t)

@t�=

1

�(1��)

Z t

0(t� ⌧)�� @w(x,⌧)

@⌧d⌧. (3)

In this context, the model given in Eq. (2) is proposed for the first time, to thebest of our knowledge. The fractional power series technique [17–23] will be usedto extract the analytical supportive approximate solutions. Next, we will explain thenecessary steps of applying the fractional power series to solve Eq. (2) subject to theinitial conditions

w(x,0) = f(x),

D�t w(x,0) = gi(x) : i= 1,2, (4)

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3 Chaotic and solitonic solutions for a Korteweg-de Vries equation Article no. 117

where g1(x) is the initial velocity for the first-mode wave and g2(x) is the initialvelocity for the second-mode wave.

We should point out here that the type of the fractional derivative consideredin this work is of Caputo sense. However, many new definitions have been pro-posed as modifications of Caputo fractional derivative and have been implementedin many works. For example, the Caputo-Fabrizio derivative has been applied to thegroundwater flow within Confined Aquifer [24]. Also, the Liouville-Caputo (LC),Caputo-Fabrizio (CF), and Atangana-Baleanu (AB) fractional-order time derivativeshave been considered to study the time-fractional versions of both KdV and Burgers’equations [25, 26]. Fractional derivatives with non-local and non-singular kernelshave been established and implemented in many applications [27]. Finally, the newproperties of the new conformable derivative have been studied in Ref. [28].

The original contribution of the current work is twofold. We investigate thephysical structures of the two-mode KdV equation by means of the elegant Kudrya-shov method. Then, we study the stability behavior under time evolution of thederived two waves when considering the Caputo time-fractional derivative in themodel under investigation.

The organization of this paper is as follows. In Sec. 2, we illustrate how toutilize the fractional power scheme for producing supportive approximate solutionof the time-fractional two-mode KdV equation. The solitonic behavior of the two-waves solution of the TMKdV equation is discussed in Sec. 3. Then, in Sec. 4,we validate the proposed scheme by testing a few numerical examples. Finally, wesummarize in Sec. 5 the main results obtained in this work.

2. THE FRACTIONAL POWER SERIES METHOD

First, we write the solution of (2) in the following form

w(x,t) =mX

j=0

�j(x)tj�

�(j�+1)+Rm(x,t), (5)

where Rm(x,t) =P1

j=m+1�j(x)tj�

�(j�+1) . We require that Rm(x,t) is to be verysmall for n�m+1 over the intervals (x,t) 2 (a,b)⇥ (0,T ) : T < 1. Accordingly,we rewrite (5) as

w(x,t) =mX

j=0

�j(x)tj�

�(j�+1). (6)

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Article no. 117 Marwan Alquran et al. 4

By Caputo’s derivative, the fractional derivative of order 0< � < 1 for the exponentfunction has the following rule

D�t t

� =

8>><

>>:

�(�+1)

�(���+1)t���, � � �

0, � = 0.

(7)

By (7) and (6), it is easy to conclude the following

D2�t w(x,t) =

m�2X

j=0

�j+2(x)tj�

�(j�+1). (8)

Next, we implement the relations given in (6)-(8) to be inserted in (2) to reach at

L(x,t,�,m) =m�2X

j=0

�j+2(x)tj�

�(j�+1)�s2

mX

j=0

�00j (x)

tj�

�(j�+1)

+(D�t �↵sDx)

0

@mX

j=0

�j(x)tj�

�(j�+1)

1

A

0

@mX

j=0

�0j(x)

tj�

�(j�+1)

1

A

+(D�t ��sDx)

mX

j=0

�000j (x)

tj�

�(j�+1)= 0,

(9)

where wwx =⇣Pm

j=0�j(x)tj�

�(j�+1)

⌘⇣Pmj=0�

0j(x)

tj�

�(j�+1)

⌘. Applying the product

of two finite series, we have the following expansion

wwx =A(x,t) =2mX

j=0

jX

i=0

✓�i(x)�0

j�i(x)

�(i�+1)�((j� i)�+1)

◆tj�

�m�1X

j=0

2m�jX

i=m+1

✓(�i(x)�j(x))0

�(i�+1)�(j�+1)

◆t(j+i)�. (10)

Therefore, implementing the operator D�t on A(x,t) = wwx leads to

D�t (wwx) =B(x,t) =

2m�1X

j=0

j+1X

i=0

✓�i(x)�0

j�i+1(x)�((j+1)�+1)

�(i�+1)�((j� i)�+1)�(j�+1)

◆tj�

+m�1X

j=0

2m�j�1X

i=m

✓(�i+1(x)�j(x))

0�((j+ i+1)�+1)

�((i+1)�+1)�(j�+1)�((j+ i)�+1)

◆t(j+i)�,

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5 Chaotic and solitonic solutions for a Korteweg-de Vries equation Article no. 117

Now, we combine (8) with the resulting formulas in (9), (10), and (11) to reach at thefollowing function

L(x,t,�,m) =m�2X

j=0

�j+2(x)tj�

�(j�+1)�s2

mX

j=0

�00j (x)

tj�

�(j�+1)+B(x,t)

+↵s@A(x,t)

@x+

m�1X

j=0

�000j+1(x)

tj�

�(j�+1)��s

mX

j=0

�0000j (x)

tj�

�(j�+1)

= 0.

(12)

It is clear from (6) that �0(x) =w(x,0) and �1(x) =D�t w(x,0). Thus, to determine

�m(x) : m= 2,3,4, ..., we solve, recursively, the following equation

D(m�2)�t L(x,t,�,m) = 0, m= 2,3,4, . . . . (13)

3. SOLITONIC SOLUTIONS FOR THE TWO-MODE KDV EQUATION

We aim in this Section to seek some solutions for the two-mode KdV equation(1) to show the physical structures of such type of nonlinear equations. Commonlyin similar studies, we seek the solution of the form

w(x,t) =nX

k=0

akYk, Y = tanh(µ(x� ct)) or coth(µ(x� ct)). (14)

The above suggested solution is derived from the well-known tanh-coth-func-tion method [29–34]. Substitution of (14) in (1) and collecting the coefficients ofY i yields a system of nonlinear algebraic equations in terms of the unknowns ak, µ,and c. But the resulting system can not be solved unless we add a reasonable extraconditions. We may assume that ↵ = � = h. Therefore, six different solutions areobtained:

w(x,t) = a0+a1 tanh(µ(x�hst)), h=±1, (15)w(x,t) = a0+a1 coth(µ(x�hst)), h=±1, (16)

and

w(x,t) = 4µ2�12µ2 tanh⇣µ(x+(2µ2±

ps2�4hsµ2+4µ4)t)

⌘, (17)

w(x,t) = 4µ2�12µ2 coth⇣µ(x+(2µ2±

ps2�4hsµ2+4µ4)t)

⌘, |h| 1, (18)

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Article no. 117 Marwan Alquran et al. 6

and

w(x,t) = 12µ2�12µ2 tanh⇣µ(x� (2µ2⌥

ps2+4hsµ2+4µ4)t)

⌘, (19)

w(x,t) = 12µ2�12µ2 coth⇣µ(x� (2µ2⌥

ps2+4hsµ2+4µ4)t)

⌘, |h| 1.

(20)

The above parameters a0, a1, and µ are free parameters. Figure 1 representsthe solitonic behavior of the two-waves depicted in (15) and their phase interactionsupon increasing the phase velocity s. In the next Section, we study the propagationof these moving two-waves if the integer-order time derivative is replaced by thefractional-order time derivative.

!1.0

!0.5

0.0

0.5

1.0

x

!1.0

!0.5

0.0

0.5

1.0

t

!1.0

!0.5

0.0

0.5

1.0

!1.0

!0.5

0.0

0.5

1.0

x

!1.0

!0.5

0.0

0.5

1.0

t

!1.0

!0.5

0.0

0.5

1.0

!1.0

!0.5

0.0

0.5

1.0

x

!1.0

!0.5

0.0

0.5

1.0

t

!1.0

!0.5

0.0

0.5

1.0

Fig. 1 – The interaction of the two waves depicted in (15) upon increasing the phase velocity s, wherea0 = a1 = µ= 1 and s= 1, 3, 5, respectively.

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7 Chaotic and solitonic solutions for a Korteweg-de Vries equation Article no. 117

4. DYNAMICS OF THE TIME-FRACTIONAL TWO-MODE KDV EQUATION

In this Section, we apply the fractional power series scheme, which was in-troduced earlier, to solve the following time-fractional version of two-mode KdVequation

�D2�

t �s2D2x

�w+(D�

t �↵sDx)wwx+(D�t ��sDx)wxxx = 0, � < 1, (21)

subject to

w(x,0) = f(x),

D�t w(x,0) = gi(x) : i= 1,2, (22)

where g1(x) and g2(x) are regarded as the initial velocities, for the right-mode waveand left-mode wave, respectively.

In order to investigate the efficiency of the proposed method and then to ex-plore the physical structure of the new model given in (21), we study the followingexample. Consider the equation (21) subject to

w(x,0) = tanh(µx),

D�t w(x,0) = s h µ sech2(µx). (23)

Based on the result obtained in (15), the exact solution of the problem (21) and(23) is w(x,t) = tanh(µ(x� sht)) when ↵ = � = h : h = ±1 and the fractionalorder is � = 1. Now, following the structure of the fractional power series illustratedin (12) and (13), we present the first four terms of the sequence {�i(x)}1i=0

�0(x) = tanh(µx),

�1(x) = s h µ sech2(µx),

�2(x) = �2s2µ2 sech2(µx)tanh(µx),

�3(x) = �2(1+�)(sh(�3cosh(µx)+cosh(3µx))�4sinh(µx))+2�(1+2�)sinh(µx)�2(1+�)/(s2µ3 sech5(µx))

.(24)

Table 1

The right-mode wave ”h= 1” w6(x,t) against w(x,t) at some mesh points when the fractional orderis � = 1.

x|t 0.10 0.15 0.20 0.25 0.30 0.35

�0.5 2.77⇥10�9 4.63⇥10�8 3.39⇥10�7 1.57⇥10�6 5.46⇥10�6 1.55⇥10�5

�0.3 5.81⇥10�10 1.25⇥10�8 1.13⇥10�7 6.28⇥10�7 2.58⇥10�6 8.51⇥10�6

�0.1 4.80⇥10�9 8.32⇥10�8 6.31⇥10�7 3.04⇥10�6 1.09⇥10�5 3.24⇥10�5

0.1 4.44⇥10�9 7.4⇥10�8 5.39⇥10�7 2.5⇥10�6 8.67⇥10�6 2.26⇥10�5

0.3 2.81⇥10�12 2.27⇥10�9 3.38⇥10�8 2.38⇥10�7 1.11⇥10�6 4.02⇥10�6

To this end, we consider w6(x,t) =P6

i=0�i(x)ti�

�(i�+1) being the supportive

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Article no. 117 Marwan Alquran et al. 8

!1.0 !0.5 0.0 0.5 1.0

0.0

0.2

0.4

0.6

0.8

1.0

!a" w6!x , t", Μ#1, s#1, h#1, Σ#1

!1.0 !0.5 0.0 0.5 1.0

0.0

0.2

0.4

0.6

0.8

1.0

! b" w6!x , t", Μ#1, s#1, h#!1, Σ#1

Fig. 2 – Contour plots of w6(x,t) when h=±1 and a0 = a1 = µ= s= � = 1.

!c" w6!x , t", Μ"1, s"1, h"1, Σ"1

$1.0

$0.5

0.0

0.5

1.0

x

$1.0

$0.5

0.0

0.5

1.0

t

$1.0

$0.5

0.0

0.5

1.0

!d" w6!x , t", Μ"1, s"1, h"#1, Σ"1

#1.0

#0.5

0.0

0.5

1.0

x

#1.0

#0.5

0.0

0.5

1.0

t

#1.0

#0.5

0.0

0.5

1.0

!e" Original two!mode waves, Μ#1, s#1, Σ#1

!1.0

!0.5

0.0

0.5

1.0

x

!1.0

!0.5

0.0

0.5

1.0

t

!1.0!0.5

0.0

0.5

1.0

Fig. 3 – (c) The right-mode wave ”h= 1” w6(x,t). (d) The left-mode wave ”h=�1” w6(x,t). (e)Both right-left waves are depicted in w(x,t).

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9 Chaotic and solitonic solutions for a Korteweg-de Vries equation Article no. 117

approximate solution of (21) and (23). Now, we are ready to give some graphicalanalysis regarding the obtained approximate solution. Figure 2 represents the contourplots of the two waves of w6(x,t) for the case of � = 1. We observe that both waveshave the same physical shapes and the reflexive relation. For clarity, we give thename of right-mode wave when h= 1 and left-mode wave when h=�1.

To study the efficiency of the proposed method we present the plots of both leftand right-mode waves depicted in w6(x,t) against the exact solution w(x,t), see Fig.3. Moreover, the obtained absolute errors evaluated at some mesh points are givenin Table 1. This comparison is achieved by comparing ”in particular” the right-modewave w6(x,t) against the exact solution w(x,t) when h= 1.

Σ " 0.80

Σ " 0.85

Σ " .90

Σ " 0.95

Σ " 0.98

Σ " 1

!0.4 !0.2 0.2 0.4t

2

4

6

8

10

12

!I" Profile solutions of w6!0.1 , t" when h"1

!0.4 !0.2 0.2 0.4t

2

4

6

8

10

!II" Profile solutions of w6!0.1 , t" when h"!1

Fig. 4 – (I) Profile solutions of w6(x,t) when h= 1. (II) Profile solutions of w6(x,t) when h=�1.

Finally, Fig. 4 shows the effect of the fractional order � on the propagationof the two-mode waves. It can be seen that there is a homotopy mapping of thesolution as � varies over (0,1]. For large values of �, this homotopy preserves thesame physical shape. In other words, the solution is stable when � approaches oneand coincides with the integer-case derivative when � = 1.

5. CONCLUSION

In this work we have introduced a new physical model, namely the two-modeKdV equation that describes the propagation of two directional symmetrical wavesmoving simultaneously. We have generalized this dynamical model to include thefractional-order time derivative instead of the common integer-order time derivative.The fractional power series method was used to find a supportive approximate solu-tion for this fractional physical model.

The validity and accuracy of the proposed method was verified using illustra-tive graphs and a supporting Table. We have conclued this study by examining theeffect of the fractional-order derivative on ensuring the stability of the corresponding(c) 2020 RRP 72(0) 117 - v.2.0*2020.7.20 —ATG

Article no. 117 Marwan Alquran et al. 10

waves of the two-mode KdV equation. As a future work, we aim to apply the method-ology of this work to study other types of fractional two-mode evolution equations.

Acknowledgements. The second and the third authors would like to express their sincere ap-preciation to the Ajman University, Ajman, UAE, for providing the financial support.

REFERENCES

1. Y. Qiu, B.A. Malomed, D. Mihalache, X. Zhu, L. Zhang, and Y. He, Soliton dynamics in a frac-tional complex Ginzburg-Landau model, Chaos Solitons & Fractals 131, 109471 (2020).

2. P. Li, B.A. Malomed, and D. Mihalache, Symmetry breaking of spatial Kerr solitons in fractionaldimension, Chaos Solitons & Fractals 132, 109602 (2020).

3. P. Li, J. Li, B. Han, H. Ma, and D. Mihalache, PT-symmetric optical modes and spontaneoussymmetry breaking in the space-fractional Schrodinger equation, Rom. Rep. Phys. 71, 106 (2019).

4. M. Ali, M. Alquran, I. Jaradat, N. Abu Afouna, and D. Baleanu, Dynamics of integer-fractionaltime-derivative for the new two-mode Kuramoto-Sivashinsky model, Rom. Rep. Phys. 72, 103(2020).

5. D. Baleanu, A. Jajarmi, and J.H. Asad, Classical and fractional aspects of two coupled pendulums,Rom. Rep. Phys. 71, 103 (2019).

6. M.A. Abdelkawy, An improved collocation technique for distributed-order fractional partial dif-ferential equations, Rom. Rep. Phys. 72, 104 (2020).

7. M. Alquran, I. Jaradat, D. Baleanu, and R. Abdel-Muhsen, An analytical study of (2 + 1)-dimensional physical models embedded entirely in fractal space, Rom. J. Phys. 64, 103 (2019).

8. M. Alquran, I. Jaradat, D. Baleanu, and M. Syam, The Duffing model endowed with fractional timederivative and multiple pantograph time delays, Rom. J. Phys. 64, 107 (2019).

9. B.A. Malomed and D. Mihalache, Nonlinear waves in optical and matter-wave media: A topicalsurvey of recent theoretical and experimental results, Rom. J. Phys. 64, 106 (2019).

10. R. Hirota and J. Satsuma, Soliton solutions of a coupled Korteweg-de Vries equation, Phys. Lett.A 85(8-9), 407–408 (1981).

11. S.V. Korsunsky, Soliton solutions for a second-order KdV equation, Phys. Lett. A 185, 174–176(1994).

12. C.T. Lee, Multi-soliton solutions of the two-mode KdV, PhD thesis (Oxford University, Oxford2007).

13. A.M. Wazwaz, Multiple soliton solutions and other exact solutions for a two-mode KdV equation,Math. Meth. Appl. Sci. 40(6), 1277–1283 (2017).

14. C.C. Lee, C.T. Lee, J.J. Liu, and W.Y. Wang, Quasi-solitons of the two-mode Korteweg-de Vriesequation, Eur. Phys. J. Appl. Phys. 52, 11301 (2010).

15. O. Yassin and M. Alquran, Constructing new solutions for some types of two-mode nonlinearequations, Appl. Math. Inf. Sci. 12(2), 361–367 (2018).

16. M. Alquran and A. Jarrah, Jacobi elliptic function solutions for a two-mode KdV equation, J. KingSaud Univ. Sci. 31(4), 485–489 (2019).

17. H.M. Jaradat, I. Jaradat, M. Alquran, M.M.M. Jaradat, Z. Mustafa, K. Abohassan, and R. Ab-delkarim, Approximate solutions to the generalized time-fractional Ito system, Italian J. Pure ApplMath. 37, 699–710 (2017).

18. M. Alquran, I. Jaradat, and S. Sivasundaram, Elegant scheme for solving Caputo-time-fractionalintegro-differential equations, Nonlinear Stud. 25(2), 385–393 (2018).

(c) 2020 RRP 72(0) 117 - v.2.0*2020.7.20 —ATG

11 Chaotic and solitonic solutions for a Korteweg-de Vries equation Article no. 117

19. I. Jaradat, M. Al-Dolat, K. Al-Zoubi, and M. Alquran, Theory and applications of a more generalform for fractional power series expansion, Chaos Solitons & Fractals 108, 107–110 (2018).

20. I. Jaradat, M. Alquran, and M. Al-Dolat, Analytic solution of homogeneous time-invariant frac-tional IVP, Adv. Differ. Equ. 2018, 143 (2018).

21. I. Jaradat, M. Alquran, and R. Abdel-Muhsen, An analytical framework of 2D diffusion, wave-like, telegraph, and Burgers’ models with twofold Caputo derivatives ordering, Nonl. Dyn. 93(4),1911–1922 (2018).

22. A. Jaradat, M.S.M. Noorani, M. Alquran, and H.M. Jaradat, A novel method for solving Caputo-time-fractional dispersive long wave Wu-Zhang system, Nonl. Dyn. Syst. Theory 18(2), 182–190(2018).

23. M. Alquran and I. Jaradat, A novel scheme for solving Caputo time-fractional nonlinear equations:theory and application, Nonl. Dyn. 91(4), 2389–2395 (2018).

24. A. Atangana and D. Baleanu, Caputo-Fabrizio derivative applied to groundwater flow within con-fined aquifer, J. Eng. Mech. 143(5), D4016005 (2017).

25. K.M. Saad, D. Baleanu, and A. Atangana, New fractional derivatives applied to the Korteweg-de Vries and Korteweg-de Vries-Burger’s equations, J. Comput. Appl. Math. 37(4), 5203–5216(2018).

26. K.M. Saad, A. Atangana, and D. Baleanu, New fractional derivatives with non-singular kernelapplied to the Burgers equation, Chaos 28(6), 063109 (2018).

27. A. Atangana and D. Baleanu, New fractional derivatives with non-local and non-singular kernel:Theory and application to heat transfer model, Therm. Sci. 20(2), 763–769 (2016).

28. A. Atangana, D. Baleanu, and A. Alsaedi, New properties of conformable derivative, Open Math.13, 889–898 (2015).

29. W. Malfliet and W. Hereman, The tanh method: I. Exact solutions of nonlinear evolution and waveequations, Phys. Scr. 54, 563–568 (1996).

30. I. Jaradat, M. Alquran, and M. Ali, A numerical study on weak-dissipative two-mode perturbedBurgers’ and Ostrovsky models: right-left moving waves, Eur. Phys. J. Plus 133, 164 (2018).

31. A.M. Wazwaz, A variety of distinct kinds of multiple soliton solutions for a (3+1)-dimensionalnonlinear evolution equation, Math. Meth. Appl. Sci. 36(3), 349–357 (2013).

32. M. Alquran and K. Al-Khaled, The tanh and sine-cosine methods for higher order equations ofKorteweg-de Vries type, Phys. Scr. 84, 025010 (2011).

33. I. Abu-Irwaq, M. Alquran, I. Jaradat, and D. Baleanu, New dual-mode Kadomtsev-Petviashvilimodel with strong-weak surface tension: analysis and application, Adv. Diff. Equ. 2018, 433(2018).

34. M. Alquran, I. Jaradat, and D. Baleanu, Shapes and dynamics of dual-mode Hirota-Satsuma cou-pled KdV equations: Exact traveling wave solutions and analysis, Chin. J. Phys. 58, 49–56 (2019).

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