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Dark soliton and periodic wave solutions of nonlinear evolution equations

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Abstract

This paper studies the Kudryashov-Sinelshchikov and Jimbo-Miwa equations. Subsequently, we formally derive the dark (topological) soliton solutions for these equations. By using the sine-cosine method, some additional periodic solutions are derived. The physical parameters in the soliton solutions of the ansatz method, amplitude, inverse width and velocity, are obtained as functions of the dependent model coefficients.

PACS Codes:02.30.Jr, 02.70.Wz, 05.45.Yv, 94.05.Fg.

1 Introduction

In recent years, many powerful methods to construct exact solutions of nonlinear partial differential equations have been established and developed, which led to one of the most exciting advances of nonlinear science and theoretical physics. Particularly, the existence of soliton-type solutions for nonlinear models is of great importance because of their potential application in many physics areas such as nonlinear optics, plasmas, fluid mechanics, condensed matter and many more. Remarkably, the interest in dark and bright solitons has been growing steadily in recent years [13]. In fact, many kinds of exact soliton solutions have been obtained by using, for example, the tanh-sech method [46], extended tanh method [79], homogeneous balance method [10, 11], first integral method [12, 13], Jacobi elliptic function method [14, 15], ( G G )-expansion method [16, 17], F-expansion method [18, 19], Hirota bilinear method [20, 21], multiple exp-function method [22] and transformed rational function method [23] and so on.

In 2010, Kudryashov and Sinelshchikov [24] obtained a more common nonlinear partial differential equation for describing the pressure waves in a mixture liquid and gas bubbles taking into consideration the viscosity of the liquid and the heat transfer. The equation reads as follows [25]:

u t +αu u x + u x x x ( u u x x ) x b u x u x x =0.
(1.1)

In this equation, u is a density and α, b are real parameters. Ryabov [26] obtained some exact solutions for b=3 and b=4 using a modification of the truncated expansion method. Solutions are derived in a more straightforward manner and cast into a simpler form, and some new types of solutions which contain solitary wave and periodic wave solutions are presented in [27].

On the other hand, there is the Jimbo-Miwa equation (JM)

u x x x y +3 u y u x x +3 u x u x y +2 u y t 3 u x z =0.
(1.2)

This equation was first introduced by Jimbo and Miwa [28] and it is known that this model is not Painlevé integrable. For many years, many scientists have researched it and certain explicit solutions have been obtained [2931]. Exact three-wave solutions including periodic cross-kink wave solutions, doubly periodic solitary wave solutions and breather type of two-solitary wave solutions for the Jimbo-Miwa equation have been obtained using the generalized three-wave method in [32].

The layout of this paper is organized as follows. In Section 2, we give the description of the sine-cosine method and we apply this method to the Kudryashov-Sinelshchikov (KS) and Jimbo-Miwa equations. We apply the ansatz method to the KS and JM equations in Section 3. Conclusions are given in the last section.

2 Sine-cosine method

In this section, the sine-cosine method will be first described and then subsequently applied to solve the Kudryashov-Sinelshchikov and Jimbo-Miwa equations.

2.1 Brief of the method

The sine-cosine method was first proposed by Wazwaz in 2004 [33]. This method has been applied to various kinds of nonlinear problems arising in the applied sciences [3437].

  1. 1.

    We introduce the wave variable ξ=x+yct into the PDE

    P(u, u t , u x , u y , u t t , u x x , u y y , u x t , u x y , u t y ,)=0,
    (2.1)

where u(x,y,t) is a traveling wave solution. This enables us to use the following changes:

t = c ξ , 2 t 2 = c 2 2 ξ 2 , x = ξ , 2 x 2 = 2 ξ 2 , y = ξ , 2 y 2 = 2 ξ 2 , .
(2.2)

One can immediately reduce nonlinear PDE (2.1) into the nonlinear ODE

Q(u, u ξ , u ξ ξ , u ξ ξ ξ ,)=0.
(2.3)

Ordinary differential equation (2.3) is then integrated as long as all terms contain derivatives, where we neglect integration constants.

  1. 2.

    The solutions of many nonlinear equations can be expressed in the form [34]

    u(x,t)={ λ cos β ( μ ξ ) , | ξ | π 2 μ , 0 , otherwise ,
    (2.4)

or in the form

u(x,t)={ λ sin β ( μ ξ ) , | ξ | π μ , 0 , otherwise ,
(2.5)

where λ, μ and β0 are parameters that will be determined, μ and c are the wave number and the wave speed respectively. We use

u ( ξ ) = λ cos β ( μ ξ ) , u n ( ξ ) = λ n cos n β ( μ ξ ) , ( u n ) ξ = n μ β λ n sin ( μ ξ ) cos n β 1 ( μ ξ ) , ( u n ) ξ ξ = n 2 μ 2 β 2 λ n cos n β ( μ ξ ) + n μ 2 λ n β ( n β 1 ) cos n β 2 ( μ ξ ) ,
(2.6)

and the derivatives of (2.5) become

u ( ξ ) = λ sin β ( μ ξ ) , u n ( ξ ) = λ n sin n β ( μ ξ ) , ( u n ) ξ = n μ β λ n cos ( μ ξ ) sin n β 1 ( μ ξ ) , ( u n ) ξ ξ = n 2 μ 2 β 2 λ n sin n β ( μ ξ ) + n μ 2 λ n β ( n β 1 ) sin n β 2 ( μ ξ ) ,
(2.7)

and so on for other derivatives.

3. We substitute (2.6) or (2.7) into the reduced equation obtained above in (2.3), balance the terms of the cosine functions when (2.7) is used, or balance the terms of the sine functions when (2.6) is used, and solve the resulting system of algebraic equations by using the computerized symbolic calculations. We next collect all terms with the same power in cos k (μξ) or sin k (μξ) and set to zero their coefficients to get a system of algebraic equations among the unknowns μ, β and λ. We obtained all possible values of the parameters μ, β and λ [33].

2.2 Application of the sine-cosine method to the Kudryashov-Sinelshchikov equation

We begin first with Eq. (1.1). Using the wave variable ξ=xvt, Eq. (1.1) is carried to the ODE

( u v ) +α ( u 2 2 ) + u ( u u ) b 2 [ ( u ) 2 ] =0,
(2.8)

where by integrating once we obtain

uv+α u 2 2 + u u u b 2 ( u ) 2 +k=0,
(2.9)

where k is the integration constant.

Substituting (2.6) into (2.9) gives

(2.10)

Equating the exponents and the coefficients of each pair of the cosine functions, we find the following system of algebraic equations:

( β 1 ) 0 , β 2 = 2 β , 1 2 α λ 2 + 4 λ 2 μ 2 + 2 b λ 2 μ 2 = 0 , v λ 4 λ μ 2 2 λ 2 μ 2 2 b λ 2 μ 2 = 0 , 2 λ μ 2 + k = 0 .
(2.11)

Solving the system (2.11) yields

β = 2 , μ = α 8 + 4 b , λ = 2 k ( b + 2 ) α , v = a + 2 k + 3 k b + b 2 k b + 2 .
(2.12)

The result (2.11) can be easily obtained if we also use the sine method (2.5). Consequently, for α 8 + 4 b <0, the following periodic solutions can be obtained:

u(x,t)= ( 2 k ( b + 2 ) α ) sec 2 [ α 8 + 4 b ( x v t ) ] ,
(2.13)

where | α 8 + 4 b (xvt)|< π 2 , and

u(x,t)= ( 2 k ( b + 2 ) α ) csc 2 [ α 8 + 4 b ( x v t ) ] ,
(2.14)

where 0< α 8 + 4 b (xvt)<π.

However, for α 8 + 4 b >0, we obtain the soliton solutions

u(x,t)= ( 2 k ( b + 2 ) α ) sech 2 [ α 8 + 4 b ( x v t ) ]
(2.15)

and

u(x,t)= ( 2 k ( b + 2 ) α ) csch 2 [ α 8 + 4 b ( x v t ) ] .
(2.16)

All the solutions reported in this paper have been verified with Maple by putting them back into original Eq. (1.1), which cannot be obtained by the methods [2527]. To the best of our knowledge, these solutions are new and have not been reported yet.

2.3 Application of the sine-cosine method to the Jimbo-Miwa equation

We begin second with Eq. (1.2). Using the wave variable ξ=x+y+zvt, Eq. (1.2) is carried to the ODE

u +3 [ ( u ) 2 ] (2v+3) u =0,
(2.17)

where by integrating once we obtain

u +3 ( u ) 2 (2v+3) u =0,
(2.18)

which is obtained upon setting the constant of integration to zero. Setting u =ρ, Eq. (2.18) becomes

ρ +3 ρ 2 (2v+3)ρ=0.
(2.19)

Substituting (2.4) into (2.19) gives

(2.20)

Equating the exponents and the coefficients of each pair of the cosine functions, we find the following system of algebraic equations:

( β 1 ) 0 , β 2 = 2 β , 6 λ μ 2 + 3 λ 2 = 0 , 4 λ μ 2 2 λ v 3 λ = 0 .
(2.21)

Solving the system (2.21) yields

β = 2 , μ = 1 2 2 v 3 , λ = v + 3 2 .
(2.22)

The result (2.21) can be easily obtained if we also use the sine method (2.5). Consequently, for 2v+3<0, the following periodic solutions can be obtained:

u(x,y,z,t)= ( v + 3 2 ) sec 2 [ 1 2 2 v 3 ( x + y + z v t ) ] ,
(2.23)

where | 1 2 2 v 3 (x+y+zvt)|< π 2 , and

u(x,y,z,t)= ( v + 3 2 ) csc 2 [ 1 2 2 v 3 ( x + y + z v t ) ] ,
(2.24)

where 0< 1 2 2 v 3 (x+y+zvt)<π.

However, for 2v+3>0, we obtain the soliton solutions

u(x,y,z,t)= ( v + 3 2 ) sech 2 [ 1 2 2 v 3 ( x + y + z v t ) ]
(2.25)

and

u(x,y,z,t)= ( v 3 2 ) csch 2 [ 1 2 2 v + 3 ( x + y + z v t ) ] .
(2.26)

Comparing the above results with the relevant ones in [2831], it can be seen that some of the obtained results are new and the rest of solutions are the same.

3 Ansatz method

In this section, the ansatz method will be used to carry out the integration of the Kudryashov-Sinelshchikov and Jimbo-Miwa equations. The search is going to be for a topological 1-soliton solution which is also known as a kink solution or a shock wave solution. This will be demonstrated in the following two subsections. For both equations, arbitrary constant coefficients will be considered. There are many applications of this method [3844].

3.1 Application of the ansatz method to the Kudryashov-Sinelshchikov equation

In this section the search is going to be for a topological 1-soliton solution to the Kudryashov-Sinelshchikov equation given by (1.1). To start off, the hypothesis is given by [45, 46]

(3.1)
(3.2)

Here, A and B are free parameters and v is the velocity of the wave in (3.1) and (3.2). The exponent p is unknown at this point and its values will fall out in the process of deriving the solution of this equation. Thus from (3.1) we get

(3.3)
(3.4)
(3.5)
(3.6)
(3.7)

Substituting Eqs. (3.3)-(3.7) into (1.1), we have

(3.8)

From (3.8), equating the exponents 2p1 and p+1 gives

2p1=p+1

so that

p=2.
(3.9)

It should be noted that the same value of p is yielded when the exponent pairs 2p3 and p1, 2p+1 and p+3 are equated with each other, respectively.

(3.10)
(3.11)
(3.12)
(3.13)

If we put p=2 in (3.10)-(3.13), the system reduces to

(3.14)
(3.15)
(3.16)

Solving the above equations yields

(3.17)
(3.18)
(3.19)

Hence, finally, the 1-soliton solution to (1.1) is respectively given by

u(x,t)=A tanh 2 [ B ( x v t ) ] ,
(3.20)

where the free parameter A is given by (3.18), the velocity of the solitons v is given in (3.19).

3.2 Application of the ansatz method to the Jimbo-Miwa equation

In this section the search is going to be for a topological 1-soliton solution to the (3+1)-dimensional Jimbo-Miwa equation. Without any loss of generality, it is assumed that the dark soliton solution to (1.2) is given by

u(x,y,z,t)=A tanh p τ,
(3.21)

where

τ=B(x+y+zvt).
(3.22)

Here, A and B are free parameters and v is the velocity of the wave in (3.21) and (3.22). The exponent p is unknown at this point and its values will fall out in the process of deriving the solution of this equation. Thus from (3.21) we get

(3.23)
(3.24)
(3.25)
(3.26)
(3.27)
(3.28)
(3.29)
(3.30)
(3.31)

Substituting Eqs. (3.23)-(3.31) into (1.2), we have

(3.32)

From (3.32), equating the exponents 2p+3 and p+4 gives

2p+3=p+4

so that

p=1.

It should be noted that the same value of p is yielded when the exponent pairs 2p3 and p2, 2p+1 and p+2, 2p1 and p are equated with each other, respectively.

(3.33)
(3.34)
(3.35)

Solving the above system for p=1 gives

A = 2 B , v = 2 B 2 3 2 .
(3.36)

Thus, finally, the 1-soliton solution to (1.2) is respectively given by

u(x,y,z,t)=Atanh [ B ( x + y + z v t ) ] ,
(3.37)

where the free parameter A is given by (3.36) and the velocity of the solitons v is given in (3.36).

4 Conclusion

In this paper, the KS and JM equations are solved by the sine-cosine method as well as by the solitary wave ansatz method. There are several solutions that are obtained by the first method. The solitary wave ansatz method is used to carry out the integration of these equations. The obtained solutions may be useful for understanding of the mechanism of complicated nonlinear physical phenomena in wave interaction. In addition, we note that the solitary wave ansatz method is an efficient method for constructing exact soliton solutions for nonlinear wave equations. These results are going to be very useful for conducting research in future.

References

  1. 1.

    Triki H, Wazwaz AM: Bright and dark soliton solutions for a K(m,n)equation with t -dependent coefficients. Phys. Lett. A 2009, 373: 2162-2165. 10.1016/j.physleta.2009.04.029

  2. 2.

    Green PD, Biswas A: Bright and dark optical solitons with time-dependent coefficients in a non-Kerr law media. Commun. Nonlinear Sci. Numer. Simul. 2010, 15: 3865-3873. 10.1016/j.cnsns.2010.01.018

  3. 3.

    Biswas A, Milovic D: Bright and dark solitons of the generalized nonlinear Schrödinger’s equation. Commun. Nonlinear Sci. Numer. Simul. 2010, 15: 1473-1484. 10.1016/j.cnsns.2009.06.017

  4. 4.

    Malfliet W: Solitary wave solutions of nonlinear wave equations. Am. J. Phys. 1992, 60: 650-654. 10.1119/1.17120

  5. 5.

    Malfliet W, Hereman W: The tanh method. I: exact solutions of nonlinear evolution and wave equations. Phys. Scr. 1996, 54: 563-568. 10.1088/0031-8949/54/6/003

  6. 6.

    Wazwaz AM: The tanh method for travelling wave solutions of nonlinear equations. Appl. Math. Comput. 2004, 154(3):713-723. 10.1016/S0096-3003(03)00745-8

  7. 7.

    Abdou MA: New solitons and periodic wave solutions for nonlinear physical models. Nonlinear Dyn. 2008, 52(1-2):129-136. 10.1007/s11071-007-9265-7

  8. 8.

    Fan E: Extended tanh-function method and its applications to nonlinear equations. Phys. Lett. A 2000, 277: 212-218. 10.1016/S0375-9601(00)00725-8

  9. 9.

    Wazwaz AM: The extended tanh method for new soliton solutions for many forms of the fifth-order KdV equations. Appl. Math. Comput. 2007, 184(2):1002-1014. 10.1016/j.amc.2006.07.002

  10. 10.

    Fan E, Zhang H: A note on the homogeneous balance method. Phys. Lett. A 1998, 246: 403-406. 10.1016/S0375-9601(98)00547-7

  11. 11.

    Wang ML: Exact solutions for a compound KdV-Burgers equation. Phys. Lett. A 1996, 213: 279-287. 10.1016/0375-9601(96)00103-X

  12. 12.

    Feng ZS: The first integral method to study the Burgers-KdV equation. J. Phys. A, Math. Gen. 2002, 35: 343. 10.1088/0305-4470/35/2/312

  13. 13.

    Taghizadeh N, Mirzazadeh M: The first integral method to some complex nonlinear partial differential equations. J. Comput. Appl. Math. 2011, 235(16):4871. 10.1016/j.cam.2011.02.021

  14. 14.

    Fan E, Zhang J: Applications of the Jacobi elliptic function method to special-type nonlinear equations. Phys. Lett. A 2002, 305: 383-392. 10.1016/S0375-9601(02)01516-5

  15. 15.

    Liu S, Fu Z, Liu S, Zhao Q: Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations. Phys. Lett. A 2001, 289: 69-74. 10.1016/S0375-9601(01)00580-1

  16. 16.

    Wang ML, Li X, Zhang J:The ( G G )-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics. Phys. Lett. A 2008, 372(4):417-423. 10.1016/j.physleta.2007.07.051

  17. 17.

    Bekir A:Application of the ( G G )-expansion method for nonlinear evolution equations. Phys. Lett. A 2008, 372(19):3400-3406. 10.1016/j.physleta.2008.01.057

  18. 18.

    Abdou MA: Further improved F -expansion and new exact solutions for nonlinear evolution equations. Nonlinear Dyn. 2008, 52(3):277-288. 10.1007/s11071-007-9277-3

  19. 19.

    Zhang JL, Wang ML, Wang YM, Fang ZD: The improved F -expansion method and its applications. Phys. Lett. A 2006, 350: 103-109. 10.1016/j.physleta.2005.10.099

  20. 20.

    Wazwaz AM: Multiple soliton solutions for coupled KdV and coupled KP systems. Can. J. Phys. 2010, 87(12):1227-1232.

  21. 21.

    Wazwaz AM: Extended KP equations and extended system of KP equations: multiple-soliton solutions. Can. J. Phys. 2011, 89(7):739-743. 10.1139/p11-065

  22. 22.

    Ma WX, Huang T, Zhang Y: A multiple exp-function method for nonlinear differential equations and its application. Phys. Scr. 2010., 82: Article ID 065003

  23. 23.

    Ma WX:Comment on the (3+1)-dimensional Kadomtsev-Petviashvili equations. Commun. Nonlinear Sci. Numer. Simul. 2011, 16: 2663. 10.1016/j.cnsns.2010.10.003

  24. 24.

    Kudryashov NA, Sinelshchikov DI: Nonlinear waves in bubbly liquids with consideration for viscosity and heat transfer. Phys. Lett. A 2010, 374: 2011-2016. 10.1016/j.physleta.2010.02.067

  25. 25.

    He B, Meng Q, Zhang J, Long Y: Periodic loop solutions and their limit forms for the Kudryashov-Sinelshchikov equation. Math. Probl. Eng. 2012., 2012: Article ID 320163

  26. 26.

    Ryabov PN: Exact solutions of the Kudryashov and Sinelshchikov equation. Appl. Math. Comput. 2010, 217: 3585-3590. 10.1016/j.amc.2010.09.003

  27. 27.

    Randrüüt M: On the Kudryashov-Sinelshchikov equation for waves in bubbly liquids. Phys. Lett. A 2011, 375: 3687-3692. 10.1016/j.physleta.2011.08.048

  28. 28.

    Jimbo M, Miwa T: Solitons and infinite dimensional Lie algebras. Publ. Res. Inst. Math. Sci. 1983, 19: 943-1001. 10.2977/prims/1195182017

  29. 29.

    Senthilvelan M: On the extended applications of homogenous balance method. Appl. Math. Comput. 2001, 123: 381-388. 10.1016/S0096-3003(00)00076-X

  30. 30.

    Tang XY, Liang ZF:Variable separation solutions for the (3+1)-dimensional Jimbo-Miwa equation. Phys. Lett. A 2006, 351: 398-402. 10.1016/j.physleta.2005.11.035

  31. 31.

    Wazwaz AM: New solutions of distinct physical structures to high-dimensional nonlinear evolution equations. Appl. Math. Comput. 2008, 196: 363-370. 10.1016/j.amc.2007.06.002

  32. 32.

    Li Z, Dai Z, Liu J:Exact three-wave solutions for the (3+1)-dimensional Jimbo-Miwa equation. Comput. Math. Appl. 2011, 61: 2062-2066. 10.1016/j.camwa.2010.08.070

  33. 33.

    Wazwaz AM: A sine-cosine method for handling nonlinear wave equations. Math. Comput. Model. 2004, 40: 499-508. 10.1016/j.mcm.2003.12.010

  34. 34.

    Wazwaz AM: The tanh method and the sine-cosine method for solving the KP-MEW equation. Int. J. Comput. Math. 2005, 82(2):235-246. 10.1080/00207160412331296706

  35. 35.

    Bekir A, Cevikel AC: New solitons and periodic solutions for nonlinear physical models in mathematical physics. Nonlinear Anal., Real World Appl. 2010, 11(4):3275-3285. 10.1016/j.nonrwa.2009.10.015

  36. 36.

    Bekir A: New solitons and periodic wave solutions for some nonlinear physical models by using sine-cosine method. Phys. Scr. 2008, 77(4):501-505.

  37. 37.

    Zedan HA, Monaquel SJ: The sine-cosine method for the Davey-Stewartson equations. Appl. Math. E-Notes 2010, 10: 103-111.

  38. 38.

    Triki H, Wazwaz AM:A one-soliton solution of the ZK(m,n,k) equation with generalized evolution and time-dependent coefficients. Nonlinear Anal., Real World Appl. 2011, 12: 2822-2825. 10.1016/j.nonrwa.2011.04.008

  39. 39.

    Triki H, Wazwaz AM: Bright soliton solution to a generalized Burgers-KdV equation with time-dependent coefficients. Appl. Math. Comput. 2010, 217: 466-471. 10.1016/j.amc.2010.05.078

  40. 40.

    Triki H, Wazwaz AM: Bright and dark solitons for a generalized Korteweg de-Vries-modified Korteweg de-Vries equation with higher-order nonlinear terms and time-dependent coefficients. Can. J. Phys. 2011, 89(3):253-259. 10.1139/P11-015

  41. 41.

    Triki H, Wazwaz AM: Soliton solutions for a generalized KdV and BBM equations with time-dependent coefficients. Commun. Nonlinear Sci. Numer. Simul. 2011, 16: 1122-1126. 10.1016/j.cnsns.2010.06.024

  42. 42.

    Biswas A:1-soliton solution of the B(m,n) equation with generalized evolution. Commun. Nonlinear Sci. Numer. Simul. 2009, 14: 3226-3229. 10.1016/j.cnsns.2008.12.025

  43. 43.

    Triki H, Ismail MS:Soliton solutions of a BBM(m,n) equation with generalized evolution. Appl. Math. Comput. 2010, 217(1):48-54. 10.1016/j.amc.2010.05.063

  44. 44.

    Esfahani A: On the generalized Kadomtsev-Petviashvili equation with generalized evolution and variable coefficients. Phys. Lett. A 2010, 374: 3635-3645. 10.1016/j.physleta.2010.07.015

  45. 45.

    Biswas A, Triki H, Labidi M: Bright and dark solitons of the Rosenau-Kawahara equation with power law nonlinearity. Phys. Wave Phenom. 2011, 19(1):24-29. 10.3103/S1541308X11010067

  46. 46.

    Triki H, Wazwaz AM: Dark solitons for a combined potential KdV and Schwarzian KdV equations with t -dependent coefficients and forcing term. Appl. Math. Comput. 2011, 217: 8846-8851. 10.1016/j.amc.2011.03.050

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Correspondence to Adem Cengiz Cevikel.

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The authors declare that there is no conflict of interest among the authors.

Authors’ contributions

AÇ carried out the theoretical studies. AB carried out the computational studies and wrote the paper according to the rules of journal. ÖG found out the problem in literature, and carried out the computational studies. All authors read and approved the final manuscript.

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Keywords

  • dark soliton
  • sine-cosine method
  • ansatz method
  • Kudryashov-Sinelshchikov equation
  • Jimbo-Miwa equation