- Email: [email protected]
Soliton solutions of the Klein–Gordon–Zakharov equations with power law nonlinearity
Applied Mathematics and Computation 227 (2014) 341–346
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Soliton solutions of the Klein–Gordon–Zakharov equations with power law nonlinearity Houria Triki ⇑, Noureddine Boucerredj Department of Physics, Faculty of Sciences, Badji Mokhtar University, P.O. Box 12, 23000 Annaba, Algeria
a r t i c l e
i n f o
Keywords: Solitons Klein–Gordon–Zakharov equations Solitary wave ansatz method
a b s t r a c t In this paper, the Klein–Gordon–Zakharov equations which model the interaction between the Langmuir wave and the ion acoustic wave in a high frequency plasma, are considered. To examine the role played by the nonlinear dispersion term in the formation of solitons, a family of the considered equations with power law nonlinearity are investigated. By using p p two solitary wave ansatze in terms of sec h ðxÞ and tanh ðxÞ functions, we find exact analytical bright and dark soliton solutions for the considered model. The physical parameters in the soliton solutions are obtained as function of the dependent model coefficients. The conditions of existence of solitons are presented. These closed form solutions are helpful to well understand the mechanism of the complicated physical phenomena and dynamical processes modeled by the used model. Ó 2013 Elsevier Inc. All rights reserved.
1. Introduction Two different types of envelope solitons, bright and dark, can propagate in nonlinear dispersive media [1]. The dynamics of such shape-preserving waves is typically described by a certain class of nonlinear partial differential equations (NLPDEs). The best known examples include the cubic nonlinear Schrödinger equation, the Korteweg–de Vries equation, the Sine–Gordon equation, the Boussinesq equation, etc. The formation of solitons has been regarded as a consequence of the delicate balance between dispersion (or diffraction) and nonlinearity under certain conditions. In real applications, however, it may be difficult to produce such balances [2]. It is worth noting that the existence of soliton solutions depends on the specific nonlinear and dispersive features of the medium [2]. In other words, the key factors, which determine the closed form solutions of a given nonlinear evolution equation are the dispersion and nonlinear coefficients which can be constant or variable parameters depending on the physical situation. For this, one may need to know what kind of the existing effects that may contribute for generating soliton pulses in the medium. As is well known, solitons are universal phenomenon, appearing in a great array of contexts such as, for example, nonlinear optics, plasma physics, fluid dynamics, semiconductors and many other systems. As a matter of fact, the study of such nonlinear waves has attracted extensive attention. One of the great interests is the problem of finding exact soliton solutions of the integrable nonlinear models. Based on these exact solutions directly, we can accurately analyze the properties of propagating soliton pulses in nonlinear physical systems. Towards that goal, many powerful methods to construct exact solutions of NLPDEs have been established and developed, which lead to one of the most excited advances of nonlinear science and theoretical physics [3]. Among these methods we can cite, for example, the subsidiary ordinary differential equation method (sub-ODE for short) [3–5], the coupled amplitude-phase formulation [6], sine–cosine method [7], the Hirota’s bilinear method [8,9], and many others. These methods work even though the Painlevé test of integrability will fail [10]. ⇑ Corresponding author. E-mail address: [email protected] (H. Triki). 0096-3003/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2013.10.093
342
H. Triki, N. Boucerredj / Applied Mathematics and Computation 227 (2014) 341–346
In recent years, much attention in applied mathematics has been focused on calculating soliton solutions by means of the so called solitary wave ansatz method. This method has been successfully applied to many kinds of NLPDEs with constant and varying coefficients, such as, for example, the Kðm; nÞ equation [11,12], the BBM equation [13,14], the Bðm; nÞ equation [10], the nonlinear Schrödinger’s equation [15–17], the Kawahara equation [18] and many others. The purpose of this paper is to calculate the exact bright and dark soliton solutions for a family of the Klein–Gordon– Zakharov equations, exhibiting power law nonlinearity. The importance of the results presented here is twofold. First, exact soliton solutions to a family of the considered model and the conditions of their existence are obtained for a general case of power nonlinearly law in a simple way. The finding of explicit solutions of a given nonlinear evolution equation having any value of the exponent in the nonlinearity term are very interesting since it offer some knowledge on the general dynamical behavior of the propagation so that special cases are truly meaningful both from the physical and mathematical point of view. Second, these results confirms the existence of dark solitons only in the case where the introduced exponent is equal to unity which corresponds exactly to the standard form of the Klein–Gordon–Zakharov equations. To achieve our goal we will use the solitary wave ansatz method which has recently been applied successfully to several NLPDEs. 2. Mathematical analysis The standard form of the Klein–Gordon–Zakharov (KGZ) equations is given by [19]
utt uxx þ u þ anu ¼ 0;
ð1Þ
ntt nxx ¼ bðj uj2 Þxx :
ð2Þ
Here in (1) and (2), the variable uðx; tÞ is a complex function and nðx; tÞ is a real function. a and b are two nonzero real parameters. This system describes the interaction of the langmuir wave and the ion acoustic wave in a high frequency plasma [19]. Based on the extended hyperbolic functions method, Shang et al. [19] have recently obtained the multiple exact explicit solutions of the Klein–Gordon—Zakharov equations (1) and (2). In this paper, we are interesting in the following family of the Klein–Gordon–Zakharov equations with power law nonlinearity: 2
ð3Þ
2
ð4Þ
utt k1 uxx þ au þ bnu ¼ 0; ntt k2 nxx ¼ cðj uj2m Þxx ;
where a; b,c, k1 and k2 are nonzero real constants. The parameter m indicates the power law nonlinearity parameter. Thus for m ¼ 1, (3) and (4) respectively collapses to (1) and (2). 2.1. Bright solitons In order to solve (3) and (4), the starting hypothesis are given by
uðx; tÞ ¼
A1 ei/ p cosh 1 s
ð5Þ
nðx; tÞ ¼
A2 ; p cosh 2 s
ð6Þ
and
where
s ¼ Bðx v tÞ
ð7Þ
/ ¼ jx þ xt þ h:
ð8Þ
and
Here, in (5)–(8), A1 and A2 are the amplitudes of the u-soliton and n-soliton, respectively, while v is the velocity of the soliton and B is the inverse widths of the solitons. Also, j is the frequency of the soliton, while x is the wave number of the soliton and h is the phase constant. The exponents p1 and p2 are unknown at this point and their values will fall out in the process of deriving the solution of this model of equations. Thus from (5) and (6),
( ) A1 ðp21 B2 v 2 x2 Þ p1 ðp1 þ 1ÞA1 B2 v 2 2ixp1 A1 Bv tanh s i/ e ; utt ¼ þ p p p þ2 cosh 1 s cosh 1 s cosh 1 s
ð9Þ
H. Triki, N. Boucerredj / Applied Mathematics and Computation 227 (2014) 341–346
343
uxx ¼
( ) A1 ðp21 B2 j2 Þ p1 ðp1 þ 1ÞA1 B2 2ijp1 A1 B tanh s i/ e ; þ p p p þ2 cosh 1 s cosh 1 s cosh 1 s
ð10Þ
nu ¼
A1 A2 ei/ ; p þp cosh 1 2 s
ð11Þ
ntt ¼
p22 v 2 A2 B2 p2 ðp2 þ 1Þv 2 A2 B2 ; p p þ2 cosh 2 s cosh 2 s
ð12Þ
nxx ¼
p22 A2 B2 p2 ðp2 þ 1ÞA2 B2 ; p p þ2 cosh 2 s cosh 2 s
ð13Þ
ðj uj2m Þxx ¼
2 4p21 m2 A2m 1 B 2p1 m
cosh
s
2 2p1 mð2p1 m þ 1ÞA2m 1 B
cosh
2p1 mþ2
s
ð14Þ
:
Substituting (9)–(14) into (3) and decomposing into real and imaginary parts respectively yields 2
2
2
A1 p21 B2 ðv 2 k1 Þ A1 ðx2 k1 j2 Þ þ aA1 p1 ðp1 þ 1ÞA1 B2 ðv 2 k1 Þ bA1 A2 þ ¼0 p p þp p þ2 cosh 1 s cosh 1 2 s cosh 1 s
ð15Þ
and 2
2xp1 A1 Bv tanh s 2k1 jp1 A1 B tanh s ¼ 0; p p cosh 1 s cosh 1 s
ð16Þ
while (4) reduces to 2
2
2 2 p22 A2 B2 ðv 2 k2 Þ p2 ðp2 þ 1ÞA2 B2 ðv 2 k2 Þ 4cp21 m2 A2m 2cp1 mð2p1 m þ 1ÞA2m 1 B 1 B þ ¼ 0: p2 p þ2 2p m 2p mþ2 cosh s cosh 1 s cosh 1 s cosh 2 s
ð17Þ
Now (16) gives 2
v¼
k1 j
x
ð18Þ
:
From (15), equating the exponents p1 þ 2 and p1 þ p2 gives
p1 þ 2 ¼ p1 þ p2 ;
ð19Þ
so that
p2 ¼ 2:
ð20Þ
Now from (17), equating the exponents 2p1 m þ 2 and p2 þ 2 gives
2p1 m þ 2 ¼ p2 þ 2
ð21Þ
and therefore
p1 ¼
1 : m
ð22Þ
It needs to be noted that the same value of p1 is yielded when the exponents 2p1 m and p2 are equated with each other. We would like to point out that we must have p1 > 0 for the solution (5) to exist. This implies that m > 1 from (22). Now from p þj (15), setting the coefficients of the linearly independent functions 1=cosh 1 s to zero, where j ¼ 0; 2, gives 2
2
A1 p21 B2 ðv 2 k1 Þ A1 ðx2 k1 j2 Þ þ aA1 ¼ 0; 2
p1 ðp1 þ 1ÞA1 B2 ðv 2 k1 Þ þ bA1 A2 ¼ 0:
ð23Þ ð24Þ
Solving the above equations with making use of (18), we obtain
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2 2 2 mx u t x k1 j a B¼ k1 x2 k21 j2
ð25Þ
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H. Triki, N. Boucerredj / Applied Mathematics and Computation 227 (2014) 341–346
and
B¼
mx k1
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi bA2 : 2 ðm þ 1Þðx2 k1 j2 Þ
ð26Þ
Next, equating the two values of B from (25) and (26), gives
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 bA2 þ ðm þ 1Þðk1 j2 þ aÞ x¼ : mþ1
ð27Þ
Thus (26) introduces the constraint condition 2
bðx2 k1 j2 Þ < 0:
ð28Þ p2 þj
From (17), setting the coefficients of the linearly independent functions 1=cosh unique equation:
s to zero, for j ¼ 0; 2, yields the following
2
A2 ðv 2 k2 Þ ¼ cA2m 1 :
ð29Þ
Hence, finally, the 1-soliton solution of the Klein–Gordon–Zakharov equations (3) and (4) are given by
uðx; tÞ ¼
A1 1
coshm ½Bðx v tÞ
eiðjxþxtþhÞ
ð30Þ
:
ð31Þ
and
nðx; tÞ ¼
A2 2
cosh ½Bðx v tÞ
where the amplitudes A1 and A2 are connected by (29) and the width of the solitons are given by (25) or (26). The velocity is given by (18) and the wave number is shown in (27). 2.2. Dark solitons In this section the search is going to be for shock wave solution or topological 1-soliton solution to Klein–Gordon–Zakharov equations given by (3) and (4). To start off, the hypothesis is given by [12,15] p1
sei/
ð32Þ
p2
s;
ð33Þ
uðx; tÞ ¼ A1 tanh and
nðx; tÞ ¼ A2 tanh where
s ¼ Bðx v tÞ
ð34Þ
/ ¼ jx þ xt þ h:
ð35Þ
and Here, in (32)–(35), A1 ; A2 and B are free parameters and v is the velocity of the solitons. Also, j is the frequency of the soliton, while x is the wave number of the soliton and h is the phase constant. The exponents p1 and p2 are unknown at this point and their values will fall out in the process of deriving the solutions of these coupled equations. Therefore from (32) and (33),
utt ¼ ½p1 A1 B2 v 2 fðp1 1Þ tanh
p1 2
s 2p1 tanhp1 s þ ðp1 þ 1Þ tanhp1 þ2 sg 2ixp1 A1 Bv ðtanhp1 1 s tanhp1 þ1 sÞ
p1
x2 A1 tanh ei/ ;
ð36Þ p1 2
uxx ¼ ½p1 A1 B2 fðp1 1Þ tanh
s 2p1 tanhp1 s þ ðp1 þ 1Þ tanhp1 þ2 sg 2ijp1 A1 B ðtanhp1 1 s tanhp1 þ1 sÞ
p
j2 A1 tanh 1 ei/ ; nu ¼ A1 A2 tanh
p1 þp2
ð37Þ
sei/ ;
ð38Þ p2 2
ntt ¼ p2 A2 B2 v 2 fðp2 1Þ tanh nxx ¼ p2 A2 B2 fðp2 1Þ tanh
p2 2
s 2p2 tanhp2 s þ ðp2 þ 1Þ tanhp2 þ2 sg;
s 2p2 tanhp2 s þ ðp2 þ 1Þ tanhp2 þ2 sg;
2 ðj uj2m Þxx ¼ 2p1 mA2m 1 B fð2p1 m 1Þ tanh
2p1 m2
s 4pm tanh2p1 m s þ ð2pm þ 1Þ tanh2p1 mþ2 sg:
ð39Þ ð40Þ ð41Þ
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H. Triki, N. Boucerredj / Applied Mathematics and Computation 227 (2014) 341–346
Substituting (36)–(41) into (3) and decomposing into real and imaginary parts respectively yields 2
p1 2
p1
p1 þp2
p1 A1 B2 ðv 2 k1 Þfðp1 1Þ tanh þ aA1 tanh
s þ bA1 A2 tanh
s 2p1 tanhp1 s þ ðp1 þ 1Þ tanhp1 þ2 sg x2 A1 tanhp1 þ k21 j2 A1 tanhp1 s
¼0
ð42Þ
and p1 1
2xp1 A1 Bv ðtanh
s tanhp1 þ1 sÞ þ 2k21 jp1 A1 Bðtanhp1 1 s tanhp1 þ1 sÞ ¼ 0;
ð43Þ
while (4) reduces to 2
p2 2
s 2p2 tanhp2 s þ ðp2 þ 1Þ tanhp2 þ2 sg 2p1 m2 2 s 4p1 m tanh2p1 m s þ ð2p1 m þ 1Þ tanh2p1 mþ2 sg ¼ 0: 2cp1 mA2m 1 B fð2p1 m 1Þ tanh
p2 A2 B2 ðv 2 k2 Þfðp2 1Þ tanh
ð44Þ
From (42), equating the exponents p1 þ 2 and p1 þ p2 gives
p1 þ 2 ¼ p1 þ p2 ;
ð45Þ
so that
p2 ¼ 2:
ð46Þ
Now from (44), equating the exponents 2p1 m þ 2 and p2 þ 2 gives
2p1 m þ 2 ¼ p2 þ 2
ð47Þ
and therefore
p1 ¼
1 : m
ð48Þ
It needs to be noted that the same value of p is yielded when the exponents pairs 2p1 m and p2 ; 2p1 m2 and p2 2 are equated with each other. From (43), we get 2
k1 j
v¼
x
ð49Þ
: 2p1 mþj
Now from (44) the linearly independent functions are tanh to zero yields
s for j ¼ 2; 0; 2. Hence setting their respective coefficients
2
A2 ðv 2 k2 Þ ¼ cA2m 1 :
ð50Þ p1 þj
Now from (42), setting the coefficients of the linearly independent functions tanh 2
2p21 A1 B ðv 2 2
2
2
2
p1 A1 B ðv p1 A1 B ðv
2 k1 Þ
2 k1 Þðp1 2 k1 Þðp1
x 2 A1 þ
2 k1
j2 A1 þ aA1 ¼ 0;
s to zero, where j ¼ 2; 0; 2, gives ð51Þ
þ 1Þ þ bA1 A2 ¼ 0;
ð52Þ
1Þ ¼ 0:
ð53Þ
To solve (53), we have considered the case p1 1 ¼ 0: This yields
p1 ¼ 1:
ð54Þ
Substituting (54) into the above system and using (49) gives
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u
x ux2 k21 j2 a B¼ t k1 x2 k21 j2
ð55Þ
and
B¼
x
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi bA2
k1
2ðx2 k1 j2 Þ
2
:
ð56Þ
Next, equating the two values of B from (55) and (56), gives
x¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 bA2 þ k1 j2 þ a;
ð57Þ
which shows that the solitons will exist for 2
bA2 þ k1 j2 þ a > 0:
ð58Þ
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H. Triki, N. Boucerredj / Applied Mathematics and Computation 227 (2014) 341–346
Also, (56) implies that it is necessary to have 2
bðx2 k1 j2 Þ > 0
ð59Þ
for solitons to exist. Now equating the two values of p1 from (48) and (54) gives the condition:
m ¼ 1:
ð60Þ
Thus, for the Klein–Gordon–Zakharov equations (3) and (4), dark solitons will exist only when m ¼ 1, which corresponds to the standard form of the Klein–Gordon–Zakharov equations (1) and (2). This important observation is being made for the first time in this paper. Thus, finally, the topological 1-soliton solutions to the considered equations (3) and (4) are given by
uðx; tÞ ¼ A1 tanh½Bðx v tÞeiðjxþxtþhÞ
ð61Þ
and 2
nðx; tÞ ¼ A2 tanh ½Bðx v tÞ;
ð62Þ
where the relation between the free parameters A1 and A2 is given by (50), and the velocity of the soliton is given by (49). Finally the free parameter B is given by (55) or (56), while the wave number x is of the soliton is shown in (57). 3. Conclusion In this paper, the solitary wave ansatz method is adopted to integrate a family of the Klein–Gordon–Zakharov equations with power law nonlinearity. Both of the topological (dark) as well as non-topological (bright) soliton solutions of the considered model with power law nonlinearity, are obtained. The physical parameters in the obtained soliton solutions are calculated in course of the derivation of the special exact solutions as function of the dependent model coefficients. Conditions for the existence of solitary wave solutions have also been reported. These results indicate that the bright solitons exist for any exponent m > 1, while the dark solitons exist only when m ¼ 1. These solutions may be useful to explain some physical phenomena in dynamical systems that are described by the Klein–Gordon–Zakharov equations-type models. In future, the dual power nonlinearity term will be taken into consideration to study the influence of high order nonlinearities on the dynamical behavior of the solitons propagation. These results will be reported in future. References [1] Mark M. Scott, Mikhail P. Kostylev, Boris A. Kalinikos, Carl E. Patton, Excitation of bright and dark envelope solitons for magnetostatic waves with attractive nonlinearity, Phys. Rev. B. 71 (2005) 174440:1–174440:4. [2] Ruiyu Hao, Lu Li, Zhonghao Li, Guosheng Zhou, Exact multisoliton solutions of the higher-order nonlinear Schrödinger equation with variable coefficients, Phys. Rev. E. 70 (2004) 1–6 (066603). [3] X. Li, M. Wang, A sub-ODE method for finding exact solutions of a generalized KdV–mKdV equation with high-order nonlinear terms, Phys. Lett A. 361 (2007) 115–118. [4] H. Triki, A.M. Wazwaz, Sub-ODE method and soliton solutions for the variable-coefficient mKdV equation, Appl. Math. Comput. 214 (2009) 370–373. [5] H. Triki, T.R. Taha, The sub-ODE method and soliton solutions for a higher order dispersive cubic–quintic nonlinear Schrödinger equation, Chaos Solitons Fract. 42 (2009) 1068–1072. [6] S.L. Palacios, A. Guinea, J.M. Fernandez-Diaz, R.D. Crespo, Dark solitary waves in the nonlinear Schrödinger equation with third order dispersion, selfsteepening, and self-frequency shift, Phys. Rev. E 60 (1) (1999) 45–47. [7] A.M. Wazwaz, New solitary wave solutions to the modified Kawahara equation, Phys. Lett. A 360 (2007) 588–592. [8] A.M. Wazwaz, H. Triki, Multiple soliton solutions for the sixth-order Ramani equation and a coupled Ramani equation, Appl. Math. Comput. 216 (2010) 332–336. [9] A.M. Wazwaz, Integrable (2+1)-dimensional and (3+1)-dimensional breaking soliton equations, Phys. Scr. 81 (2010) 5 (035005). [10] A. Biswas, 1-Soliton solution of the Bðm; nÞ equation with generalized evolution, Commun. Nonlinear Sci. Numer. Simul. 14 (2009) 3226–3229. [11] Anjan Biswas, 1-Soliton solution of the Kðm; nÞ equation with generalized evolution, Phys. Lett. A 372 (2008) 4601–4602. [12] H. Triki, A.M. Wazwaz, Bright and dark soliton solutions for a Kðm; nÞ equation with t-dependent coefficients, Phys. Lett. A 373 (2009) 2162–2165. [13] H. Triki, M.S. Ismail, Soliton solutions of a BBMðm; nÞ equation with generalized evolution, Appl. Math. Comput. 217 (2010) 48–54. [14] A.M. Wazwaz, H. Triki, Soliton solutions for a generalized KdV and BBM equations with time-dependent coefficients, Commun. Nonlinear Sci. Numer. Simul. 16 (2011) 1122–1126. [15] Anjan Biswas, Daniela Milovic, Bright and dark solitons of the generalized nonlinear Schrödinger’s equation, Commun. Nonlinear Sci. Numer. Simulat. 15 (2010) 1473–1484. [16] Manirupa Saha, Amarendra K. Sarma, Anjan Biswas, Dark optical solitons in power law media with time-dependent coefficients, Phys. Lett. A 373 (2009) 4438–4441. [17] Anjan Biswas, 1-Soliton solution of (1+2) dimensional nonlinear Schrödinger’s equation in dual-power law media, Phys. Lett. A 372 (2008) 5941–5943. [18] Anjan Biswas, Solitary wave solution for the generalized Kawahara equation, Appl. Math. Lett. 22 (2009) 208–210. [19] Yadong Shang, Yong Huang, Wenjun Yuan, New exact traveling wave solutions for the Klein–Gordon–Zakharov equations, Comput. Math. Appl. 56 (2008) 1441–1450.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Soliton solutions of the Klein–Gordon–Zakharov equations with power law nonlinearity Houria Triki ⇑, Noureddine Boucerredj Department of Physics, Faculty of Sciences, Badji Mokhtar University, P.O. Box 12, 23000 Annaba, Algeria
a r t i c l e
i n f o
Keywords: Solitons Klein–Gordon–Zakharov equations Solitary wave ansatz method
a b s t r a c t In this paper, the Klein–Gordon–Zakharov equations which model the interaction between the Langmuir wave and the ion acoustic wave in a high frequency plasma, are considered. To examine the role played by the nonlinear dispersion term in the formation of solitons, a family of the considered equations with power law nonlinearity are investigated. By using p p two solitary wave ansatze in terms of sec h ðxÞ and tanh ðxÞ functions, we find exact analytical bright and dark soliton solutions for the considered model. The physical parameters in the soliton solutions are obtained as function of the dependent model coefficients. The conditions of existence of solitons are presented. These closed form solutions are helpful to well understand the mechanism of the complicated physical phenomena and dynamical processes modeled by the used model. Ó 2013 Elsevier Inc. All rights reserved.
1. Introduction Two different types of envelope solitons, bright and dark, can propagate in nonlinear dispersive media [1]. The dynamics of such shape-preserving waves is typically described by a certain class of nonlinear partial differential equations (NLPDEs). The best known examples include the cubic nonlinear Schrödinger equation, the Korteweg–de Vries equation, the Sine–Gordon equation, the Boussinesq equation, etc. The formation of solitons has been regarded as a consequence of the delicate balance between dispersion (or diffraction) and nonlinearity under certain conditions. In real applications, however, it may be difficult to produce such balances [2]. It is worth noting that the existence of soliton solutions depends on the specific nonlinear and dispersive features of the medium [2]. In other words, the key factors, which determine the closed form solutions of a given nonlinear evolution equation are the dispersion and nonlinear coefficients which can be constant or variable parameters depending on the physical situation. For this, one may need to know what kind of the existing effects that may contribute for generating soliton pulses in the medium. As is well known, solitons are universal phenomenon, appearing in a great array of contexts such as, for example, nonlinear optics, plasma physics, fluid dynamics, semiconductors and many other systems. As a matter of fact, the study of such nonlinear waves has attracted extensive attention. One of the great interests is the problem of finding exact soliton solutions of the integrable nonlinear models. Based on these exact solutions directly, we can accurately analyze the properties of propagating soliton pulses in nonlinear physical systems. Towards that goal, many powerful methods to construct exact solutions of NLPDEs have been established and developed, which lead to one of the most excited advances of nonlinear science and theoretical physics [3]. Among these methods we can cite, for example, the subsidiary ordinary differential equation method (sub-ODE for short) [3–5], the coupled amplitude-phase formulation [6], sine–cosine method [7], the Hirota’s bilinear method [8,9], and many others. These methods work even though the Painlevé test of integrability will fail [10]. ⇑ Corresponding author. E-mail address: [email protected] (H. Triki). 0096-3003/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2013.10.093
342
H. Triki, N. Boucerredj / Applied Mathematics and Computation 227 (2014) 341–346
In recent years, much attention in applied mathematics has been focused on calculating soliton solutions by means of the so called solitary wave ansatz method. This method has been successfully applied to many kinds of NLPDEs with constant and varying coefficients, such as, for example, the Kðm; nÞ equation [11,12], the BBM equation [13,14], the Bðm; nÞ equation [10], the nonlinear Schrödinger’s equation [15–17], the Kawahara equation [18] and many others. The purpose of this paper is to calculate the exact bright and dark soliton solutions for a family of the Klein–Gordon– Zakharov equations, exhibiting power law nonlinearity. The importance of the results presented here is twofold. First, exact soliton solutions to a family of the considered model and the conditions of their existence are obtained for a general case of power nonlinearly law in a simple way. The finding of explicit solutions of a given nonlinear evolution equation having any value of the exponent in the nonlinearity term are very interesting since it offer some knowledge on the general dynamical behavior of the propagation so that special cases are truly meaningful both from the physical and mathematical point of view. Second, these results confirms the existence of dark solitons only in the case where the introduced exponent is equal to unity which corresponds exactly to the standard form of the Klein–Gordon–Zakharov equations. To achieve our goal we will use the solitary wave ansatz method which has recently been applied successfully to several NLPDEs. 2. Mathematical analysis The standard form of the Klein–Gordon–Zakharov (KGZ) equations is given by [19]
utt uxx þ u þ anu ¼ 0;
ð1Þ
ntt nxx ¼ bðj uj2 Þxx :
ð2Þ
Here in (1) and (2), the variable uðx; tÞ is a complex function and nðx; tÞ is a real function. a and b are two nonzero real parameters. This system describes the interaction of the langmuir wave and the ion acoustic wave in a high frequency plasma [19]. Based on the extended hyperbolic functions method, Shang et al. [19] have recently obtained the multiple exact explicit solutions of the Klein–Gordon—Zakharov equations (1) and (2). In this paper, we are interesting in the following family of the Klein–Gordon–Zakharov equations with power law nonlinearity: 2
ð3Þ
2
ð4Þ
utt k1 uxx þ au þ bnu ¼ 0; ntt k2 nxx ¼ cðj uj2m Þxx ;
where a; b,c, k1 and k2 are nonzero real constants. The parameter m indicates the power law nonlinearity parameter. Thus for m ¼ 1, (3) and (4) respectively collapses to (1) and (2). 2.1. Bright solitons In order to solve (3) and (4), the starting hypothesis are given by
uðx; tÞ ¼
A1 ei/ p cosh 1 s
ð5Þ
nðx; tÞ ¼
A2 ; p cosh 2 s
ð6Þ
and
where
s ¼ Bðx v tÞ
ð7Þ
/ ¼ jx þ xt þ h:
ð8Þ
and
Here, in (5)–(8), A1 and A2 are the amplitudes of the u-soliton and n-soliton, respectively, while v is the velocity of the soliton and B is the inverse widths of the solitons. Also, j is the frequency of the soliton, while x is the wave number of the soliton and h is the phase constant. The exponents p1 and p2 are unknown at this point and their values will fall out in the process of deriving the solution of this model of equations. Thus from (5) and (6),
( ) A1 ðp21 B2 v 2 x2 Þ p1 ðp1 þ 1ÞA1 B2 v 2 2ixp1 A1 Bv tanh s i/ e ; utt ¼ þ p p p þ2 cosh 1 s cosh 1 s cosh 1 s
ð9Þ
H. Triki, N. Boucerredj / Applied Mathematics and Computation 227 (2014) 341–346
343
uxx ¼
( ) A1 ðp21 B2 j2 Þ p1 ðp1 þ 1ÞA1 B2 2ijp1 A1 B tanh s i/ e ; þ p p p þ2 cosh 1 s cosh 1 s cosh 1 s
ð10Þ
nu ¼
A1 A2 ei/ ; p þp cosh 1 2 s
ð11Þ
ntt ¼
p22 v 2 A2 B2 p2 ðp2 þ 1Þv 2 A2 B2 ; p p þ2 cosh 2 s cosh 2 s
ð12Þ
nxx ¼
p22 A2 B2 p2 ðp2 þ 1ÞA2 B2 ; p p þ2 cosh 2 s cosh 2 s
ð13Þ
ðj uj2m Þxx ¼
2 4p21 m2 A2m 1 B 2p1 m
cosh
s
2 2p1 mð2p1 m þ 1ÞA2m 1 B
cosh
2p1 mþ2
s
ð14Þ
:
Substituting (9)–(14) into (3) and decomposing into real and imaginary parts respectively yields 2
2
2
A1 p21 B2 ðv 2 k1 Þ A1 ðx2 k1 j2 Þ þ aA1 p1 ðp1 þ 1ÞA1 B2 ðv 2 k1 Þ bA1 A2 þ ¼0 p p þp p þ2 cosh 1 s cosh 1 2 s cosh 1 s
ð15Þ
and 2
2xp1 A1 Bv tanh s 2k1 jp1 A1 B tanh s ¼ 0; p p cosh 1 s cosh 1 s
ð16Þ
while (4) reduces to 2
2
2 2 p22 A2 B2 ðv 2 k2 Þ p2 ðp2 þ 1ÞA2 B2 ðv 2 k2 Þ 4cp21 m2 A2m 2cp1 mð2p1 m þ 1ÞA2m 1 B 1 B þ ¼ 0: p2 p þ2 2p m 2p mþ2 cosh s cosh 1 s cosh 1 s cosh 2 s
ð17Þ
Now (16) gives 2
v¼
k1 j
x
ð18Þ
:
From (15), equating the exponents p1 þ 2 and p1 þ p2 gives
p1 þ 2 ¼ p1 þ p2 ;
ð19Þ
so that
p2 ¼ 2:
ð20Þ
Now from (17), equating the exponents 2p1 m þ 2 and p2 þ 2 gives
2p1 m þ 2 ¼ p2 þ 2
ð21Þ
and therefore
p1 ¼
1 : m
ð22Þ
It needs to be noted that the same value of p1 is yielded when the exponents 2p1 m and p2 are equated with each other. We would like to point out that we must have p1 > 0 for the solution (5) to exist. This implies that m > 1 from (22). Now from p þj (15), setting the coefficients of the linearly independent functions 1=cosh 1 s to zero, where j ¼ 0; 2, gives 2
2
A1 p21 B2 ðv 2 k1 Þ A1 ðx2 k1 j2 Þ þ aA1 ¼ 0; 2
p1 ðp1 þ 1ÞA1 B2 ðv 2 k1 Þ þ bA1 A2 ¼ 0:
ð23Þ ð24Þ
Solving the above equations with making use of (18), we obtain
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2 2 2 mx u t x k1 j a B¼ k1 x2 k21 j2
ð25Þ
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H. Triki, N. Boucerredj / Applied Mathematics and Computation 227 (2014) 341–346
and
B¼
mx k1
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi bA2 : 2 ðm þ 1Þðx2 k1 j2 Þ
ð26Þ
Next, equating the two values of B from (25) and (26), gives
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 bA2 þ ðm þ 1Þðk1 j2 þ aÞ x¼ : mþ1
ð27Þ
Thus (26) introduces the constraint condition 2
bðx2 k1 j2 Þ < 0:
ð28Þ p2 þj
From (17), setting the coefficients of the linearly independent functions 1=cosh unique equation:
s to zero, for j ¼ 0; 2, yields the following
2
A2 ðv 2 k2 Þ ¼ cA2m 1 :
ð29Þ
Hence, finally, the 1-soliton solution of the Klein–Gordon–Zakharov equations (3) and (4) are given by
uðx; tÞ ¼
A1 1
coshm ½Bðx v tÞ
eiðjxþxtþhÞ
ð30Þ
:
ð31Þ
and
nðx; tÞ ¼
A2 2
cosh ½Bðx v tÞ
where the amplitudes A1 and A2 are connected by (29) and the width of the solitons are given by (25) or (26). The velocity is given by (18) and the wave number is shown in (27). 2.2. Dark solitons In this section the search is going to be for shock wave solution or topological 1-soliton solution to Klein–Gordon–Zakharov equations given by (3) and (4). To start off, the hypothesis is given by [12,15] p1
sei/
ð32Þ
p2
s;
ð33Þ
uðx; tÞ ¼ A1 tanh and
nðx; tÞ ¼ A2 tanh where
s ¼ Bðx v tÞ
ð34Þ
/ ¼ jx þ xt þ h:
ð35Þ
and Here, in (32)–(35), A1 ; A2 and B are free parameters and v is the velocity of the solitons. Also, j is the frequency of the soliton, while x is the wave number of the soliton and h is the phase constant. The exponents p1 and p2 are unknown at this point and their values will fall out in the process of deriving the solutions of these coupled equations. Therefore from (32) and (33),
utt ¼ ½p1 A1 B2 v 2 fðp1 1Þ tanh
p1 2
s 2p1 tanhp1 s þ ðp1 þ 1Þ tanhp1 þ2 sg 2ixp1 A1 Bv ðtanhp1 1 s tanhp1 þ1 sÞ
p1
x2 A1 tanh ei/ ;
ð36Þ p1 2
uxx ¼ ½p1 A1 B2 fðp1 1Þ tanh
s 2p1 tanhp1 s þ ðp1 þ 1Þ tanhp1 þ2 sg 2ijp1 A1 B ðtanhp1 1 s tanhp1 þ1 sÞ
p
j2 A1 tanh 1 ei/ ; nu ¼ A1 A2 tanh
p1 þp2
ð37Þ
sei/ ;
ð38Þ p2 2
ntt ¼ p2 A2 B2 v 2 fðp2 1Þ tanh nxx ¼ p2 A2 B2 fðp2 1Þ tanh
p2 2
s 2p2 tanhp2 s þ ðp2 þ 1Þ tanhp2 þ2 sg;
s 2p2 tanhp2 s þ ðp2 þ 1Þ tanhp2 þ2 sg;
2 ðj uj2m Þxx ¼ 2p1 mA2m 1 B fð2p1 m 1Þ tanh
2p1 m2
s 4pm tanh2p1 m s þ ð2pm þ 1Þ tanh2p1 mþ2 sg:
ð39Þ ð40Þ ð41Þ
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H. Triki, N. Boucerredj / Applied Mathematics and Computation 227 (2014) 341–346
Substituting (36)–(41) into (3) and decomposing into real and imaginary parts respectively yields 2
p1 2
p1
p1 þp2
p1 A1 B2 ðv 2 k1 Þfðp1 1Þ tanh þ aA1 tanh
s þ bA1 A2 tanh
s 2p1 tanhp1 s þ ðp1 þ 1Þ tanhp1 þ2 sg x2 A1 tanhp1 þ k21 j2 A1 tanhp1 s
¼0
ð42Þ
and p1 1
2xp1 A1 Bv ðtanh
s tanhp1 þ1 sÞ þ 2k21 jp1 A1 Bðtanhp1 1 s tanhp1 þ1 sÞ ¼ 0;
ð43Þ
while (4) reduces to 2
p2 2
s 2p2 tanhp2 s þ ðp2 þ 1Þ tanhp2 þ2 sg 2p1 m2 2 s 4p1 m tanh2p1 m s þ ð2p1 m þ 1Þ tanh2p1 mþ2 sg ¼ 0: 2cp1 mA2m 1 B fð2p1 m 1Þ tanh
p2 A2 B2 ðv 2 k2 Þfðp2 1Þ tanh
ð44Þ
From (42), equating the exponents p1 þ 2 and p1 þ p2 gives
p1 þ 2 ¼ p1 þ p2 ;
ð45Þ
so that
p2 ¼ 2:
ð46Þ
Now from (44), equating the exponents 2p1 m þ 2 and p2 þ 2 gives
2p1 m þ 2 ¼ p2 þ 2
ð47Þ
and therefore
p1 ¼
1 : m
ð48Þ
It needs to be noted that the same value of p is yielded when the exponents pairs 2p1 m and p2 ; 2p1 m2 and p2 2 are equated with each other. From (43), we get 2
k1 j
v¼
x
ð49Þ
: 2p1 mþj
Now from (44) the linearly independent functions are tanh to zero yields
s for j ¼ 2; 0; 2. Hence setting their respective coefficients
2
A2 ðv 2 k2 Þ ¼ cA2m 1 :
ð50Þ p1 þj
Now from (42), setting the coefficients of the linearly independent functions tanh 2
2p21 A1 B ðv 2 2
2
2
2
p1 A1 B ðv p1 A1 B ðv
2 k1 Þ
2 k1 Þðp1 2 k1 Þðp1
x 2 A1 þ
2 k1
j2 A1 þ aA1 ¼ 0;
s to zero, where j ¼ 2; 0; 2, gives ð51Þ
þ 1Þ þ bA1 A2 ¼ 0;
ð52Þ
1Þ ¼ 0:
ð53Þ
To solve (53), we have considered the case p1 1 ¼ 0: This yields
p1 ¼ 1:
ð54Þ
Substituting (54) into the above system and using (49) gives
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u
x ux2 k21 j2 a B¼ t k1 x2 k21 j2
ð55Þ
and
B¼
x
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi bA2
k1
2ðx2 k1 j2 Þ
2
:
ð56Þ
Next, equating the two values of B from (55) and (56), gives
x¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 bA2 þ k1 j2 þ a;
ð57Þ
which shows that the solitons will exist for 2
bA2 þ k1 j2 þ a > 0:
ð58Þ
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H. Triki, N. Boucerredj / Applied Mathematics and Computation 227 (2014) 341–346
Also, (56) implies that it is necessary to have 2
bðx2 k1 j2 Þ > 0
ð59Þ
for solitons to exist. Now equating the two values of p1 from (48) and (54) gives the condition:
m ¼ 1:
ð60Þ
Thus, for the Klein–Gordon–Zakharov equations (3) and (4), dark solitons will exist only when m ¼ 1, which corresponds to the standard form of the Klein–Gordon–Zakharov equations (1) and (2). This important observation is being made for the first time in this paper. Thus, finally, the topological 1-soliton solutions to the considered equations (3) and (4) are given by
uðx; tÞ ¼ A1 tanh½Bðx v tÞeiðjxþxtþhÞ
ð61Þ
and 2
nðx; tÞ ¼ A2 tanh ½Bðx v tÞ;
ð62Þ
where the relation between the free parameters A1 and A2 is given by (50), and the velocity of the soliton is given by (49). Finally the free parameter B is given by (55) or (56), while the wave number x is of the soliton is shown in (57). 3. Conclusion In this paper, the solitary wave ansatz method is adopted to integrate a family of the Klein–Gordon–Zakharov equations with power law nonlinearity. Both of the topological (dark) as well as non-topological (bright) soliton solutions of the considered model with power law nonlinearity, are obtained. The physical parameters in the obtained soliton solutions are calculated in course of the derivation of the special exact solutions as function of the dependent model coefficients. Conditions for the existence of solitary wave solutions have also been reported. These results indicate that the bright solitons exist for any exponent m > 1, while the dark solitons exist only when m ¼ 1. These solutions may be useful to explain some physical phenomena in dynamical systems that are described by the Klein–Gordon–Zakharov equations-type models. In future, the dual power nonlinearity term will be taken into consideration to study the influence of high order nonlinearities on the dynamical behavior of the solitons propagation. These results will be reported in future. References [1] Mark M. Scott, Mikhail P. Kostylev, Boris A. Kalinikos, Carl E. Patton, Excitation of bright and dark envelope solitons for magnetostatic waves with attractive nonlinearity, Phys. Rev. B. 71 (2005) 174440:1–174440:4. [2] Ruiyu Hao, Lu Li, Zhonghao Li, Guosheng Zhou, Exact multisoliton solutions of the higher-order nonlinear Schrödinger equation with variable coefficients, Phys. Rev. E. 70 (2004) 1–6 (066603). [3] X. Li, M. Wang, A sub-ODE method for finding exact solutions of a generalized KdV–mKdV equation with high-order nonlinear terms, Phys. Lett A. 361 (2007) 115–118. [4] H. Triki, A.M. Wazwaz, Sub-ODE method and soliton solutions for the variable-coefficient mKdV equation, Appl. Math. Comput. 214 (2009) 370–373. [5] H. Triki, T.R. Taha, The sub-ODE method and soliton solutions for a higher order dispersive cubic–quintic nonlinear Schrödinger equation, Chaos Solitons Fract. 42 (2009) 1068–1072. [6] S.L. Palacios, A. Guinea, J.M. Fernandez-Diaz, R.D. Crespo, Dark solitary waves in the nonlinear Schrödinger equation with third order dispersion, selfsteepening, and self-frequency shift, Phys. Rev. E 60 (1) (1999) 45–47. [7] A.M. Wazwaz, New solitary wave solutions to the modified Kawahara equation, Phys. Lett. A 360 (2007) 588–592. [8] A.M. Wazwaz, H. Triki, Multiple soliton solutions for the sixth-order Ramani equation and a coupled Ramani equation, Appl. Math. Comput. 216 (2010) 332–336. [9] A.M. Wazwaz, Integrable (2+1)-dimensional and (3+1)-dimensional breaking soliton equations, Phys. Scr. 81 (2010) 5 (035005). [10] A. Biswas, 1-Soliton solution of the Bðm; nÞ equation with generalized evolution, Commun. Nonlinear Sci. Numer. Simul. 14 (2009) 3226–3229. [11] Anjan Biswas, 1-Soliton solution of the Kðm; nÞ equation with generalized evolution, Phys. Lett. A 372 (2008) 4601–4602. [12] H. Triki, A.M. Wazwaz, Bright and dark soliton solutions for a Kðm; nÞ equation with t-dependent coefficients, Phys. Lett. A 373 (2009) 2162–2165. [13] H. Triki, M.S. Ismail, Soliton solutions of a BBMðm; nÞ equation with generalized evolution, Appl. Math. Comput. 217 (2010) 48–54. [14] A.M. Wazwaz, H. Triki, Soliton solutions for a generalized KdV and BBM equations with time-dependent coefficients, Commun. Nonlinear Sci. Numer. Simul. 16 (2011) 1122–1126. [15] Anjan Biswas, Daniela Milovic, Bright and dark solitons of the generalized nonlinear Schrödinger’s equation, Commun. Nonlinear Sci. Numer. Simulat. 15 (2010) 1473–1484. [16] Manirupa Saha, Amarendra K. Sarma, Anjan Biswas, Dark optical solitons in power law media with time-dependent coefficients, Phys. Lett. A 373 (2009) 4438–4441. [17] Anjan Biswas, 1-Soliton solution of (1+2) dimensional nonlinear Schrödinger’s equation in dual-power law media, Phys. Lett. A 372 (2008) 5941–5943. [18] Anjan Biswas, Solitary wave solution for the generalized Kawahara equation, Appl. Math. Lett. 22 (2009) 208–210. [19] Yadong Shang, Yong Huang, Wenjun Yuan, New exact traveling wave solutions for the Klein–Gordon–Zakharov equations, Comput. Math. Appl. 56 (2008) 1441–1450.