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Soliton perturbation theory for the quadratic nonlinear Klein–Gordon equation
Applied Mathematics and Computation 203 (2008) 153–156
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Soliton perturbation theory for the quadratic nonlinear Klein–Gordon equation Anjan Biswas a,*, Chenwi Zony b, Essaid Zerrad b a Center for Research and Education in Optical Sciences and Applications, Department of Applied Mathematics and Theoretical Physics, Delaware State University, Dover, DE 19901-2277, USA b Department of Physics and Pre-Engineering, Delaware State University, Dover, DE 19901-2277, USA
a r t i c l e
i n f o
a b s t r a c t This paper studies the adiabatic dynamics of soliton velocity due to the quadratic Klein– Gordon equations, in presence of perturbation terms. The soliton perturbation theory is exploited for 1-soliton solution of this equation. Ó 2008 Elsevier Inc. All rights reserved.
Keywords: Klein–Gordon equation Soliton perturbation Adiabatic variation
1. Introduction The Klein–Gordon equation (KGE) is a relativistic field equation for scalar particles (spin-0). It is the relativistic wave equation version of the Schrödinger’s equation. Although there are other relativistic wave equations, KGE has been the most frequently studied equation for describing the particle dynamics in quantum field theory [1–10]. Recently there has been a great deal of theoretical work devoted to the study of the exact solution of KGE for various potentials [2,3,7]. Such solutions are of interest in many fields such as Nuclear Physics, High-Energy Physics, Gravitation and in Statistical and Condensed Matter Physics. It also has applications in Nonlinear Optics where KGE plays an important role for Bose–Einstein condensates trapped in strong optical lattices formed by the interference patterns of laser beams [1,4,5,10]. The KGE with vector and scalar potentials can only be solved exactly for certain potentials such as Coulomb, Morse and harmonic oscillator etc. In this context, it provides a good description of the scalar particles. In particular, for the case of pionic bound states in the Coulomb field of the nucleii, the KGE gives a good first order approximation of the problem [3]. In this paper, the study is going to be focussed on the quadratic nonlinear Klein–Gordon equation along with its perturbation terms. 2. Mathematical analysis The quadratic nonlinear Klein–Gordon equation (qnKGE), in the dimensionless form is given by 2
2
qtt k qxx þ aq bq ¼ 0:
ð1Þ
While Eq. (1) is not integrable by the classical method of Inverse Scattering Transform (IST), 1-soliton solution can be obtained. Although IST is not applicable to many of the nonlinear evolution equations, recently developed various direct
* Corresponding author. E-mail address: [email protected] (A. Biswas). 0096-3003/$ - see front matter Ó 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2008.04.013
154
A. Biswas et al. / Applied Mathematics and Computation 203 (2008) 153–156
methods have been proposed that includes the tanh-function method, Jacobi’s elleptic function expansion method, homogenous balance method, sine–cosine method, auxiliary equation method just to name a few. Using the auxiliary equation method several travelling wave solutions, rather more than 15, to (1) was recently obtained in 2006 [8]. In this paper, the interest is going to be concentrated on the 1-soliton solution of (1) that is given by [8] A ; ð2Þ qðx; tÞ ¼ 2 cosh ½Bðx vtÞ where 3a ; 2brffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 a B¼ ; 2 k2 v2
A¼
ð3Þ ð4Þ
where A is the amplitude of the soliton and B is the inverse width of the soliton, v is the soliton velocity. Thus, from (4), one can see that the soliton for the qnKGE exists for 2
aðk v2 Þ > 0:
ð5Þ
Also from (3) and (4), the relation between the amplitude and the width of the soliton is given by B2 ¼
bA 2
6ðk v2 Þ
ð6Þ
:
The qnKGE has at least two integrals of motion which are the momentum ðPÞ and energy ðEÞ that are, respectively, given by Z 1 16 2 vA B P¼ qx qt dx ¼ ð7Þ 15 1 and E¼
Z
! 2 1 2 k 2 a 2 b 3 2A2 2 qt þ qx þ q q dx ¼ f12B2 ðv2 þ k Þ þ 15a 8bAg: 2 2 3 2 45B
1
1
ð8Þ
In order to evaluate the conserved quantities in (7) and (8), the 1-soliton solution given by (2) is utilized. 3. Perturbation terms The perturbed qnKGE that is going to be studied in this paper is given by 2
2
qtt k qxx þ aq bq ¼ R;
ð9Þ
where R represents the perturbation terms and is the perturbation parameter. In presence of these perturbation terms, the energy and the momentum do not stay constant. Instead, they undergo an adiabatic variation. The laws of variation of these conserved quantities are given by Z 1 dP ¼ qx R dx ð10Þ dt 1 and dE ¼ dt
Z
1
qt R dx:
ð11Þ
1
The law of adiabatic variation of the velocity of the soliton, which is the main focus of the paper, can be retrieved from either of the two laws of adiabatic variation given in (10) or (11). So, without any loss of generality, the momentum equation given by (10) will be used to obtain the law of variation of soliton velocity, in presence of such perturbation terms. Thus, from (7), one obtains 2
dP 16k A2 B dv ¼ dt 15 k2 v2 dt
ð12Þ
that leads to 2
dv 15ðk v2 Þ ¼ 2 dt 16k A2 B
Z
1
qx R dx:
ð13Þ
1
In this paper, the perturbation terms that are going to be taken into consideration are given by [6] R ¼ aq þ bqt þ cqx þ dqxt þ kqtt þ rqxxt þ mqxxxx :
ð14Þ
A. Biswas et al. / Applied Mathematics and Computation 203 (2008) 153–156
155
The perturbation term due to a arises in the charge density wave system where this ‘exotic’ perturbation term is observed. Also, the term due to b accounts for dissipative losses in Josephson junction theory due to tunneling of normal electrons across the dielectric barrier, while r accounts for losses due to a current along the barrier. The perturbation term due to c is generated by a small inhomogenous part of the local inductance while k the capacity inhomogeneity. The higher order spatial dispersion term is given by m. Finally, g1 arises in the system of easy plane ferromagnets when an external ac magnetic field is orthogonal plane and g2 is another important magnetic perturbation that describes weak higher anisotropy in an easy plane ferromagnet. The perturbation term due to g1 also arises in the context of microinhomogeneity in Josephson junction theory. Thus, the perturbed qnKGE that is going to be studied in this paper is given by 2
2
qtt k qxx þ aq bq ¼ ðaq þ bqt þ cqx þ dqxt þ kqtt þ rqxxt þ mqxxxx Þ:
ð15Þ
For the perturbation terms given by (14), Eq. (13) reduces to dv 2 2 ¼ fð7b 20rB2 Þv3 7cv2 k ð7b 20rB2 Þv þ 7ck g: dt 7k2
ð16Þ
Separating variables, leads to t 2
7k
¼
Z
dv 2
2
ð7b 20rB2 Þv3 7cv2 k ð7b 20rB2 Þv þ 7ck
:
ð17Þ
In order to carry out the integration in (17), one needs to consider following four cases: Case-I: When the polynomial in v has all three real roots say v ¼ v1 ; v2 and v3 one can write ð7b 20rB2 Þt 2
7k
¼
Z
dv ; ðv v1 Þðv v2 Þðv v3 Þ
ð18Þ
where v1 þ v2 þ v3 ¼
7c 7b 20rB2
;
2
v1 v2 þ v2 v3 þ v3 v1 ¼ k ; 7c : v1 v2 v3 ¼ 7b 20rB2
ð19Þ ð20Þ ð21Þ
In this case u v1 þ ðv2 v1 Þðv2 v3 Þ ln u v2 þ ðv3 v1 Þðv3 v2 Þ ln u v3 ; ðv1 v2 Þðv1 v3 Þ ln v v1 v v2 v v3 ð7b 20rB Þ 2
t¼
7k
2
where vðt ¼ 0Þ ¼ u. Case-II: When the polynomial in v has one real of multiplicity two, one can write Z ð7b 20rB2 Þt dv ; ¼ 2 ðv v1 Þ2 ðv v2 Þ 7k where v1 is the real root of multiplicity two and v3 is the second real root. In this case, the solution is given by " # 2 ðu v1 Þðv v2 Þ ðv uÞðv þ u 2v1 Þ 7k 1 þ ln t¼ : ðu v Þðv v Þ 2 1 ðv1 v2 Þð7b 20rB2 Þ v1 v2 ðu v1 Þ2 ðv v1 Þ2 Case-III: When the polynomial in v has one real of multiplicity three, one can write Z ð7b 20rB2 Þt dv ¼ ; 2 ðv v1 Þ3 7k where v1 is the real root of multiplicity three. In this case, the solution is given by vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2 u 7k ðu v1 Þ2 v ¼ v1 þ t 2 7k þ 2tðu v1 Þ2 ð20rB2 7bÞ
ð22Þ
ð23Þ
ð24Þ
ð25Þ
ð26Þ
so that lim vðtÞ ¼ v1 :
t!1
ð27Þ
156
A. Biswas et al. / Applied Mathematics and Computation 203 (2008) 153–156
Case-IV: When the polynomial in v has one real and two imaginary roots, one can write Z ð7b 20rB2 Þt dv ; ¼ 2 ðv v1 Þðv2 þ c2 Þ 7k where v1 is the real root and c2 is the product of the imaginary roots. In this case, one can write the solution as 2
2 2 u v1 1 7k ðv21 þ c2 Þ ln u þ c v1 tan1 cðu vÞ : ln t¼ v v 2 uv þ c2 v2 þ c2 c 1 ð7b 20rB2 Þ
ð28Þ
ð29Þ
It is important to note here that the solution is implicit in v, namely t is expressed as a function of v and not the other way around, except for the third case. In this case, one can, in addition, obtain the limiting value of the soliton velocity, for large time. 4. Conclusions In this paper, the qnKGE is studied along with the perturbation terms. These perturbation terms that are studied are all local and consequently the adiabatic variation of the velocity of the soliton is obtained in presence of such perturbation terms. There are four cases that are studied depending on the polynomial function of the velocity of the soliton. These cases exhaustively cover all the possible scenarios. In future, the other types of nonlinear Klein–Gordon equations are going to be studied. The results of those research will be reported in future. Acknowledgements This research of the first (AB) author were fully supported by NSF-CREST Grant No: HRD-0630388 and the support is genuinely and sincerely appreciated. This author, who is the corresponding author, is also extremely thankful to the referee for pointing out the algebraic errors in the calculations of this paper. The research work of the third author (EZ) is financially supported by the Army Research Office (ARO) under the award number W911NF-05-1-0451 and this support is thankfully appreciated. References [1] G.L. Alfimov, P.G. Kevrekidis, V.V. Konotop, M. Salerno, Wannier functions analysis of the nonlinear Schrödinger’s equation with a periodic potential, Physical Review E 66 (4) (2002) 046608. [2] G. Backenstoss, Pionic Atoms, Annual Review of Nuclear Science 20 (1970) 467–508. [3] M. Bawin, M. Jaminon, Pion form factor and the Klein–Gordon Equattion, Physical Review C 30 (1) (1984) 331–334. [4] F.S. Cataliotti, S. Burger, C. Fort, P. Maddaloni, F. Minardi, A. Trombettoni, A. Smerzi, M. Inguscio, Josephson junction arrays with Bose–Einstein condensates, Science 293 (5531) (2001) 843–846. [5] F.S. Cataliotti, L. Fallani, F. Ferlaino, C. Fort, P. Maddaloni, M. Inguscio, Superfluid current disruption in a chain of weakly coupled Bose–Einstein condensates, New Journal of Physics 5 (2001) 71.1–71.7. [6] Y.S. Kivshar, B.A. Malomed, Dynamics of solitons in nearly integrable systems, Reviews of Modern Physics 61 (4) (1989) 763–915. [7] Y.R. Kwon, F. Tabakin, Hadronic atoms in momentum space, Physical Review C 18 (2) (1978) 932–943. [8] Sirendaoreji, A new auxillary equation and exact travelling wave solutions of nonlinear equations, Physics Letters A 356 (2006) 124–130. [9] Sirendaoreji, Exact travelling wave solutions for four forms of nonlinear Klein–Gordon equations, Physics Letters A 363 (2007) 440–447. [10] F. Yasuk, A. Durmus, I. Boztosun, Exact analytical solution of the relativistic Klein–Gordon equation with noncentral equal scalar and vector potential, Journal of Mathematical Physics 47 (2006) 082302.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Soliton perturbation theory for the quadratic nonlinear Klein–Gordon equation Anjan Biswas a,*, Chenwi Zony b, Essaid Zerrad b a Center for Research and Education in Optical Sciences and Applications, Department of Applied Mathematics and Theoretical Physics, Delaware State University, Dover, DE 19901-2277, USA b Department of Physics and Pre-Engineering, Delaware State University, Dover, DE 19901-2277, USA
a r t i c l e
i n f o
a b s t r a c t This paper studies the adiabatic dynamics of soliton velocity due to the quadratic Klein– Gordon equations, in presence of perturbation terms. The soliton perturbation theory is exploited for 1-soliton solution of this equation. Ó 2008 Elsevier Inc. All rights reserved.
Keywords: Klein–Gordon equation Soliton perturbation Adiabatic variation
1. Introduction The Klein–Gordon equation (KGE) is a relativistic field equation for scalar particles (spin-0). It is the relativistic wave equation version of the Schrödinger’s equation. Although there are other relativistic wave equations, KGE has been the most frequently studied equation for describing the particle dynamics in quantum field theory [1–10]. Recently there has been a great deal of theoretical work devoted to the study of the exact solution of KGE for various potentials [2,3,7]. Such solutions are of interest in many fields such as Nuclear Physics, High-Energy Physics, Gravitation and in Statistical and Condensed Matter Physics. It also has applications in Nonlinear Optics where KGE plays an important role for Bose–Einstein condensates trapped in strong optical lattices formed by the interference patterns of laser beams [1,4,5,10]. The KGE with vector and scalar potentials can only be solved exactly for certain potentials such as Coulomb, Morse and harmonic oscillator etc. In this context, it provides a good description of the scalar particles. In particular, for the case of pionic bound states in the Coulomb field of the nucleii, the KGE gives a good first order approximation of the problem [3]. In this paper, the study is going to be focussed on the quadratic nonlinear Klein–Gordon equation along with its perturbation terms. 2. Mathematical analysis The quadratic nonlinear Klein–Gordon equation (qnKGE), in the dimensionless form is given by 2
2
qtt k qxx þ aq bq ¼ 0:
ð1Þ
While Eq. (1) is not integrable by the classical method of Inverse Scattering Transform (IST), 1-soliton solution can be obtained. Although IST is not applicable to many of the nonlinear evolution equations, recently developed various direct
* Corresponding author. E-mail address: [email protected] (A. Biswas). 0096-3003/$ - see front matter Ó 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2008.04.013
154
A. Biswas et al. / Applied Mathematics and Computation 203 (2008) 153–156
methods have been proposed that includes the tanh-function method, Jacobi’s elleptic function expansion method, homogenous balance method, sine–cosine method, auxiliary equation method just to name a few. Using the auxiliary equation method several travelling wave solutions, rather more than 15, to (1) was recently obtained in 2006 [8]. In this paper, the interest is going to be concentrated on the 1-soliton solution of (1) that is given by [8] A ; ð2Þ qðx; tÞ ¼ 2 cosh ½Bðx vtÞ where 3a ; 2brffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 a B¼ ; 2 k2 v2
A¼
ð3Þ ð4Þ
where A is the amplitude of the soliton and B is the inverse width of the soliton, v is the soliton velocity. Thus, from (4), one can see that the soliton for the qnKGE exists for 2
aðk v2 Þ > 0:
ð5Þ
Also from (3) and (4), the relation between the amplitude and the width of the soliton is given by B2 ¼
bA 2
6ðk v2 Þ
ð6Þ
:
The qnKGE has at least two integrals of motion which are the momentum ðPÞ and energy ðEÞ that are, respectively, given by Z 1 16 2 vA B P¼ qx qt dx ¼ ð7Þ 15 1 and E¼
Z
! 2 1 2 k 2 a 2 b 3 2A2 2 qt þ qx þ q q dx ¼ f12B2 ðv2 þ k Þ þ 15a 8bAg: 2 2 3 2 45B
1
1
ð8Þ
In order to evaluate the conserved quantities in (7) and (8), the 1-soliton solution given by (2) is utilized. 3. Perturbation terms The perturbed qnKGE that is going to be studied in this paper is given by 2
2
qtt k qxx þ aq bq ¼ R;
ð9Þ
where R represents the perturbation terms and is the perturbation parameter. In presence of these perturbation terms, the energy and the momentum do not stay constant. Instead, they undergo an adiabatic variation. The laws of variation of these conserved quantities are given by Z 1 dP ¼ qx R dx ð10Þ dt 1 and dE ¼ dt
Z
1
qt R dx:
ð11Þ
1
The law of adiabatic variation of the velocity of the soliton, which is the main focus of the paper, can be retrieved from either of the two laws of adiabatic variation given in (10) or (11). So, without any loss of generality, the momentum equation given by (10) will be used to obtain the law of variation of soliton velocity, in presence of such perturbation terms. Thus, from (7), one obtains 2
dP 16k A2 B dv ¼ dt 15 k2 v2 dt
ð12Þ
that leads to 2
dv 15ðk v2 Þ ¼ 2 dt 16k A2 B
Z
1
qx R dx:
ð13Þ
1
In this paper, the perturbation terms that are going to be taken into consideration are given by [6] R ¼ aq þ bqt þ cqx þ dqxt þ kqtt þ rqxxt þ mqxxxx :
ð14Þ
A. Biswas et al. / Applied Mathematics and Computation 203 (2008) 153–156
155
The perturbation term due to a arises in the charge density wave system where this ‘exotic’ perturbation term is observed. Also, the term due to b accounts for dissipative losses in Josephson junction theory due to tunneling of normal electrons across the dielectric barrier, while r accounts for losses due to a current along the barrier. The perturbation term due to c is generated by a small inhomogenous part of the local inductance while k the capacity inhomogeneity. The higher order spatial dispersion term is given by m. Finally, g1 arises in the system of easy plane ferromagnets when an external ac magnetic field is orthogonal plane and g2 is another important magnetic perturbation that describes weak higher anisotropy in an easy plane ferromagnet. The perturbation term due to g1 also arises in the context of microinhomogeneity in Josephson junction theory. Thus, the perturbed qnKGE that is going to be studied in this paper is given by 2
2
qtt k qxx þ aq bq ¼ ðaq þ bqt þ cqx þ dqxt þ kqtt þ rqxxt þ mqxxxx Þ:
ð15Þ
For the perturbation terms given by (14), Eq. (13) reduces to dv 2 2 ¼ fð7b 20rB2 Þv3 7cv2 k ð7b 20rB2 Þv þ 7ck g: dt 7k2
ð16Þ
Separating variables, leads to t 2
7k
¼
Z
dv 2
2
ð7b 20rB2 Þv3 7cv2 k ð7b 20rB2 Þv þ 7ck
:
ð17Þ
In order to carry out the integration in (17), one needs to consider following four cases: Case-I: When the polynomial in v has all three real roots say v ¼ v1 ; v2 and v3 one can write ð7b 20rB2 Þt 2
7k
¼
Z
dv ; ðv v1 Þðv v2 Þðv v3 Þ
ð18Þ
where v1 þ v2 þ v3 ¼
7c 7b 20rB2
;
2
v1 v2 þ v2 v3 þ v3 v1 ¼ k ; 7c : v1 v2 v3 ¼ 7b 20rB2
ð19Þ ð20Þ ð21Þ
In this case u v1 þ ðv2 v1 Þðv2 v3 Þ ln u v2 þ ðv3 v1 Þðv3 v2 Þ ln u v3 ; ðv1 v2 Þðv1 v3 Þ ln v v1 v v2 v v3 ð7b 20rB Þ 2
t¼
7k
2
where vðt ¼ 0Þ ¼ u. Case-II: When the polynomial in v has one real of multiplicity two, one can write Z ð7b 20rB2 Þt dv ; ¼ 2 ðv v1 Þ2 ðv v2 Þ 7k where v1 is the real root of multiplicity two and v3 is the second real root. In this case, the solution is given by " # 2 ðu v1 Þðv v2 Þ ðv uÞðv þ u 2v1 Þ 7k 1 þ ln t¼ : ðu v Þðv v Þ 2 1 ðv1 v2 Þð7b 20rB2 Þ v1 v2 ðu v1 Þ2 ðv v1 Þ2 Case-III: When the polynomial in v has one real of multiplicity three, one can write Z ð7b 20rB2 Þt dv ¼ ; 2 ðv v1 Þ3 7k where v1 is the real root of multiplicity three. In this case, the solution is given by vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2 u 7k ðu v1 Þ2 v ¼ v1 þ t 2 7k þ 2tðu v1 Þ2 ð20rB2 7bÞ
ð22Þ
ð23Þ
ð24Þ
ð25Þ
ð26Þ
so that lim vðtÞ ¼ v1 :
t!1
ð27Þ
156
A. Biswas et al. / Applied Mathematics and Computation 203 (2008) 153–156
Case-IV: When the polynomial in v has one real and two imaginary roots, one can write Z ð7b 20rB2 Þt dv ; ¼ 2 ðv v1 Þðv2 þ c2 Þ 7k where v1 is the real root and c2 is the product of the imaginary roots. In this case, one can write the solution as 2
2 2 u v1 1 7k ðv21 þ c2 Þ ln u þ c v1 tan1 cðu vÞ : ln t¼ v v 2 uv þ c2 v2 þ c2 c 1 ð7b 20rB2 Þ
ð28Þ
ð29Þ
It is important to note here that the solution is implicit in v, namely t is expressed as a function of v and not the other way around, except for the third case. In this case, one can, in addition, obtain the limiting value of the soliton velocity, for large time. 4. Conclusions In this paper, the qnKGE is studied along with the perturbation terms. These perturbation terms that are studied are all local and consequently the adiabatic variation of the velocity of the soliton is obtained in presence of such perturbation terms. There are four cases that are studied depending on the polynomial function of the velocity of the soliton. These cases exhaustively cover all the possible scenarios. In future, the other types of nonlinear Klein–Gordon equations are going to be studied. The results of those research will be reported in future. Acknowledgements This research of the first (AB) author were fully supported by NSF-CREST Grant No: HRD-0630388 and the support is genuinely and sincerely appreciated. This author, who is the corresponding author, is also extremely thankful to the referee for pointing out the algebraic errors in the calculations of this paper. The research work of the third author (EZ) is financially supported by the Army Research Office (ARO) under the award number W911NF-05-1-0451 and this support is thankfully appreciated. References [1] G.L. Alfimov, P.G. Kevrekidis, V.V. Konotop, M. Salerno, Wannier functions analysis of the nonlinear Schrödinger’s equation with a periodic potential, Physical Review E 66 (4) (2002) 046608. [2] G. Backenstoss, Pionic Atoms, Annual Review of Nuclear Science 20 (1970) 467–508. [3] M. Bawin, M. Jaminon, Pion form factor and the Klein–Gordon Equattion, Physical Review C 30 (1) (1984) 331–334. [4] F.S. Cataliotti, S. Burger, C. Fort, P. Maddaloni, F. Minardi, A. Trombettoni, A. Smerzi, M. Inguscio, Josephson junction arrays with Bose–Einstein condensates, Science 293 (5531) (2001) 843–846. [5] F.S. Cataliotti, L. Fallani, F. Ferlaino, C. Fort, P. Maddaloni, M. Inguscio, Superfluid current disruption in a chain of weakly coupled Bose–Einstein condensates, New Journal of Physics 5 (2001) 71.1–71.7. [6] Y.S. Kivshar, B.A. Malomed, Dynamics of solitons in nearly integrable systems, Reviews of Modern Physics 61 (4) (1989) 763–915. [7] Y.R. Kwon, F. Tabakin, Hadronic atoms in momentum space, Physical Review C 18 (2) (1978) 932–943. [8] Sirendaoreji, A new auxillary equation and exact travelling wave solutions of nonlinear equations, Physics Letters A 356 (2006) 124–130. [9] Sirendaoreji, Exact travelling wave solutions for four forms of nonlinear Klein–Gordon equations, Physics Letters A 363 (2007) 440–447. [10] F. Yasuk, A. Durmus, I. Boztosun, Exact analytical solution of the relativistic Klein–Gordon equation with noncentral equal scalar and vector potential, Journal of Mathematical Physics 47 (2006) 082302.