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Analysis of non-linear Klein–Gordon equations using Lie symmetry
Applied Mathematics Letters 23 (2010) 1397–1400
Contents lists available at ScienceDirect
Applied Mathematics Letters journal homepage: www.elsevier.com/locate/aml
Analysis of non-linear Klein–Gordon equations using Lie symmetry Chaudry Masood Khalique, Anjan Biswas ∗ International Institute for Symmetry Analysis and Mathematical Modeling, Department of Mathematical Sciences, North-West University, Mafikeng Campus, Private Bag X2046, Mmabatho 2735, South Africa
article
info
Article history: Received 28 December 2009 Accepted 21 July 2010 Keywords: Integrability Lie symmetry
abstract This work obtains the stationary solutions of the non-linear Klein–Gordon equations in 1 + 1 dimensions. The technique that is used to carry out the analysis is the Lie symmetry approach. There are five types of non-linearity that are studied in this work. In each case, the analysis yields non-trivial stationary solutions; it is the first time that this has been seen. © 2010 Elsevier Ltd. All rights reserved.
1. Introduction The Klein–Gordon equation (KGE), which is also known as Schrödinger’s relativistic wave equation, arises in the study of quantum mechanics [1–10]. The KGE is used as the equation of motion of a massive spinless particle in classical and quantum field theory. This work will study the KGE in 1 + 1 dimensions—one time dimension and two spatial dimensions—namely in x- and y-coordinates. There will be five forms of non-linearity that will be considered. In each of these forms, an exact one-soliton solution will be obtained. The solution will be a topological or a non-topological soliton depending of the type of non-linearity in question. The KGE falls into the category of non-linear evolution equations (NLEEs). There are various techniques for carrying out the integration of these NLEEs. Some of the methods are the inverse scattering transform (IST) method, the F -expansion method, the Adomian decomposition method, the G0 /G method, the tanh–coth method, the sine–cosine method, the exponential function method, the Lie symmetry method and many more. In particular, the KGE has also been studied numerically by He’s variational iteration method. However, one needs to be careful before applying analytical techniques of integration. This could lead to incorrect results as pointed out before. In this work, the Lie symmetry approach will be used to carry out the integration of the KGE in 1 + 1 dimensions. 2. Mathematical analysis The KGE in 1 + 1 dimensions that will be studied in this work is given by [1–10] qtt − k2 qxx = F (q)
(1)
where the dependent variable q(x, t ) represents the quantized field describing the particle. Here, k is a real number. In (1), the non-linear function F (q) is continuous. The function F (q) can be written in terms of the potential function U (q) as [2] F (q) =
∂U . ∂q
(2)
∗ Corresponding address: Department of Mathematical Sciences, Applied Mathematics Research Center, Center for Research and Education in Optical Sciences and Applications, Delaware State University, Dover, DE 19901-2277, USA. Tel.: +1 302 857 7913; fax: +1 302 857 7054. E-mail address: [email protected] (A. Biswas). 0893-9659/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.aml.2010.07.006
1398
C.M. Khalique, A. Biswas / Applied Mathematics Letters 23 (2010) 1397–1400
This potential function U (q) has at least two minima, q1 and q3 , and a maximum at q2 such that U (q1 ) = U (q3 ). The solutions to (1) for various forms of the non-linear function F are known as solitons. For some form of the function F , (1) leads to nontopological solitons, while in some other forms, (1) gives kink solutions that are known as topological solitons or topological defects, which take the system from one asymptotically stable state to another. The starting hypothesis is q(x, t ) = φ(x − w t )
(3)
which is also known as the traveling wave hypothesis. In (3), w is the velocity of the wave profile. In this work, the focus is going to be on recovering stationary solutions to the KGE. Therefore, the velocity w is set to zero. Thus (1) reduces to the form
φ 00 = G(φ)
(4)
where G(φ) = F (φ)/k . Eq. (4) has a single Lie point symmetry, namely X = ∂/∂ x. This symmetry will be used to integrate Eq. (4) once. It can be easily seen that the two invariants are 2
u=φ
(5)
v = φ0.
(6)
and Treating u as the independent variable and v as the dependent variable, (4) can be rewritten as dv du
=
φ 00 G(u) = . φ0 v
(7)
Separating variables and carrying out the integration leads to
v2 = 2
Z
G(u)du + c .
(8)
This equation will now be further analysed for the various forms of the functional G(u) that are discussed in the following sections. In this work, the following five forms of the function F (q) will be considered, and will lead to the various solutions. They are [4–7] F (q) = aq − bq2
(9)
F (q) = aq − bq
(10)
3
F (q) = aq − bqn
(11)
F (q) = aq − bq + cq
(12)
F (q) = aq − bq1−n + cqn+1 .
(13)
2n−1
n
These five cases will be respectively labeled as Forms I–V. In all of these five forms, a, b and c are real valued constants. The domain restrictions in all of these five cases will be discussed when they are studied in detail in the following sections. 3. Form I This case, by virtue of Eq. (4), reduces to
φ 00 = aφ + bφ 2
(14)
so (8) gives
dφ
2
dx
2
= au2 + Bu3
(15)
3
which after integrating twice leads to
φ(x) = −
3a
1
2b cosh2
√
x a 2
(16)
where without any loss of generality, the integration constant is taken to be zero. This is a non-topological one-soliton solution of Form I. The only restriction that is necessary is that a>0 while the second parameter b can be arbitrary.
(17)
C.M. Khalique, A. Biswas / Applied Mathematics Letters 23 (2010) 1397–1400
1399
4. Form II This case, by virtue of Eq. (4), reduces to
φ 00 = aφ + bφ 3
(18)
so (8) gives
dφ
2
dx
1
= au2 + Bu4
(19)
2
which after integrating twice leads to
√ φ 2a
1
q x = √ log 2 2a 2a + bφ 2 + 4a a
(20)
where without any loss of generality, the integration constant is taken to be zero. This is an implicit solution of Form II in terms of logarithmic functions. Here, once again, the restriction given by (17) is necessary for the stationary solution to exist. 5. Form III This case, by virtue of Eq. (4), reduces to
φ 00 = aφ + bφ n
(21)
so (8) gives
dφ
2
dx
= au2 +
2b n+1
Bun+1
(22)
which after integrating twice leads to
φ(x) = −
n−1 1
(n + 1)a 2b
1 2
cosh
n
(23)
√ o (n−1)x a 2
where without any loss of generality, the integration constant is taken to be zero. This is again a non-topological one-soliton solution and it is again necessary that (17) holds for these solitons to exist. 6. Form IV This case, by virtue of Eq. (4), reduces to
φ 00 = aφ + bφ n + c φ 2n−1
(24)
so (8) gives
dφ
2
dx
= au2 +
2b n+1
Bun+1 +
c n
φ 2n
(25)
which cannot be integrated for general n. The following two solutions respectively represent the results for the special cases when n = 2 and n = 3:
√
1 φ 6a x = √ log nq o a 2 6a 6a + 4bφ + 3c φ 2 + 6a + 2bφ
x=
1
√ log
2 a
√ φ 2 6a
q 2 6a 6a + 3bφ 2 + 2c φ 4 + 12a + 3bφ 2
(26)
(27)
where without any loss of generality, the integration constant is taken to be zero. This is again an implicit solution of Form II in terms of logarithmic functions. Here, once again, the restriction given by (17) is necessary for the stationary solution to exist.
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C.M. Khalique, A. Biswas / Applied Mathematics Letters 23 (2010) 1397–1400
7. Form V This case, by virtue of Eq. (4), reduces to
φ 00 = aφ + bφ 1−n + c φ n+1
(28)
so (8) gives
dφ
2
dx
= au2 +
2b n+1
Bun+1 +
c n
φ 2n
(29)
which cannot be integrated for general n. Again, the following two solutions respectively represent the results for special cases—when n = 3 and n = 4:
√
41
s
16c 2 φ 6 1− √ p 2 3 25a + 80bc − 5a −c 5aφ 3 − 10b + 2c φ 6 ! √ r 2 + 80bc c 25a 5a − 3 × F i sinh−1 2 − φ 2 √ 25a2 + 80bc − 5a 5a + 25a2 + 80bc 21 21 √ 21 9a2 +12bc −3a 2c φ 4 3c φ 4 3 √ i 1− √ +2 − c 9a2 +12bc −3a 9a2 +12bc +3a p x = − 2 3aφ 4 − 3b + c φ 8 ! √ r √ c 9a2 + 12bc −1 2 3a − × F i sinh 2 − 2 φ √ 9a + 12bc − 3a 3a + 9a2 + 12bc x = −
25a2 + 80bc − 5a
i 5
(30)
(31)
where in (30) and (31), the incomplete elliptic integral of the first kind is defined as F (x|k) =
x
Z 0
dt
q
1 − t2
(32) 1 − k2 t 2
and k is the modulus of the elliptic function. Again, without any loss of generality, the integration constant is taken to be zero. 8. Conclusions This work obtains the exact one-soliton solution for five different forms of the KGE in 1 + 1 dimensions. The Lie symmetry is used to carry out the integration of the KGE which would otherwise fail the Painlevé test of integrability. These solutions will make an impact in the field of quantum mechanics. In future, this technique will be applied to the 1 + 2-dimensional KGE as well as the coupled KGE. Such work is under way and will be reported in future publications. Acknowledgement The research work of the second author (AB) was fully supported by NSF-CREST Grant No: HRD-0630388 and this support is genuinely and sincerely appreciated. References [1] K.C. Basak, P.C. Ray, R.K. Bera, Solution of non-linear Klein–Gordon equation with a quadratic non-linear term by Adomian decomposition method, Communications in Nonlinear Science and Numerical Simulation 14 (3) (2009) 718–723. [2] A. Biswas, C. Zony, E. Zerrad, Soliton perturbation theory for the quadratic nonlinear Klein–Gordon equations, Applied Mathematics and Computation 203 (1) (2008) 153–156. [3] R.E. Sammelson, A. Heidari, S.F. Tayyari, An analytical approach to the Klein–Gordon and Dirac relativistic oscillators in non-commutative space under a constant magnetic field, Communications in Nonlinear Science and Numerical Simulation 15 (5) (2010) 1368–1371. [4] R. Sassaman, A. Biswas, Soliton perturbation theory for phi-four model and nonlinear Klein–Gordon equations, Communications in Nonlinear Science and Numerical Simulation 14 (8) (2009) 3226–3229. [5] R. Sassaman, A. Biswas, Topological and non-topological solitons of the generalized Klein–Gordon equations, Applied Mathematics and Computation 215 (1) (2009) 212–220. [6] R. Sassaman, A. Biswas, Topological and non-topological solitons of the generalized Klein–Gordon equations, Applied Mathematics and Computation 215 (1) (2009) 212–220. [7] R. Sassaman, A. Biswas, Topological and non-topological solitons of the Klein–Gordon equations in 1 + 2 dimensions, Nonlinear Dynamics 61 (1–2) (2010) 23–28. [8] Sirendaoreji, Exact travelling wave solutions for four forms of nonlinear Klein–Gordon equations, Physics Letters A 363 (2007) 440–447. [9] A.M. Wazwaz, New travelling wave solutions to the Boussinesq and the Klein–Gordon equations, Communications in Nonlinear Science and Numerical Simulation 13 (5) (2008) 889–901. [10] Y. Zheng, S. Lai, A study on three types of nonlinear Klein–Gordon equations, Dynamics of Continuous, Discrete and Impulsive Systems. Series B 16 (2) (2009) 271–279.
Contents lists available at ScienceDirect
Applied Mathematics Letters journal homepage: www.elsevier.com/locate/aml
Analysis of non-linear Klein–Gordon equations using Lie symmetry Chaudry Masood Khalique, Anjan Biswas ∗ International Institute for Symmetry Analysis and Mathematical Modeling, Department of Mathematical Sciences, North-West University, Mafikeng Campus, Private Bag X2046, Mmabatho 2735, South Africa
article
info
Article history: Received 28 December 2009 Accepted 21 July 2010 Keywords: Integrability Lie symmetry
abstract This work obtains the stationary solutions of the non-linear Klein–Gordon equations in 1 + 1 dimensions. The technique that is used to carry out the analysis is the Lie symmetry approach. There are five types of non-linearity that are studied in this work. In each case, the analysis yields non-trivial stationary solutions; it is the first time that this has been seen. © 2010 Elsevier Ltd. All rights reserved.
1. Introduction The Klein–Gordon equation (KGE), which is also known as Schrödinger’s relativistic wave equation, arises in the study of quantum mechanics [1–10]. The KGE is used as the equation of motion of a massive spinless particle in classical and quantum field theory. This work will study the KGE in 1 + 1 dimensions—one time dimension and two spatial dimensions—namely in x- and y-coordinates. There will be five forms of non-linearity that will be considered. In each of these forms, an exact one-soliton solution will be obtained. The solution will be a topological or a non-topological soliton depending of the type of non-linearity in question. The KGE falls into the category of non-linear evolution equations (NLEEs). There are various techniques for carrying out the integration of these NLEEs. Some of the methods are the inverse scattering transform (IST) method, the F -expansion method, the Adomian decomposition method, the G0 /G method, the tanh–coth method, the sine–cosine method, the exponential function method, the Lie symmetry method and many more. In particular, the KGE has also been studied numerically by He’s variational iteration method. However, one needs to be careful before applying analytical techniques of integration. This could lead to incorrect results as pointed out before. In this work, the Lie symmetry approach will be used to carry out the integration of the KGE in 1 + 1 dimensions. 2. Mathematical analysis The KGE in 1 + 1 dimensions that will be studied in this work is given by [1–10] qtt − k2 qxx = F (q)
(1)
where the dependent variable q(x, t ) represents the quantized field describing the particle. Here, k is a real number. In (1), the non-linear function F (q) is continuous. The function F (q) can be written in terms of the potential function U (q) as [2] F (q) =
∂U . ∂q
(2)
∗ Corresponding address: Department of Mathematical Sciences, Applied Mathematics Research Center, Center for Research and Education in Optical Sciences and Applications, Delaware State University, Dover, DE 19901-2277, USA. Tel.: +1 302 857 7913; fax: +1 302 857 7054. E-mail address: [email protected] (A. Biswas). 0893-9659/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.aml.2010.07.006
1398
C.M. Khalique, A. Biswas / Applied Mathematics Letters 23 (2010) 1397–1400
This potential function U (q) has at least two minima, q1 and q3 , and a maximum at q2 such that U (q1 ) = U (q3 ). The solutions to (1) for various forms of the non-linear function F are known as solitons. For some form of the function F , (1) leads to nontopological solitons, while in some other forms, (1) gives kink solutions that are known as topological solitons or topological defects, which take the system from one asymptotically stable state to another. The starting hypothesis is q(x, t ) = φ(x − w t )
(3)
which is also known as the traveling wave hypothesis. In (3), w is the velocity of the wave profile. In this work, the focus is going to be on recovering stationary solutions to the KGE. Therefore, the velocity w is set to zero. Thus (1) reduces to the form
φ 00 = G(φ)
(4)
where G(φ) = F (φ)/k . Eq. (4) has a single Lie point symmetry, namely X = ∂/∂ x. This symmetry will be used to integrate Eq. (4) once. It can be easily seen that the two invariants are 2
u=φ
(5)
v = φ0.
(6)
and Treating u as the independent variable and v as the dependent variable, (4) can be rewritten as dv du
=
φ 00 G(u) = . φ0 v
(7)
Separating variables and carrying out the integration leads to
v2 = 2
Z
G(u)du + c .
(8)
This equation will now be further analysed for the various forms of the functional G(u) that are discussed in the following sections. In this work, the following five forms of the function F (q) will be considered, and will lead to the various solutions. They are [4–7] F (q) = aq − bq2
(9)
F (q) = aq − bq
(10)
3
F (q) = aq − bqn
(11)
F (q) = aq − bq + cq
(12)
F (q) = aq − bq1−n + cqn+1 .
(13)
2n−1
n
These five cases will be respectively labeled as Forms I–V. In all of these five forms, a, b and c are real valued constants. The domain restrictions in all of these five cases will be discussed when they are studied in detail in the following sections. 3. Form I This case, by virtue of Eq. (4), reduces to
φ 00 = aφ + bφ 2
(14)
so (8) gives
dφ
2
dx
2
= au2 + Bu3
(15)
3
which after integrating twice leads to
φ(x) = −
3a
1
2b cosh2
√
x a 2
(16)
where without any loss of generality, the integration constant is taken to be zero. This is a non-topological one-soliton solution of Form I. The only restriction that is necessary is that a>0 while the second parameter b can be arbitrary.
(17)
C.M. Khalique, A. Biswas / Applied Mathematics Letters 23 (2010) 1397–1400
1399
4. Form II This case, by virtue of Eq. (4), reduces to
φ 00 = aφ + bφ 3
(18)
so (8) gives
dφ
2
dx
1
= au2 + Bu4
(19)
2
which after integrating twice leads to
√ φ 2a
1
q x = √ log 2 2a 2a + bφ 2 + 4a a
(20)
where without any loss of generality, the integration constant is taken to be zero. This is an implicit solution of Form II in terms of logarithmic functions. Here, once again, the restriction given by (17) is necessary for the stationary solution to exist. 5. Form III This case, by virtue of Eq. (4), reduces to
φ 00 = aφ + bφ n
(21)
so (8) gives
dφ
2
dx
= au2 +
2b n+1
Bun+1
(22)
which after integrating twice leads to
φ(x) = −
n−1 1
(n + 1)a 2b
1 2
cosh
n
(23)
√ o (n−1)x a 2
where without any loss of generality, the integration constant is taken to be zero. This is again a non-topological one-soliton solution and it is again necessary that (17) holds for these solitons to exist. 6. Form IV This case, by virtue of Eq. (4), reduces to
φ 00 = aφ + bφ n + c φ 2n−1
(24)
so (8) gives
dφ
2
dx
= au2 +
2b n+1
Bun+1 +
c n
φ 2n
(25)
which cannot be integrated for general n. The following two solutions respectively represent the results for the special cases when n = 2 and n = 3:
√
1 φ 6a x = √ log nq o a 2 6a 6a + 4bφ + 3c φ 2 + 6a + 2bφ
x=
1
√ log
2 a
√ φ 2 6a
q 2 6a 6a + 3bφ 2 + 2c φ 4 + 12a + 3bφ 2
(26)
(27)
where without any loss of generality, the integration constant is taken to be zero. This is again an implicit solution of Form II in terms of logarithmic functions. Here, once again, the restriction given by (17) is necessary for the stationary solution to exist.
1400
C.M. Khalique, A. Biswas / Applied Mathematics Letters 23 (2010) 1397–1400
7. Form V This case, by virtue of Eq. (4), reduces to
φ 00 = aφ + bφ 1−n + c φ n+1
(28)
so (8) gives
dφ
2
dx
= au2 +
2b n+1
Bun+1 +
c n
φ 2n
(29)
which cannot be integrated for general n. Again, the following two solutions respectively represent the results for special cases—when n = 3 and n = 4:
√
41
s
16c 2 φ 6 1− √ p 2 3 25a + 80bc − 5a −c 5aφ 3 − 10b + 2c φ 6 ! √ r 2 + 80bc c 25a 5a − 3 × F i sinh−1 2 − φ 2 √ 25a2 + 80bc − 5a 5a + 25a2 + 80bc 21 21 √ 21 9a2 +12bc −3a 2c φ 4 3c φ 4 3 √ i 1− √ +2 − c 9a2 +12bc −3a 9a2 +12bc +3a p x = − 2 3aφ 4 − 3b + c φ 8 ! √ r √ c 9a2 + 12bc −1 2 3a − × F i sinh 2 − 2 φ √ 9a + 12bc − 3a 3a + 9a2 + 12bc x = −
25a2 + 80bc − 5a
i 5
(30)
(31)
where in (30) and (31), the incomplete elliptic integral of the first kind is defined as F (x|k) =
x
Z 0
dt
q
1 − t2
(32) 1 − k2 t 2
and k is the modulus of the elliptic function. Again, without any loss of generality, the integration constant is taken to be zero. 8. Conclusions This work obtains the exact one-soliton solution for five different forms of the KGE in 1 + 1 dimensions. The Lie symmetry is used to carry out the integration of the KGE which would otherwise fail the Painlevé test of integrability. These solutions will make an impact in the field of quantum mechanics. In future, this technique will be applied to the 1 + 2-dimensional KGE as well as the coupled KGE. Such work is under way and will be reported in future publications. Acknowledgement The research work of the second author (AB) was fully supported by NSF-CREST Grant No: HRD-0630388 and this support is genuinely and sincerely appreciated. References [1] K.C. Basak, P.C. Ray, R.K. Bera, Solution of non-linear Klein–Gordon equation with a quadratic non-linear term by Adomian decomposition method, Communications in Nonlinear Science and Numerical Simulation 14 (3) (2009) 718–723. [2] A. Biswas, C. Zony, E. Zerrad, Soliton perturbation theory for the quadratic nonlinear Klein–Gordon equations, Applied Mathematics and Computation 203 (1) (2008) 153–156. [3] R.E. Sammelson, A. Heidari, S.F. Tayyari, An analytical approach to the Klein–Gordon and Dirac relativistic oscillators in non-commutative space under a constant magnetic field, Communications in Nonlinear Science and Numerical Simulation 15 (5) (2010) 1368–1371. [4] R. Sassaman, A. Biswas, Soliton perturbation theory for phi-four model and nonlinear Klein–Gordon equations, Communications in Nonlinear Science and Numerical Simulation 14 (8) (2009) 3226–3229. [5] R. Sassaman, A. Biswas, Topological and non-topological solitons of the generalized Klein–Gordon equations, Applied Mathematics and Computation 215 (1) (2009) 212–220. [6] R. Sassaman, A. Biswas, Topological and non-topological solitons of the generalized Klein–Gordon equations, Applied Mathematics and Computation 215 (1) (2009) 212–220. [7] R. Sassaman, A. Biswas, Topological and non-topological solitons of the Klein–Gordon equations in 1 + 2 dimensions, Nonlinear Dynamics 61 (1–2) (2010) 23–28. [8] Sirendaoreji, Exact travelling wave solutions for four forms of nonlinear Klein–Gordon equations, Physics Letters A 363 (2007) 440–447. [9] A.M. Wazwaz, New travelling wave solutions to the Boussinesq and the Klein–Gordon equations, Communications in Nonlinear Science and Numerical Simulation 13 (5) (2008) 889–901. [10] Y. Zheng, S. Lai, A study on three types of nonlinear Klein–Gordon equations, Dynamics of Continuous, Discrete and Impulsive Systems. Series B 16 (2) (2009) 271–279.