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Topological and non-topological solitons of the generalized Klein–Gordon equations
Applied Mathematics and Computation 215 (2009) 212–220
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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Topological and non-topological solitons of the generalized Klein–Gordon equations Ryan Sassaman a, Anjan Biswas b,* a
Department of Applied Mathematics and Theoretical Physics, Delaware State University, Dover, DE 19901-2277, USA 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
a r t i c l e
i n f o
a b s t r a c t This paper obtains the 1-soliton solution of five various forms of the generalized nonlinear Klein–Gordon equations. The solitary wave ansatz is used to obtain the soliton solutions of each of these cases. Both topological as well as non-topological soliton solutions are obtained depending on the type of nonlinearity in question. The conserved quantities are also calculated for each of these five forms of generalized nonlinear Klein–Gordon equations. Each of these forms reduce to the previously known results, as special cases. Ó 2009 Elsevier Inc. All rights reserved.
Keywords: Solitons Integrability Integrals of motion
1. Introduction The Klein–Gordon equation (KGE) is a very important equation in the area of Theoretical Physics. In particular, it is studied in the context of Quantum Mechanics [1–15]. The U4 model, U6 model, quadratic nonlinear KGE and the various other forms of the nonlinear KGE are all well studied in various papers. The closed form soliton solutions are all well known and recently soliton perturbation theory was applied to study the adiabatic variation of the soliton velocity for various perturbation terms applied to these equations [2,10]. This paper is going to study the generalized KGE (gKGE) that will reduce to all these known forms of KGE as a special case. Lately, in the past decade there are various techniques developed that are used to carry out the integration of many such generalized forms of nonlinear evolution equations. They are the Adomian decomposition method, G0 =G method, tan h–cot h method, sine–cosine method, exponential function method and many more. But one needs to be very careful in applying these various techniques of integration, especially the exponential function method integration. These could lead to multiple solutions of such nonlinear evolution equations and many of such solutions are several forms of a particular solution. This fact was pointed out in 2009 [8]. In this paper, the focus is going to be on obtaining the 1-soliton solution only for these five forms of the generalized KGE. The solitary wave ansatz is going to be exploited to carry out the integration of these various forms of the generalized KGE. 2. Mathematical analysis The gKGE is modeled by the equation 2
ðqm Þtt k ðqm Þxx þ FðqÞ ¼ 0;
* Corresponding author. E-mail address: [email protected] (A. Biswas). 0096-3003/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2009.05.001
ð1Þ
R. Sassaman, A. Biswas / Applied Mathematics and Computation 215 (2009) 212–220
213
where the dependent variable qðx; tÞ represents the quantized field describing the particle. Here, k is a real number and m is a positive integer and in fact, m P 1. In particular, the special case m ¼ 1 the gKGE falls back to the case of the regular KGE [1– 7,9–15]. In (1), the nonlinear function FðqÞ is continuous. Hence, this equation falls into the category of nonlinear evolution equations. Wazwaz studied this Eq. (1) extensively for the past few years and also obtained the soliton solutions for various forms of the function FðqÞ [12–15]. He used various mathematical techniques for integration. They are the tan h method, sine–cosine method and many others. In fact, Wazwaz named these Eq. (1) as the KGðm; mÞ equation [12,13] in a similar setting like that of the Kðm; nÞ equation that is the generalized version of the Korteweg-de Vries (KdV) equation. In this paper, Eq. (1) will be solved using the soliton ansatz method, for five different forms of the nonlinear function FðqÞ and the corresponding conserved quantities will be calculated. The function FðqÞ can be written in terms of the potential function UðqÞ as
FðqÞ ¼
@U : @q
ð2Þ
This potential function UðqÞ has at least two minima, q1 and q3 and a maxima at q2 such that Uðq1 Þ ¼ Uðq3 Þ. The solutions to (1) for various forms of the nonlinear function F are known as solitons. For some form of the function F, (1) leads to non-topological solitons while in some other forms, (1) gives kink solutions that are known as topological solitons or topological defects, which takes the system from one asymptotically stable state to another. In this paper, the following five forms of the function FðqÞ will be considered that will lead to the various forms of soliton solutions. They are 2m
ð3Þ
3m
ð4Þ
FðqÞ ¼ aqm bq ; m
FðqÞ ¼ aq bq ; n
FðqÞ ¼ aqm bq ; m
n
ð5Þ 2nm
FðqÞ ¼ aq bq þ cq nm
FðqÞ ¼ aqm bq
ð6Þ
;
þ cqnþm :
ð7Þ
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 details in the following sections. The KGE, given by (1), has at least two conserved quantities. They are the momentum (P) and the energy (E) that are respectively given by
P¼
Z
1
qt qx dx
ð8Þ
1
and
E¼
Z
" # 2 1 2 k 2 qt þ qx þ f ðqÞ dx; 2 1 2 1
ð9Þ
where f ðqÞ is the anti-derivative of FðqÞ that is given by
f ðqÞ ¼
Z
q
FðsÞds:
ð10Þ
3. Form-I In this case by virtue of Eqs. (1) and (3), the gKGE is given by 2
2m
ðqm Þtt k ðqm Þxx þ aqm bq
¼ 0:
ð11Þ
This is the generalized form of the quadratic nonlinear KGE. The special case m ¼ 1 is the quadratic nonlinear KGE which was studied along with its perturbation terms [2] in 2008. In order to solve (1), the solitary wave solution ansatz to (11) is [2]
qðx; tÞ ¼
A p ; cos h s
ð12Þ
where
s ¼ Bðx v tÞ:
ð13Þ
Here A is the soliton amplitude and B is the inverse width of the soliton and v is the soliton velocity. The unknown index p will be determined in terms of m during the course of derivation of the solution of this Eq. (11). From (12), it is possible to obtain
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m2 p2 v 2 Am B2 mpðmp þ 1Þv 2 Am B2 ; mp mpþ2 cos h s cos h s m2 p2 Am B2 mpðmp þ 1Þv 2 Am B2 ðqm Þxx ¼ ; mp mpþ2 cos h s cos h s aAm aqm ¼ mp ; cos h s 2m bA 2m bq ¼ : 2mp cos h s ðqm Þtt ¼
ð14Þ ð15Þ ð16Þ ð17Þ
Substituting (14)–(17) into (11) yields 2
2
2m
m2 p2 v 2 Am B2 mpðmp þ 1Þv 2 Am B2 k m2 p2 Am B2 k mpðmp þ 1Þv 2 Am B2 aAm bA þ ¼ 0: mp mp mp mpþ2 mpþ2 cos h s cos h s cos h s cos h2mp s cos h s cos h s
ð18Þ
Now, from (18) equating the exponents mp þ 2 and 2mp gives
mp þ 2 ¼ 2mp;
ð19Þ
which yields
p¼
2 : m
ð20Þ mpþj
Finally setting the coefficients of 1= cos h
A¼
3a 2b
s, in (18), to zero where j ¼ 1; 2 gives
1 m
ð21Þ
and
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u a B¼u t 2 4 k v2
ð22Þ 2
From (22) it is important to note that aðk v 2 Þ > 0 so that consequently from (21) ab > 0 if m is even. However, ab could be negative if m is odd. Thus, finally, the 1-soliton solution of (11) is given by
qðx; tÞ ¼
A 2
cos hm s
ð23Þ
;
where the amplitude (A) and the inverse width (B) are respectively given by (21) and (22) with the appropriate domain restriction that is discussed after (22). 3.1. Integrals of motion The two conserved quantities for thre gKGE given by (11) are as follows
P¼
Z
1
qt qx dx ¼
1
4v A2 B C 12 C m2 mðm þ 4Þ C 12 þ m2
ð24Þ
and
! 2 1 2 k 2 a b q t þ qx þ qmþ1 q2mþ1 dx 2 mþ1 2m þ 1 2 1 2 2
1 1 m 2A B v 2 þ k C 1 C 2 C 2 C m 2Amþ1 a 2ðm þ 1ÞbA 21 2m þ ; ¼ mðm þ 4Þ ðm þ 2ÞB m þ 1 ð2m þ 1Þð3m þ 2Þ C 12 þ m1 C 2þm
E¼
Z
1
ð25Þ
where in (24) and (25), CðuÞ is the Euler’s gamma function. These conserved quantities are calculated by using the 1-soliton solution of (11) that is given by (23). 4. Form-II In this case, Eqs. (1) and (4) together gives 2
3m
ðqm Þtt k ðqm Þxx þ aqm bq
¼ 0:
ð26Þ
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This is the generalized form of the phi-four model. The special case m ¼ 1 reduces to the U4 equation that was studied along with its perturbation terms [10–13]. In order to solve (1), the same solitary wave solution as given by (12) is assumed [10– 13]. Here, 3m
3m
bq
bA
¼
cos h
3mp
s
ð27Þ
:
Substituting (14)–(16) and (27) into (26) yields 2
2
3m
m2 p2 v 2 Am B2 mpðmp þ 1Þv 2 Am B2 k m2 p2 Am B2 k mpðmp þ 1Þv 2 Am B2 aAm bA þ ¼ 0: mp mp mp mpþ2 mpþ2 cos h s cos h s cos h s cos h3mp s cos h s cos h s
ð28Þ
Now, from (28) equating the exponents mp þ 2 and 3mp gives
mp þ 2 ¼ 3mp;
ð29Þ
which yields
p¼
1 : m
ð30Þ mpþj
Finally, setting the coefficients of 1= cos h
s, in (28), to zero where j ¼ 1; 2 gives
1 2m 2a A¼ b
ð31Þ
and
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u a : B¼u t 2 k v2
ð32Þ 2
From (32) it is important to note that aðk v 2 Þ > 0 and also from (31) ab > 0. Thus finally, the 1-soliton solution of (26) is given by
qðx; tÞ ¼
A 1
cos hm s
ð33Þ
;
where the amplitude (A) and the inverse width (B) are respectively given by (31) and (32). 4.1. Integrals of motion The two conserved quantities for thre gKGE given by (26) are as follows
P¼
Z
1
qt qx dx ¼
1
v A2 B
C 12 C m1 mðm þ 2Þ C 12 þ m1
ð34Þ
and
! 2 1 2 k 2 a b qt þ qx þ qmþ1 q4mþ1 dx 2 mþ1 4m þ 1 2 1 ( ) 2 2 2m 1 A B v2 þ k C 1 C 1 C 12 C 2m þ 12 2Amþ1 a ðm þ 1ÞbA 2 m ¼ þ : 1 ðm þ 2ÞB m þ 1 ð2m þ 1Þð3m þ 1Þ 2mðm þ 2Þ C 12 þ m1 C 2m
E¼
Z
1
ð35Þ
These conserved quantities are calculated by using the 1-soliton solution of (26) that is given by (33). 5. Form-III Here, Eqs. (1) and (5) together imply 2
n
ðqm Þtt k ðqm Þxx þ aqm bq ¼ 0:
ð36Þ
This is the generalized form of the nonlinear KGE. The special case m ¼ 1 reduces to the first type of nonlinear KGE that was studied along with its perturbation terms [10]. In particular the case m ¼ 1 with n ¼ 3 is called the U6 model that appears in solid state Physics, Condensed Matter Physics as well as Quantum Field Theory [6]. In order to solve (1), the same solitary wave solution as given by (12) is assumed [10]. Here, the unknown index p will be determined in terms of m and n during the course of derivation of the solution of this Eq. (36). Now,
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R. Sassaman, A. Biswas / Applied Mathematics and Computation 215 (2009) 212–220 n
n
bq ¼
bA np : cos h s
ð37Þ
Substituting (14)–(16) and (37) into (36) yields 2
2
n
m2 p2 v 2 Am B2 mpðmp þ 1Þv 2 Am B2 k m2 p2 Am B2 k mpðmp þ 1Þv 2 Am B2 aAm bA þ mp mp mp np ¼ 0: mpþ2 mpþ2 cos h s cos h s cos h s cos h s cos h s cos h s
ð38Þ
Now, from (38) equating the exponents mp þ 2 and np gives
mp þ 2 ¼ np;
ð39Þ
which yields
p¼
2 : nm
ð40Þ mpþj
Finally, setting the coefficients of 1= cos h
s, in (38), to zero where j ¼ 1; 2 yields
1 aðn þ mÞ nm A¼ 2bm
ð41Þ
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u aðn mÞ2 : B¼t 2 4m2 k v 2
ð42Þ
and
2
From (42), it is important to note that aðk v 2 Þ > 0 and also from (41) ab > 0 if n m is even; however, ab could be negative if n m is odd. Thus, finally, the 1-soliton solution of (36) is given by
qðx; tÞ ¼
A 2
cos hnm s
ð43Þ
;
where the amplitude (A) and the inverse width (B) are respectively given by (41) and (42). 5.1. Integrals of motion The two conserved quantities for thre gKGE given by (36) are as follows
P¼
Z
1
qt qx dx ¼
1
2 C 12 C nm 4 v A2 B 2 ðn mÞðn m þ 4Þ C 12 þ nm
ð44Þ
and
! 2 1 2 k 2 a b mþ1 nþ1 dx E¼ q þ qx þ q q 2 t mþ1 nþ1 2 1 2 2 mþ1 1 nþ1 1 nþ1 2A2 B v 2 þ k C 12 C nm þ2 C 12 C nm þ aAmþ1 C 12 C nm bA 1 1 nþ1 2 : ¼ þ 2 ðn mÞðn m þ 4Þ C 2 þ nm ðm þ 1ÞB C 12 þ mþ1 ðn þ 1ÞB C 2 þ nm nm Z
1
ð45Þ
These conserved quantities are calculated by using the 1-soliton solution of (36) that is given by (43). 6. Form-IV In this case, Eqs. (1) and (6) together implies 2
n
ðqm Þtt k ðqm Þxx þ aqm bq þ cq2nm ¼ 0:
ð46Þ
This is the generalized form of the second type of nonlinear KGE [9,10]. The special case m ¼ 1 collapses back to the case of nonlinear KGE of the second form which was studied along with its perturbation terms [10]. In order to solve (1), the solitary wave solution ansatz to (46) is [10]
qðx; tÞ ¼
A p: ðD þ cos hsÞ
where A is the soliton amplitude and B is the inverse width of the soliton and constant that depends on a, b, m and n. From (47), it is possible to obtain
ð47Þ
v is the soliton velocity and D is also another
R. Sassaman, A. Biswas / Applied Mathematics and Computation 215 (2009) 212–220
2 2 m 2 p2 v 2 Am B2 mpð2mp þ 1Þv 2 DAm B2 mpðmp þ 1Þv A B D 1 ðq Þtt ¼ þ ; mp mpþ1 mpþ2 ðD þ cos hsÞ ðD þ cos hsÞ ðD þ cos hsÞ m 2 2 p2 Am B2 mpð2mp þ 1ÞDAm B2 mpðmp þ 1ÞA B D 1 m ðq Þxx ¼ þ ; mp mpþ1 mpþ2 ðD þ cos hsÞ ðD þ cos hsÞ ðD þ cos hsÞ m
aAm mp ; ðD þ cos hsÞ
aqm ¼
217
ð48Þ
ð49Þ ð50Þ
n
n
bq ¼
bA np ; ðD þ cos hsÞ
cq2nm ¼
ð51Þ
cA2nm ðD þ cos hsÞ
ð2nmÞp
ð52Þ
:
Substituting (48)–(52) into (47) yields
2 2 m 2 2 p2 v 2 Am B2 mpð2mp þ 1Þv 2 DAm B2 mpðmp þ 1Þv A B D 1 k p2 Am B2 þ mp mp mpþ1 mpþ2 ðD þ cos hsÞ ðD þ cos hsÞ ðD þ cos hsÞ ðD þ cos hsÞ 2 m 2 2 2 n k mpð2mp þ 1ÞDAm B2 k mpðmp þ 1ÞA B D 1 aAm bA þ þ mp np mpþ1 mpþ2 ðD þ cos hsÞ ðD þ cos hsÞ ðD þ cos hsÞ ðD þ cos hsÞ þ
cA2nm ðD þ cos hsÞ
ð2nmÞp
ð53Þ
:
Now, from (53), equating the exponents mp þ 1 and np gives
mp þ 1 ¼ np;
ð54Þ
which yields
p¼
1 : nm
ð55Þ
It needs to be noted that the same result is obtained on equating the exponents mp þ 2 and ð2n mÞp. Finally, setting the s coefficients of 1=ðD þ cos h Þmpþj , in (53), to zero where j ¼ 0; 1; 2 gives
1 amðn þ mÞD nm ; b vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u a B ¼ ðn mÞu t 2 k v2 A¼
ð56Þ ð57Þ
and
pffiffiffi b n D ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 2 nb acmðm þ nÞ2
ð58Þ 2
From (57) it is important to note that aðk v 2 Þ > 0 and from (56) abD > 0 if n m is even; however abD could be negative if n m is odd. Thus, finally, the 1-soliton solution of (46) is given by
qðx; tÞ ¼
A 1
ðD þ cos hsÞnm
ð59Þ
;
where the amplitude (A) and the inverse width (B) are respectively given by (56) and (57). 6.1. Integrals of motion The two conserved quantities for thre gKGE given by (46) are as follows
P¼
Z
1
1
and
qt qx dx ¼
2 v A2 B 2
ðn mÞ 2
2 nm
F
2 2 2 3 1D 2 3 þ 2; ; þ ; B ; nm nm nm 2 2 nm 2
ð60Þ
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R. Sassaman, A. Biswas / Applied Mathematics and Computation 215 (2009) 212–220
! 2 1 2 k 2 a b c q t þ qx þ qmþ1 qnþ1 þ q2nmþ1 dx 2 mþ1 nþ1 2n m þ 1 2 1 2 2 A B v2 þ k 2 2 2 3 1D 2 3 ¼ F þ 2; ; þ ; B ; 2 nm nm nm 2 2 nm 2 ðn mÞ2 2nm mþ1 2aA mþ1 mþ1 mþ1 1 1D mþ1 1 F þ ; ; þ ; B ; mþ1 2 nm 2 ðm þ 1Þ2nm B n m n m n m 2 nþ1 2bA nþ1 nþ1 nþ1 1 1D nþ1 1 B F ; ; þ ; ; nþ1 2 nm 2 ðm þ 1Þ2nm B n m n m n m 2 2nmþ1 2cA 2n m þ 1 2n m þ 1 2n m þ 1 1 1 D 2n m þ 1 1 þ F ; ; þ ; B ; ; 2nmþ1 nm nm nm 2 2 nm 2 ð2n m þ 1Þ2 nm B
E¼
Z
1
ð61Þ
where in (60) and (61) Fða; b; c; zÞ is the Gauss’ hypergeometric function defined as
Fða; b; c; zÞ ¼
1 CðcÞ X Cða þ nÞCðb þ nÞ zn CðaÞCðbÞ n¼0 Cðc þ nÞ n!
ð62Þ
and Bðl; mÞ is the beta function. These conserved quantities are calculated by using the 1-soliton solution of (46) that is given by (59). It needs to be noted that Rabbe’s test of convergence that the series in Gauss’ hypergeometric function, defined in (62), converges provided
c < a þ b:
ð63Þ
This leads to, from (60), the fact that solitons for this form will exist for
nm> 4
ð64Þ
in (46). 7. Form V In this case, Eqs. (1) and (7) together imply that the generalized KGE is 2
mn
ðqm Þtt k ðqm Þxx þ aqm bq
þ cqnþm ¼ 0:
ð65Þ
This is the generalized form of the quadratic nonlinear KGE. The special case m ¼ 1 reduces to the case of quadratic nonlinear KGE of the third form which was studied along with its perturbation terms [10]. In this case, it is necesary to have n > 0 and in particular n – 2; 4 for solitons to exist. In order to solve (1), the solitary wave solution ansatz to (65) is [10] p
qðx; tÞ ¼ A tan h s;
ð66Þ
since the special case of (65) supports toplogical solitons. In this case A and B are free parameters and v is the soliton velocity. From (66), it is possible to obtain
n o mp2 ðqm Þtt ¼ mpv 2 Am B2 ðmp 1Þ tan h s 2 tan hmp s þ ðmp þ 1Þ tan hmpþ2 s ; n o mp2 ðqm Þxx ¼ mpAm B2 ðmp 1Þ tan h s 2 tan hmp s þ ðmp þ 1Þ tan hmpþ2 s ; m
m
aq ¼ aA tan h mn
bq
mn
¼ bA
mp
s;
ð68Þ ð69Þ
ðmnÞp
ð70Þ
ðnþmÞp
ð71Þ
tan h
cqnþm ¼ cAnþm tan h
ð67Þ
s; s:
Substituting (67)–(71) into (65) yields
n o mp2 mpv 2 Am B2 ðmp 1Þ tan h s 2 tan hmp s þ ðmp þ 1Þ tan hmpþ2 s n o 2 mp2 mpk Am B2 ðmp 1Þ tan h s 2 tan hmp s þ ðmp þ 1Þ tan hmpþ2 s þ aAm tan h
mp
s bAmn tan hðmnÞp s þ cAnþm tan hðnþmÞp s ¼ 0:
ð72Þ
Now from (72) equating the exponents mp 2 and ðm nÞp gives
mp þ 2 ¼ ðm nÞp;
ð73Þ
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R. Sassaman, A. Biswas / Applied Mathematics and Computation 215 (2009) 212–220
which yields
p¼
2 : n
ð74Þ
It needs to be noted that the same result is obtained on equating the exponents mp þ 2 and ðm þ nÞp. Finally, setting the mpþj s, in (72), to zero where j ¼ 2; 0; 2 gives coefficients of tan h
A¼
1n 4mb ð2m nÞa
ð75Þ
and
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u an2 u : B¼t 2 8m2 v 2 k
ð76Þ
In addition, the constraint relation between the nonlinear coefficients a, b, c and the exponents m and n given by
4m2 a2 þ 4bc ¼ n2 a2
ð77Þ 2
should be valid, in (65), for the topological solitons to exist, in this case. From (76), it is important to note that aðv 2 k Þ > 0 and from (75) ab > 0 if n is even provided n < 2m; and ab could be negative if n is odd. Thus, finally, the 1-soliton solution of (65) is given by 2
qðx; tÞ ¼ A tan hn s;
ð78Þ
where the free parameters A and B are respectively given by (75) and (76). 7.1. Integrals of motion The two conserved quantities for (65) are given by
P¼
Z
1
qt qx dx ¼
1
16v A2 B n2 16
ð79Þ
and
! 2 1 2 k 2 a b c qt þ qx þ qmþ1 qmnþ1 þ qnþmþ1 dx 2 mþ1 mnþ1 nþmþ1 2 1 2 2 8A B v 2 þ k 2naAmþ1 mþ1 1 mþ1 3 ¼ þ þ ; 1; þ ;1 F 2 16 n n 2 n 2 ðm þ 1Þð2m þ n þ 2ÞB mnþ1 2nbA mnþ1 1 mnþ1 3 þ ; 1; þ ;1 F n 2 n 2 ðm n þ 1Þð2m n þ 2ÞB 2ncAmþnþ1 mþnþ1 1 mþnþ1 3 þ ; 1; þ ;1 : F þ n 2 n 2 ðm þ n þ 1Þð2m þ 3n þ 2ÞB
E¼
Z
1
ð80Þ
These conserved quantities are calculated by using the 1-soliton solution of (65) that is given by (78). 8. Conclusions This paper obtains the exact 1-soliton solution of five different forms of the generalized KGE. Each of these solutions reduce to the special cases for m ¼ 1 that has been studied, so far [1–15]. The conserved quantities are also calculated for each of these five generalized forms of KGE. These generalized KGEs and their respective soliton solutions will make a major impact in the study of Theoretical Physics. In future, the soliton perturbation theory for these generalized KGE will be developed. The quasi-stationary soliton solution, for the perturbation terms, will be obtained. In addition, the quasi-particle theory will also be developed. Such work is under way and will be reported in future publications. Acknowledgements The research work of the second author (AB) was fully supported by NSF-CREST Grant No. HRD-0630388 and Army Research Office (ARO) along with the Air Force Office of Scientific Research (AFOSR) under the award number: W54428-RT-ISP and these supports are genuinely and sincerely appreciated.
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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] G. Chen, Solution of the Klein–Gordon for exponential scalar and vector potentials, Physics Letters A 339 (3–5) (2005) 300–303. [4] X. Deng, M. Zhao, X. Li, Travelling wave solutions for a nonlinear variant of the phi-four equation, Mathematical and Computer Modelling 49 (3–4) (2009) 617–622. [5] A. Ebaid, Exact solutions for the generalized Klein–Gordon equation via a transformation and Exp-function method and comparison with Adomian’s method, Journal of Computational and Applied Mathematics 223 (1) (2009) 278–290. [6] A. Elgarayahi, New periodic wave solutions for the shallow water equations and the generalized Klein–Gordon equation, Communications in Nonlinear Science and Numerical Simulation 13 (5) (2008) 877–888. [7] D. Feng, J. Li, Exact explicit travelling wave solutions for the ðn þ 1Þ-dimensional U6 field model, Physics Letters A 369 (2007) 255–261. [8] N.A. Kudryashov, Seven common errors in finding exact solutions of nonlinear differential equations, Communications in Nonlinear Science and Numerical Simulation 14 (9–10) (2009). [9] Mustafa Inc., New compacton and solitary pattern solutions of the nonlinear modified dispersive Klein–Gordon equations, Chaos, Solitons & Fractals 33 (4) (2007) 1275–1284. [10] R. Sassaman, A. Biswas, Soliton perturbation theory for phi-four model and nonlinear Klein–Gordon equations, Communications in Nonlinear Science and Numerical Simulations 14 (8) (2009) 3226–3229. [11] Sirendaoreji, Exact travelling wave solutions for four forms of nonlinear Klein–Gordon equations, Physics Letters A 363 (2007) 440–447. [12] A.M. Wazwaz, Solutions of compact and noncompact structures for nonlinear Klein–Gordon type equation, Applied Mathematics and Computation 134 (2–3) (2003) 487–500. [13] A.M. Wazwaz, The tan h and sine–cosine methods for compact and noncompact solutions of the nonlinear Klein–Gordon equation, Applied Mathematics and Computation 167 (2) (2005) 1179–1195. [14] A.M. Wazwaz, Generalized forms of the phi-four equation with compactons, solitons and periodic solutions, Mathematics and Computers in Simulations 69 (5–6) (2005) 580–588. [15] 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.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Topological and non-topological solitons of the generalized Klein–Gordon equations Ryan Sassaman a, Anjan Biswas b,* a
Department of Applied Mathematics and Theoretical Physics, Delaware State University, Dover, DE 19901-2277, USA 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
a r t i c l e
i n f o
a b s t r a c t This paper obtains the 1-soliton solution of five various forms of the generalized nonlinear Klein–Gordon equations. The solitary wave ansatz is used to obtain the soliton solutions of each of these cases. Both topological as well as non-topological soliton solutions are obtained depending on the type of nonlinearity in question. The conserved quantities are also calculated for each of these five forms of generalized nonlinear Klein–Gordon equations. Each of these forms reduce to the previously known results, as special cases. Ó 2009 Elsevier Inc. All rights reserved.
Keywords: Solitons Integrability Integrals of motion
1. Introduction The Klein–Gordon equation (KGE) is a very important equation in the area of Theoretical Physics. In particular, it is studied in the context of Quantum Mechanics [1–15]. The U4 model, U6 model, quadratic nonlinear KGE and the various other forms of the nonlinear KGE are all well studied in various papers. The closed form soliton solutions are all well known and recently soliton perturbation theory was applied to study the adiabatic variation of the soliton velocity for various perturbation terms applied to these equations [2,10]. This paper is going to study the generalized KGE (gKGE) that will reduce to all these known forms of KGE as a special case. Lately, in the past decade there are various techniques developed that are used to carry out the integration of many such generalized forms of nonlinear evolution equations. They are the Adomian decomposition method, G0 =G method, tan h–cot h method, sine–cosine method, exponential function method and many more. But one needs to be very careful in applying these various techniques of integration, especially the exponential function method integration. These could lead to multiple solutions of such nonlinear evolution equations and many of such solutions are several forms of a particular solution. This fact was pointed out in 2009 [8]. In this paper, the focus is going to be on obtaining the 1-soliton solution only for these five forms of the generalized KGE. The solitary wave ansatz is going to be exploited to carry out the integration of these various forms of the generalized KGE. 2. Mathematical analysis The gKGE is modeled by the equation 2
ðqm Þtt k ðqm Þxx þ FðqÞ ¼ 0;
* Corresponding author. E-mail address: [email protected] (A. Biswas). 0096-3003/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2009.05.001
ð1Þ
R. Sassaman, A. Biswas / Applied Mathematics and Computation 215 (2009) 212–220
213
where the dependent variable qðx; tÞ represents the quantized field describing the particle. Here, k is a real number and m is a positive integer and in fact, m P 1. In particular, the special case m ¼ 1 the gKGE falls back to the case of the regular KGE [1– 7,9–15]. In (1), the nonlinear function FðqÞ is continuous. Hence, this equation falls into the category of nonlinear evolution equations. Wazwaz studied this Eq. (1) extensively for the past few years and also obtained the soliton solutions for various forms of the function FðqÞ [12–15]. He used various mathematical techniques for integration. They are the tan h method, sine–cosine method and many others. In fact, Wazwaz named these Eq. (1) as the KGðm; mÞ equation [12,13] in a similar setting like that of the Kðm; nÞ equation that is the generalized version of the Korteweg-de Vries (KdV) equation. In this paper, Eq. (1) will be solved using the soliton ansatz method, for five different forms of the nonlinear function FðqÞ and the corresponding conserved quantities will be calculated. The function FðqÞ can be written in terms of the potential function UðqÞ as
FðqÞ ¼
@U : @q
ð2Þ
This potential function UðqÞ has at least two minima, q1 and q3 and a maxima at q2 such that Uðq1 Þ ¼ Uðq3 Þ. The solutions to (1) for various forms of the nonlinear function F are known as solitons. For some form of the function F, (1) leads to non-topological solitons while in some other forms, (1) gives kink solutions that are known as topological solitons or topological defects, which takes the system from one asymptotically stable state to another. In this paper, the following five forms of the function FðqÞ will be considered that will lead to the various forms of soliton solutions. They are 2m
ð3Þ
3m
ð4Þ
FðqÞ ¼ aqm bq ; m
FðqÞ ¼ aq bq ; n
FðqÞ ¼ aqm bq ; m
n
ð5Þ 2nm
FðqÞ ¼ aq bq þ cq nm
FðqÞ ¼ aqm bq
ð6Þ
;
þ cqnþm :
ð7Þ
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 details in the following sections. The KGE, given by (1), has at least two conserved quantities. They are the momentum (P) and the energy (E) that are respectively given by
P¼
Z
1
qt qx dx
ð8Þ
1
and
E¼
Z
" # 2 1 2 k 2 qt þ qx þ f ðqÞ dx; 2 1 2 1
ð9Þ
where f ðqÞ is the anti-derivative of FðqÞ that is given by
f ðqÞ ¼
Z
q
FðsÞds:
ð10Þ
3. Form-I In this case by virtue of Eqs. (1) and (3), the gKGE is given by 2
2m
ðqm Þtt k ðqm Þxx þ aqm bq
¼ 0:
ð11Þ
This is the generalized form of the quadratic nonlinear KGE. The special case m ¼ 1 is the quadratic nonlinear KGE which was studied along with its perturbation terms [2] in 2008. In order to solve (1), the solitary wave solution ansatz to (11) is [2]
qðx; tÞ ¼
A p ; cos h s
ð12Þ
where
s ¼ Bðx v tÞ:
ð13Þ
Here A is the soliton amplitude and B is the inverse width of the soliton and v is the soliton velocity. The unknown index p will be determined in terms of m during the course of derivation of the solution of this Eq. (11). From (12), it is possible to obtain
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R. Sassaman, A. Biswas / Applied Mathematics and Computation 215 (2009) 212–220
m2 p2 v 2 Am B2 mpðmp þ 1Þv 2 Am B2 ; mp mpþ2 cos h s cos h s m2 p2 Am B2 mpðmp þ 1Þv 2 Am B2 ðqm Þxx ¼ ; mp mpþ2 cos h s cos h s aAm aqm ¼ mp ; cos h s 2m bA 2m bq ¼ : 2mp cos h s ðqm Þtt ¼
ð14Þ ð15Þ ð16Þ ð17Þ
Substituting (14)–(17) into (11) yields 2
2
2m
m2 p2 v 2 Am B2 mpðmp þ 1Þv 2 Am B2 k m2 p2 Am B2 k mpðmp þ 1Þv 2 Am B2 aAm bA þ ¼ 0: mp mp mp mpþ2 mpþ2 cos h s cos h s cos h s cos h2mp s cos h s cos h s
ð18Þ
Now, from (18) equating the exponents mp þ 2 and 2mp gives
mp þ 2 ¼ 2mp;
ð19Þ
which yields
p¼
2 : m
ð20Þ mpþj
Finally setting the coefficients of 1= cos h
A¼
3a 2b
s, in (18), to zero where j ¼ 1; 2 gives
1 m
ð21Þ
and
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u a B¼u t 2 4 k v2
ð22Þ 2
From (22) it is important to note that aðk v 2 Þ > 0 so that consequently from (21) ab > 0 if m is even. However, ab could be negative if m is odd. Thus, finally, the 1-soliton solution of (11) is given by
qðx; tÞ ¼
A 2
cos hm s
ð23Þ
;
where the amplitude (A) and the inverse width (B) are respectively given by (21) and (22) with the appropriate domain restriction that is discussed after (22). 3.1. Integrals of motion The two conserved quantities for thre gKGE given by (11) are as follows
P¼
Z
1
qt qx dx ¼
1
4v A2 B C 12 C m2 mðm þ 4Þ C 12 þ m2
ð24Þ
and
! 2 1 2 k 2 a b q t þ qx þ qmþ1 q2mþ1 dx 2 mþ1 2m þ 1 2 1 2 2
1 1 m 2A B v 2 þ k C 1 C 2 C 2 C m 2Amþ1 a 2ðm þ 1ÞbA 21 2m þ ; ¼ mðm þ 4Þ ðm þ 2ÞB m þ 1 ð2m þ 1Þð3m þ 2Þ C 12 þ m1 C 2þm
E¼
Z
1
ð25Þ
where in (24) and (25), CðuÞ is the Euler’s gamma function. These conserved quantities are calculated by using the 1-soliton solution of (11) that is given by (23). 4. Form-II In this case, Eqs. (1) and (4) together gives 2
3m
ðqm Þtt k ðqm Þxx þ aqm bq
¼ 0:
ð26Þ
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R. Sassaman, A. Biswas / Applied Mathematics and Computation 215 (2009) 212–220
This is the generalized form of the phi-four model. The special case m ¼ 1 reduces to the U4 equation that was studied along with its perturbation terms [10–13]. In order to solve (1), the same solitary wave solution as given by (12) is assumed [10– 13]. Here, 3m
3m
bq
bA
¼
cos h
3mp
s
ð27Þ
:
Substituting (14)–(16) and (27) into (26) yields 2
2
3m
m2 p2 v 2 Am B2 mpðmp þ 1Þv 2 Am B2 k m2 p2 Am B2 k mpðmp þ 1Þv 2 Am B2 aAm bA þ ¼ 0: mp mp mp mpþ2 mpþ2 cos h s cos h s cos h s cos h3mp s cos h s cos h s
ð28Þ
Now, from (28) equating the exponents mp þ 2 and 3mp gives
mp þ 2 ¼ 3mp;
ð29Þ
which yields
p¼
1 : m
ð30Þ mpþj
Finally, setting the coefficients of 1= cos h
s, in (28), to zero where j ¼ 1; 2 gives
1 2m 2a A¼ b
ð31Þ
and
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u a : B¼u t 2 k v2
ð32Þ 2
From (32) it is important to note that aðk v 2 Þ > 0 and also from (31) ab > 0. Thus finally, the 1-soliton solution of (26) is given by
qðx; tÞ ¼
A 1
cos hm s
ð33Þ
;
where the amplitude (A) and the inverse width (B) are respectively given by (31) and (32). 4.1. Integrals of motion The two conserved quantities for thre gKGE given by (26) are as follows
P¼
Z
1
qt qx dx ¼
1
v A2 B
C 12 C m1 mðm þ 2Þ C 12 þ m1
ð34Þ
and
! 2 1 2 k 2 a b qt þ qx þ qmþ1 q4mþ1 dx 2 mþ1 4m þ 1 2 1 ( ) 2 2 2m 1 A B v2 þ k C 1 C 1 C 12 C 2m þ 12 2Amþ1 a ðm þ 1ÞbA 2 m ¼ þ : 1 ðm þ 2ÞB m þ 1 ð2m þ 1Þð3m þ 1Þ 2mðm þ 2Þ C 12 þ m1 C 2m
E¼
Z
1
ð35Þ
These conserved quantities are calculated by using the 1-soliton solution of (26) that is given by (33). 5. Form-III Here, Eqs. (1) and (5) together imply 2
n
ðqm Þtt k ðqm Þxx þ aqm bq ¼ 0:
ð36Þ
This is the generalized form of the nonlinear KGE. The special case m ¼ 1 reduces to the first type of nonlinear KGE that was studied along with its perturbation terms [10]. In particular the case m ¼ 1 with n ¼ 3 is called the U6 model that appears in solid state Physics, Condensed Matter Physics as well as Quantum Field Theory [6]. In order to solve (1), the same solitary wave solution as given by (12) is assumed [10]. Here, the unknown index p will be determined in terms of m and n during the course of derivation of the solution of this Eq. (36). Now,
216
R. Sassaman, A. Biswas / Applied Mathematics and Computation 215 (2009) 212–220 n
n
bq ¼
bA np : cos h s
ð37Þ
Substituting (14)–(16) and (37) into (36) yields 2
2
n
m2 p2 v 2 Am B2 mpðmp þ 1Þv 2 Am B2 k m2 p2 Am B2 k mpðmp þ 1Þv 2 Am B2 aAm bA þ mp mp mp np ¼ 0: mpþ2 mpþ2 cos h s cos h s cos h s cos h s cos h s cos h s
ð38Þ
Now, from (38) equating the exponents mp þ 2 and np gives
mp þ 2 ¼ np;
ð39Þ
which yields
p¼
2 : nm
ð40Þ mpþj
Finally, setting the coefficients of 1= cos h
s, in (38), to zero where j ¼ 1; 2 yields
1 aðn þ mÞ nm A¼ 2bm
ð41Þ
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u aðn mÞ2 : B¼t 2 4m2 k v 2
ð42Þ
and
2
From (42), it is important to note that aðk v 2 Þ > 0 and also from (41) ab > 0 if n m is even; however, ab could be negative if n m is odd. Thus, finally, the 1-soliton solution of (36) is given by
qðx; tÞ ¼
A 2
cos hnm s
ð43Þ
;
where the amplitude (A) and the inverse width (B) are respectively given by (41) and (42). 5.1. Integrals of motion The two conserved quantities for thre gKGE given by (36) are as follows
P¼
Z
1
qt qx dx ¼
1
2 C 12 C nm 4 v A2 B 2 ðn mÞðn m þ 4Þ C 12 þ nm
ð44Þ
and
! 2 1 2 k 2 a b mþ1 nþ1 dx E¼ q þ qx þ q q 2 t mþ1 nþ1 2 1 2 2 mþ1 1 nþ1 1 nþ1 2A2 B v 2 þ k C 12 C nm þ2 C 12 C nm þ aAmþ1 C 12 C nm bA 1 1 nþ1 2 : ¼ þ 2 ðn mÞðn m þ 4Þ C 2 þ nm ðm þ 1ÞB C 12 þ mþ1 ðn þ 1ÞB C 2 þ nm nm Z
1
ð45Þ
These conserved quantities are calculated by using the 1-soliton solution of (36) that is given by (43). 6. Form-IV In this case, Eqs. (1) and (6) together implies 2
n
ðqm Þtt k ðqm Þxx þ aqm bq þ cq2nm ¼ 0:
ð46Þ
This is the generalized form of the second type of nonlinear KGE [9,10]. The special case m ¼ 1 collapses back to the case of nonlinear KGE of the second form which was studied along with its perturbation terms [10]. In order to solve (1), the solitary wave solution ansatz to (46) is [10]
qðx; tÞ ¼
A p: ðD þ cos hsÞ
where A is the soliton amplitude and B is the inverse width of the soliton and constant that depends on a, b, m and n. From (47), it is possible to obtain
ð47Þ
v is the soliton velocity and D is also another
R. Sassaman, A. Biswas / Applied Mathematics and Computation 215 (2009) 212–220
2 2 m 2 p2 v 2 Am B2 mpð2mp þ 1Þv 2 DAm B2 mpðmp þ 1Þv A B D 1 ðq Þtt ¼ þ ; mp mpþ1 mpþ2 ðD þ cos hsÞ ðD þ cos hsÞ ðD þ cos hsÞ m 2 2 p2 Am B2 mpð2mp þ 1ÞDAm B2 mpðmp þ 1ÞA B D 1 m ðq Þxx ¼ þ ; mp mpþ1 mpþ2 ðD þ cos hsÞ ðD þ cos hsÞ ðD þ cos hsÞ m
aAm mp ; ðD þ cos hsÞ
aqm ¼
217
ð48Þ
ð49Þ ð50Þ
n
n
bq ¼
bA np ; ðD þ cos hsÞ
cq2nm ¼
ð51Þ
cA2nm ðD þ cos hsÞ
ð2nmÞp
ð52Þ
:
Substituting (48)–(52) into (47) yields
2 2 m 2 2 p2 v 2 Am B2 mpð2mp þ 1Þv 2 DAm B2 mpðmp þ 1Þv A B D 1 k p2 Am B2 þ mp mp mpþ1 mpþ2 ðD þ cos hsÞ ðD þ cos hsÞ ðD þ cos hsÞ ðD þ cos hsÞ 2 m 2 2 2 n k mpð2mp þ 1ÞDAm B2 k mpðmp þ 1ÞA B D 1 aAm bA þ þ mp np mpþ1 mpþ2 ðD þ cos hsÞ ðD þ cos hsÞ ðD þ cos hsÞ ðD þ cos hsÞ þ
cA2nm ðD þ cos hsÞ
ð2nmÞp
ð53Þ
:
Now, from (53), equating the exponents mp þ 1 and np gives
mp þ 1 ¼ np;
ð54Þ
which yields
p¼
1 : nm
ð55Þ
It needs to be noted that the same result is obtained on equating the exponents mp þ 2 and ð2n mÞp. Finally, setting the s coefficients of 1=ðD þ cos h Þmpþj , in (53), to zero where j ¼ 0; 1; 2 gives
1 amðn þ mÞD nm ; b vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u a B ¼ ðn mÞu t 2 k v2 A¼
ð56Þ ð57Þ
and
pffiffiffi b n D ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 2 nb acmðm þ nÞ2
ð58Þ 2
From (57) it is important to note that aðk v 2 Þ > 0 and from (56) abD > 0 if n m is even; however abD could be negative if n m is odd. Thus, finally, the 1-soliton solution of (46) is given by
qðx; tÞ ¼
A 1
ðD þ cos hsÞnm
ð59Þ
;
where the amplitude (A) and the inverse width (B) are respectively given by (56) and (57). 6.1. Integrals of motion The two conserved quantities for thre gKGE given by (46) are as follows
P¼
Z
1
1
and
qt qx dx ¼
2 v A2 B 2
ðn mÞ 2
2 nm
F
2 2 2 3 1D 2 3 þ 2; ; þ ; B ; nm nm nm 2 2 nm 2
ð60Þ
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R. Sassaman, A. Biswas / Applied Mathematics and Computation 215 (2009) 212–220
! 2 1 2 k 2 a b c q t þ qx þ qmþ1 qnþ1 þ q2nmþ1 dx 2 mþ1 nþ1 2n m þ 1 2 1 2 2 A B v2 þ k 2 2 2 3 1D 2 3 ¼ F þ 2; ; þ ; B ; 2 nm nm nm 2 2 nm 2 ðn mÞ2 2nm mþ1 2aA mþ1 mþ1 mþ1 1 1D mþ1 1 F þ ; ; þ ; B ; mþ1 2 nm 2 ðm þ 1Þ2nm B n m n m n m 2 nþ1 2bA nþ1 nþ1 nþ1 1 1D nþ1 1 B F ; ; þ ; ; nþ1 2 nm 2 ðm þ 1Þ2nm B n m n m n m 2 2nmþ1 2cA 2n m þ 1 2n m þ 1 2n m þ 1 1 1 D 2n m þ 1 1 þ F ; ; þ ; B ; ; 2nmþ1 nm nm nm 2 2 nm 2 ð2n m þ 1Þ2 nm B
E¼
Z
1
ð61Þ
where in (60) and (61) Fða; b; c; zÞ is the Gauss’ hypergeometric function defined as
Fða; b; c; zÞ ¼
1 CðcÞ X Cða þ nÞCðb þ nÞ zn CðaÞCðbÞ n¼0 Cðc þ nÞ n!
ð62Þ
and Bðl; mÞ is the beta function. These conserved quantities are calculated by using the 1-soliton solution of (46) that is given by (59). It needs to be noted that Rabbe’s test of convergence that the series in Gauss’ hypergeometric function, defined in (62), converges provided
c < a þ b:
ð63Þ
This leads to, from (60), the fact that solitons for this form will exist for
nm> 4
ð64Þ
in (46). 7. Form V In this case, Eqs. (1) and (7) together imply that the generalized KGE is 2
mn
ðqm Þtt k ðqm Þxx þ aqm bq
þ cqnþm ¼ 0:
ð65Þ
This is the generalized form of the quadratic nonlinear KGE. The special case m ¼ 1 reduces to the case of quadratic nonlinear KGE of the third form which was studied along with its perturbation terms [10]. In this case, it is necesary to have n > 0 and in particular n – 2; 4 for solitons to exist. In order to solve (1), the solitary wave solution ansatz to (65) is [10] p
qðx; tÞ ¼ A tan h s;
ð66Þ
since the special case of (65) supports toplogical solitons. In this case A and B are free parameters and v is the soliton velocity. From (66), it is possible to obtain
n o mp2 ðqm Þtt ¼ mpv 2 Am B2 ðmp 1Þ tan h s 2 tan hmp s þ ðmp þ 1Þ tan hmpþ2 s ; n o mp2 ðqm Þxx ¼ mpAm B2 ðmp 1Þ tan h s 2 tan hmp s þ ðmp þ 1Þ tan hmpþ2 s ; m
m
aq ¼ aA tan h mn
bq
mn
¼ bA
mp
s;
ð68Þ ð69Þ
ðmnÞp
ð70Þ
ðnþmÞp
ð71Þ
tan h
cqnþm ¼ cAnþm tan h
ð67Þ
s; s:
Substituting (67)–(71) into (65) yields
n o mp2 mpv 2 Am B2 ðmp 1Þ tan h s 2 tan hmp s þ ðmp þ 1Þ tan hmpþ2 s n o 2 mp2 mpk Am B2 ðmp 1Þ tan h s 2 tan hmp s þ ðmp þ 1Þ tan hmpþ2 s þ aAm tan h
mp
s bAmn tan hðmnÞp s þ cAnþm tan hðnþmÞp s ¼ 0:
ð72Þ
Now from (72) equating the exponents mp 2 and ðm nÞp gives
mp þ 2 ¼ ðm nÞp;
ð73Þ
219
R. Sassaman, A. Biswas / Applied Mathematics and Computation 215 (2009) 212–220
which yields
p¼
2 : n
ð74Þ
It needs to be noted that the same result is obtained on equating the exponents mp þ 2 and ðm þ nÞp. Finally, setting the mpþj s, in (72), to zero where j ¼ 2; 0; 2 gives coefficients of tan h
A¼
1n 4mb ð2m nÞa
ð75Þ
and
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u an2 u : B¼t 2 8m2 v 2 k
ð76Þ
In addition, the constraint relation between the nonlinear coefficients a, b, c and the exponents m and n given by
4m2 a2 þ 4bc ¼ n2 a2
ð77Þ 2
should be valid, in (65), for the topological solitons to exist, in this case. From (76), it is important to note that aðv 2 k Þ > 0 and from (75) ab > 0 if n is even provided n < 2m; and ab could be negative if n is odd. Thus, finally, the 1-soliton solution of (65) is given by 2
qðx; tÞ ¼ A tan hn s;
ð78Þ
where the free parameters A and B are respectively given by (75) and (76). 7.1. Integrals of motion The two conserved quantities for (65) are given by
P¼
Z
1
qt qx dx ¼
1
16v A2 B n2 16
ð79Þ
and
! 2 1 2 k 2 a b c qt þ qx þ qmþ1 qmnþ1 þ qnþmþ1 dx 2 mþ1 mnþ1 nþmþ1 2 1 2 2 8A B v 2 þ k 2naAmþ1 mþ1 1 mþ1 3 ¼ þ þ ; 1; þ ;1 F 2 16 n n 2 n 2 ðm þ 1Þð2m þ n þ 2ÞB mnþ1 2nbA mnþ1 1 mnþ1 3 þ ; 1; þ ;1 F n 2 n 2 ðm n þ 1Þð2m n þ 2ÞB 2ncAmþnþ1 mþnþ1 1 mþnþ1 3 þ ; 1; þ ;1 : F þ n 2 n 2 ðm þ n þ 1Þð2m þ 3n þ 2ÞB
E¼
Z
1
ð80Þ
These conserved quantities are calculated by using the 1-soliton solution of (65) that is given by (78). 8. Conclusions This paper obtains the exact 1-soliton solution of five different forms of the generalized KGE. Each of these solutions reduce to the special cases for m ¼ 1 that has been studied, so far [1–15]. The conserved quantities are also calculated for each of these five generalized forms of KGE. These generalized KGEs and their respective soliton solutions will make a major impact in the study of Theoretical Physics. In future, the soliton perturbation theory for these generalized KGE will be developed. The quasi-stationary soliton solution, for the perturbation terms, will be obtained. In addition, the quasi-particle theory will also be developed. Such work is under way and will be reported in future publications. Acknowledgements The research work of the second author (AB) was fully supported by NSF-CREST Grant No. HRD-0630388 and Army Research Office (ARO) along with the Air Force Office of Scientific Research (AFOSR) under the award number: W54428-RT-ISP and these supports are genuinely and sincerely appreciated.
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