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Soliton perturbation theory for the generalized Benjamin–Bona–Mahoney equation
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Communications in Nonlinear Science and Numerical Simulation 13 (2008) 703–706 www.elsevier.com/locate/cnsns
Soliton perturbation theory for the generalized Benjamin–Bona–Mahoney equation Anjan Biswas a
a,*
, Swapan Konar
b
Department of Applied Mathematics and Theoretical Physics, Delaware State University, 1200 N. DuPont Highway, Dover, DE 19901-2277, USA b Department of Applied Physics, Birla Institute of Technology, Mesra, Ranchi 835215, India Received 11 July 2006; accepted 11 July 2006 Available online 7 September 2006
Abstract The soliton perturbation theory is used to study the adiabatic parameter dynamics of solitons due to the Benjamin– Bona–Mahoney equations in presence of perturbation terms. The change in the velocity is also obtained in this paper. Ó 2006 Elsevier B.V. All rights reserved. PACS: 02.30.Jr; 02.30.Ik MSC: 35Q51; 35Q53; 37K10 Keywords: Solitons; Soliton perturbation theory; Adiabatic parameter dynamics; Integrals of motion
1. Introduction The dimensionless form of the generalized Benjamin–Bona–Mahoney (gBBM) equation that is going to be studied in this paper is given by qt þ qx þ aqn qx þ qxxx ¼ 0
ð1Þ
where a is a constant parameter and n P 1. For n = 1, (1) reduces to the BBM equation that is also known as the regularized long wave equation. Eq. (1) appears in the study of surface water waves in certain regimes. There were several different approaches to study the BBM equation of the type that is given by (1). Some of these techniques include the Inverse Scattering Transform, Backlund Transform, Hirota’s bilinear method. However, the tanh method that was introduced in 1992 [5,8], that is improved in 2003 [2], serves as a powerful technique to integrate the nonlinear evolution equations, even if the Painleve test of integrability fails. In the
*
Corresponding author. Tel.: +1 302 659 0169; fax: +1 302 857 7517. E-mail address: [email protected] (A. Biswas).
1007-5704/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2006.07.005
704
A. Biswas, S. Konar / Communications in Nonlinear Science and Numerical Simulation 13 (2008) 703–706
case of gBBM equation that is given by (1), the 1-soliton solution that is obtained by the powerful tanh method is given by [9] qðx; tÞ ¼
A
ð2Þ
2 n
cosh Bðx xÞ
where
ðn þ 1Þðn þ 2Þðv 1Þ A¼ 2a n pffiffiffiffiffiffiffiffiffiffiffi v1 B¼ 2
1n ð3Þ ð4Þ
Here A represents the amplitude of the soliton that is given by (2) while B is the width of the soliton while x represents the center position of the soliton and therefore the velocity of the soliton is given by v¼
dx dt
ð5Þ
From (4) one can see that it is necessary to have v P 1. 2. Mathematical properties Eq. (1) has at least two integrals of motion [10] that are known as linear momentum (M) and energy (E). These are respectively given by Z 1 A C 12 C 1n M¼ qdx ¼ ð6Þ B C 12 þ 1n 1 and E¼
Z
1
q2 dx ¼
1
A2 C 12 C 2n B C 12 þ 2n
ð7Þ
These conserved quantities are calculated by using the 1-soliton solution given by (2). Also, in (6) and (7), C(x) is the usual gamma function that is defined as Z 1 CðxÞ ¼ et tx dt ð8Þ 0
The center of the soliton x is given by the definition R1 R1 xqdx xqdx 1 R x ¼ 1 ¼ 1 M qdx 1 where M is defined in (6). Thus, the velocity of the soliton is given by R1 R1 xqt dx xq dx dx 1 v¼ ¼ R1 ¼ 1 t M dt qdx 1
ð9Þ
ð10Þ
On using (1) and (6), the velocity of the soliton reduces to v¼
2aAn þ n þ 2 nþ2
ð11Þ
3. Perturbation terms The perturbed gBBM equation that is going to be studied in this paper is given by qt þ qx þ aqn qx þ qxxx ¼ R
ð12Þ
A. Biswas, S. Konar / Communications in Nonlinear Science and Numerical Simulation 13 (2008) 703–706
705
where in (12), is the perturbation parameter and 0 < 1 [1,3], while R gives the perturbation terms. In presence of perturbation terms, the momentum and the energy of the soliton do not stay conserved. Instead, they undergo adiabatic changes that lead to the adiabatic deformation of the soliton amplitude, width and a slow change in the velocity [3,4]. Using (8), the law of adiabatic deformation of the soliton energy is given by [1–4] Z 1 dE ¼ 2 qR dx ð13Þ dt 1 while the adiabatic law of change of the velocity of the soliton is given by [1–4] Z 1 2aAn þ n þ 2 þ v¼ xR dx nþ2 M 1
ð14Þ
In this paper, the perturbation terms that are going to be considered are R ¼ aq þ bqxx þ cqx qxx þ dqm qx þ kqxxx þ mqqx qxx þ rq3x þ nqx qxxxx þ gqxx qxxx þ qqxxxx þ wqxxxxx þ jqqxxxx ð15Þ In R, dissipation gives rise to the first two terms and so a and b are small dissipative coefficients [3]. Also, d or w represent the coefficient of higher order nonlinear dispersive term [3] and m is a positive integer with 1 6 m 6 4 [1,7]. The coefficient of q provide a higher order stabilizing term and must therefore be taken into account [3]. The perturbation term given by coefficient of g was recently considered [6] while the remaining perturbation terms arise in the context of extended version of integrable equations [7]. 3.1. Applications In presence of these perturbation terms, the adiabatic variation of the energy of the soliton is given by # " 2 C 2n dE 1 aA C 2n 4bA2 B 16qð3n þ 2ÞA2 B3 C 2n 32jA3 B3 C 3n þ þ 2 ¼ 2C dt 2 nðn þ 4Þ C 12 þ 2n B C 12 þ 2n nðn þ 4Þð3n þ 4Þ C 12 þ 2n n ðn þ 6Þ C 12 þ 3n ð16Þ The law of the change of velocity for the given perturbation terms in (15) is given by " C 1n þ 12 C 2n C mþ1 C 2n 2aAn þ n þ 2 2cAB2 dAm 6kAB2 n þ þ v¼ nþ2 nðn þ 4Þ C 12 þ 2n m þ 1 C 12 þ mþ1 nðn þ 4Þ C 12 þ 2n C 1n n C 3n C 3n C 2n 4mAB2 4rðn þ 9ÞA2 B2 24nAB4 þ 2 þ 3ðn þ 3Þðn þ 6Þ C 12 þ 3n 3nðn þ 3Þðn þ 6Þ C 12 þ 3n n ðn þ 4Þð3n þ 4Þ C 12 þ 2n # C 2n 72gAB4 þ 2 n ðn þ 4Þð3n þ 4Þ C 12 þ 2n
ð17Þ
4. Conclusions In this paper, soliton perturbation theory is used to study the perturbed gBBM equation. This theory is used to establish the adiabatic parameter dynamics of the soliton energy. Also, it is shown that the velocity undergoes a slow change due to these perturbation terms. In future, it is possible to extend these perturbation terms to include other perturbation terms that include the non-local ones too. The quasi-stationary aspects of the perturbed soliton in presence of such perturbation terms will be studied and reported in future publications. Acknowledgement This research of the first author (AB) was fully supported by NSF Grant No. HRD-970668 and the support is very thankfully appreciated.
706
A. Biswas, S. Konar / Communications in Nonlinear Science and Numerical Simulation 13 (2008) 703–706
References [1] Biswas A, Konar S. Soliton perturbation theory for the compound KdV equation. Int J Theor Phys, in press. [2] Fan E, Hon YC. Applications of the extended tanh method to ‘special’ types of nonlinear equations. Appl Math Comput 2003;141 (2–3):351–8. [3] Kivshar Y, Malomed BA. Dynamics of solitons in nearly integrable systems. Rev Mod Phys 1989;61(4):763–915. [4] Kodama Y, Ablowitz MJ. Perturbations of solitons and solitary waves. Stud Appl Math 1981;64:225–45. [5] Malfliet W. Solitary wave solutions of nonlinear wave equations. Am J Phys 1992;60(7):650–4. [6] Niu X, Pan Z. New approximate solution of the perturbed KdV equation. Phys Lett A 2006;349:192–7. [7] Osborne AR. Approximate asymptotic integration of a higher order water-wave equation using the inverse scattering transform. Nonlinear Process Geophys 1997;4(1):29–53. [8] Parkes EJ, Duffy BR. An automated tanh function method for finding solitary wave solutions to non-linear evolution equations. Comput Phys Commun 1996;98(3):288–300. [9] Wazwaz AM. New travelling wave solutions of different physical structures to generalized BBM equation. Phys Lett A 2006;355:358–62. [10] Zhidkov PE. Korteweg–de Vries and nonlinear Schro¨dinger’s equations: qualitative theory. New York, NY: Springer Verlag; 2001.
Communications in Nonlinear Science and Numerical Simulation 13 (2008) 703–706 www.elsevier.com/locate/cnsns
Soliton perturbation theory for the generalized Benjamin–Bona–Mahoney equation Anjan Biswas a
a,*
, Swapan Konar
b
Department of Applied Mathematics and Theoretical Physics, Delaware State University, 1200 N. DuPont Highway, Dover, DE 19901-2277, USA b Department of Applied Physics, Birla Institute of Technology, Mesra, Ranchi 835215, India Received 11 July 2006; accepted 11 July 2006 Available online 7 September 2006
Abstract The soliton perturbation theory is used to study the adiabatic parameter dynamics of solitons due to the Benjamin– Bona–Mahoney equations in presence of perturbation terms. The change in the velocity is also obtained in this paper. Ó 2006 Elsevier B.V. All rights reserved. PACS: 02.30.Jr; 02.30.Ik MSC: 35Q51; 35Q53; 37K10 Keywords: Solitons; Soliton perturbation theory; Adiabatic parameter dynamics; Integrals of motion
1. Introduction The dimensionless form of the generalized Benjamin–Bona–Mahoney (gBBM) equation that is going to be studied in this paper is given by qt þ qx þ aqn qx þ qxxx ¼ 0
ð1Þ
where a is a constant parameter and n P 1. For n = 1, (1) reduces to the BBM equation that is also known as the regularized long wave equation. Eq. (1) appears in the study of surface water waves in certain regimes. There were several different approaches to study the BBM equation of the type that is given by (1). Some of these techniques include the Inverse Scattering Transform, Backlund Transform, Hirota’s bilinear method. However, the tanh method that was introduced in 1992 [5,8], that is improved in 2003 [2], serves as a powerful technique to integrate the nonlinear evolution equations, even if the Painleve test of integrability fails. In the
*
Corresponding author. Tel.: +1 302 659 0169; fax: +1 302 857 7517. E-mail address: [email protected] (A. Biswas).
1007-5704/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2006.07.005
704
A. Biswas, S. Konar / Communications in Nonlinear Science and Numerical Simulation 13 (2008) 703–706
case of gBBM equation that is given by (1), the 1-soliton solution that is obtained by the powerful tanh method is given by [9] qðx; tÞ ¼
A
ð2Þ
2 n
cosh Bðx xÞ
where
ðn þ 1Þðn þ 2Þðv 1Þ A¼ 2a n pffiffiffiffiffiffiffiffiffiffiffi v1 B¼ 2
1n ð3Þ ð4Þ
Here A represents the amplitude of the soliton that is given by (2) while B is the width of the soliton while x represents the center position of the soliton and therefore the velocity of the soliton is given by v¼
dx dt
ð5Þ
From (4) one can see that it is necessary to have v P 1. 2. Mathematical properties Eq. (1) has at least two integrals of motion [10] that are known as linear momentum (M) and energy (E). These are respectively given by Z 1 A C 12 C 1n M¼ qdx ¼ ð6Þ B C 12 þ 1n 1 and E¼
Z
1
q2 dx ¼
1
A2 C 12 C 2n B C 12 þ 2n
ð7Þ
These conserved quantities are calculated by using the 1-soliton solution given by (2). Also, in (6) and (7), C(x) is the usual gamma function that is defined as Z 1 CðxÞ ¼ et tx dt ð8Þ 0
The center of the soliton x is given by the definition R1 R1 xqdx xqdx 1 R x ¼ 1 ¼ 1 M qdx 1 where M is defined in (6). Thus, the velocity of the soliton is given by R1 R1 xqt dx xq dx dx 1 v¼ ¼ R1 ¼ 1 t M dt qdx 1
ð9Þ
ð10Þ
On using (1) and (6), the velocity of the soliton reduces to v¼
2aAn þ n þ 2 nþ2
ð11Þ
3. Perturbation terms The perturbed gBBM equation that is going to be studied in this paper is given by qt þ qx þ aqn qx þ qxxx ¼ R
ð12Þ
A. Biswas, S. Konar / Communications in Nonlinear Science and Numerical Simulation 13 (2008) 703–706
705
where in (12), is the perturbation parameter and 0 < 1 [1,3], while R gives the perturbation terms. In presence of perturbation terms, the momentum and the energy of the soliton do not stay conserved. Instead, they undergo adiabatic changes that lead to the adiabatic deformation of the soliton amplitude, width and a slow change in the velocity [3,4]. Using (8), the law of adiabatic deformation of the soliton energy is given by [1–4] Z 1 dE ¼ 2 qR dx ð13Þ dt 1 while the adiabatic law of change of the velocity of the soliton is given by [1–4] Z 1 2aAn þ n þ 2 þ v¼ xR dx nþ2 M 1
ð14Þ
In this paper, the perturbation terms that are going to be considered are R ¼ aq þ bqxx þ cqx qxx þ dqm qx þ kqxxx þ mqqx qxx þ rq3x þ nqx qxxxx þ gqxx qxxx þ qqxxxx þ wqxxxxx þ jqqxxxx ð15Þ In R, dissipation gives rise to the first two terms and so a and b are small dissipative coefficients [3]. Also, d or w represent the coefficient of higher order nonlinear dispersive term [3] and m is a positive integer with 1 6 m 6 4 [1,7]. The coefficient of q provide a higher order stabilizing term and must therefore be taken into account [3]. The perturbation term given by coefficient of g was recently considered [6] while the remaining perturbation terms arise in the context of extended version of integrable equations [7]. 3.1. Applications In presence of these perturbation terms, the adiabatic variation of the energy of the soliton is given by # " 2 C 2n dE 1 aA C 2n 4bA2 B 16qð3n þ 2ÞA2 B3 C 2n 32jA3 B3 C 3n þ þ 2 ¼ 2C dt 2 nðn þ 4Þ C 12 þ 2n B C 12 þ 2n nðn þ 4Þð3n þ 4Þ C 12 þ 2n n ðn þ 6Þ C 12 þ 3n ð16Þ The law of the change of velocity for the given perturbation terms in (15) is given by " C 1n þ 12 C 2n C mþ1 C 2n 2aAn þ n þ 2 2cAB2 dAm 6kAB2 n þ þ v¼ nþ2 nðn þ 4Þ C 12 þ 2n m þ 1 C 12 þ mþ1 nðn þ 4Þ C 12 þ 2n C 1n n C 3n C 3n C 2n 4mAB2 4rðn þ 9ÞA2 B2 24nAB4 þ 2 þ 3ðn þ 3Þðn þ 6Þ C 12 þ 3n 3nðn þ 3Þðn þ 6Þ C 12 þ 3n n ðn þ 4Þð3n þ 4Þ C 12 þ 2n # C 2n 72gAB4 þ 2 n ðn þ 4Þð3n þ 4Þ C 12 þ 2n
ð17Þ
4. Conclusions In this paper, soliton perturbation theory is used to study the perturbed gBBM equation. This theory is used to establish the adiabatic parameter dynamics of the soliton energy. Also, it is shown that the velocity undergoes a slow change due to these perturbation terms. In future, it is possible to extend these perturbation terms to include other perturbation terms that include the non-local ones too. The quasi-stationary aspects of the perturbed soliton in presence of such perturbation terms will be studied and reported in future publications. Acknowledgement This research of the first author (AB) was fully supported by NSF Grant No. HRD-970668 and the support is very thankfully appreciated.
706
A. Biswas, S. Konar / Communications in Nonlinear Science and Numerical Simulation 13 (2008) 703–706
References [1] Biswas A, Konar S. Soliton perturbation theory for the compound KdV equation. Int J Theor Phys, in press. [2] Fan E, Hon YC. Applications of the extended tanh method to ‘special’ types of nonlinear equations. Appl Math Comput 2003;141 (2–3):351–8. [3] Kivshar Y, Malomed BA. Dynamics of solitons in nearly integrable systems. Rev Mod Phys 1989;61(4):763–915. [4] Kodama Y, Ablowitz MJ. Perturbations of solitons and solitary waves. Stud Appl Math 1981;64:225–45. [5] Malfliet W. Solitary wave solutions of nonlinear wave equations. Am J Phys 1992;60(7):650–4. [6] Niu X, Pan Z. New approximate solution of the perturbed KdV equation. Phys Lett A 2006;349:192–7. [7] Osborne AR. Approximate asymptotic integration of a higher order water-wave equation using the inverse scattering transform. Nonlinear Process Geophys 1997;4(1):29–53. [8] Parkes EJ, Duffy BR. An automated tanh function method for finding solitary wave solutions to non-linear evolution equations. Comput Phys Commun 1996;98(3):288–300. [9] Wazwaz AM. New travelling wave solutions of different physical structures to generalized BBM equation. Phys Lett A 2006;355:358–62. [10] Zhidkov PE. Korteweg–de Vries and nonlinear Schro¨dinger’s equations: qualitative theory. New York, NY: Springer Verlag; 2001.