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Solitary wave solution for the generalized KdV equation with time-dependent damping and dispersion
Commun Nonlinear Sci Numer Simulat 14 (2009) 3503–3506
Contents lists available at ScienceDirect
Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns
Short communication
Solitary wave solution for the generalized KdV equation with time-dependent damping and dispersion Anjan Biswas * 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
a r t i c l e
i n f o
Article history: Received 3 September 2008 Received in revised form 19 September 2008 Accepted 19 September 2008 Available online 10 October 2008
a b s t r a c t The solitary wave solution of the generalized KdV equation is obtained in this paper in presence of time-dependent damping and dispersion. The approach is from a solitary wave ansatze that leads to the exact solution. A particular example is also considered to complete the analysis. Ó 2008 Elsevier B.V. All rights reserved.
MSC: 35Q51 35Q53 37K10 PACS codes: 02.30.Jr 02.30.Ik Keywords: Solitary wave Integrability Dispersion Damping
1. Introduction The study of nonlinear evolution equation is going on for the past few decades. It is an important area of study in the fields of Physics and Mathematics. In particular, in the last couple of decades there has been a large number of papers published that talks about the integrability aspects of various nonlinear evolution equations that proved to be a failure by the Painleve test of integrability [1–10]. Thus, those equations failed to be integrable by the classical method of integration, namely the inverse scattering transform (IST) which is the nonlinear analog of Fourier transform that is used to integrate linear partial differential equations. In the present times, there are a bunch of new techniques for carrying out the integration of various nonlinear evolution equations. These new methods include the Wadati trace method, pseudo-spectral method, tanh-sech method, sine-cosine method and the Riccati equation expansion method, exponential function method and many more. Without these modern methods of integrability, many such equations would not have been solved.
* Tel.: +1 302 659 0169; fax: +1 302 857 7517. E-mail address: [email protected] 1007-5704/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2008.09.026
3504
A. Biswas / Commun Nonlinear Sci Numer Simulat 14 (2009) 3503–3506
In this paper, the solitary wave ansatze will be exploited to find the 1-soliton solution of the generalized Korteweg–de Vries equation (gKdV), in presence of linear, time-dependent damping and time-dependent dispersion.
2. Mathematical analysis The gKdV equation with time-dependent damping and dispersion that is going to be studied in this paper is given by
qt þ qn qx þ aðtÞq þ bðtÞqxxx ¼ 0
ð1Þ
Here the first term is the evolution term, while the second term represents the nonlinear term. The third term is the linear damping with a time-dependent coefficient a(t) while the fourth term is the dispersion term with time-dependent coefficient b(t). In (1), a and b 2 R while n 2 Z+. This equation arises in various physical situations including the study of coastal waves in ocean, liquid drops and bubbles [3]. It also arises in the context of atmospheric blocking phenomenon and in particular in the issue of dipole blocking [3,9]. It is not possible to integrate this equation by the IST as the Painleve test will fail. However, a particular form of (1) was recently studied by the aid of Lie transform [3]. In this paper, the solitary wave ansatze will enable to find the solution to (1) for a general connection between the functions a(t) and b(t) as long as they are Riemann integrable. Without any loss of generality it is assumed that the solitary wave solution to (1) is given by
qðx; tÞ ¼
AðtÞ p cosh ½BðtÞðx v ðtÞtÞ
ð2Þ
where A represents the amplitude of the soliton, while B is the inverse width of the soliton and t represents the velocity of the soliton. It needs to be noted that since damping and dispersion have time-dependent coefficients, one needs to have, in general,
A ¼ AðtÞ
ð3Þ
B ¼ BðtÞ
ð4Þ
v ¼ v ðtÞ
ð5Þ
and
Thus from (2), one gets,
dA 1 dv tanh s pA dB s tanh s þ pAB v þ t dt coshp s dt coshp s B dt coshp s tanh s qn qx ¼ pAnþ1 B pðnþ1Þ cosh s aðtÞA aðtÞq ¼ p cosh s tanh s tanh s bðtÞqxxx ¼ pðp þ 1Þðp þ 2ÞbðtÞAB3 p3 bðtÞAB3 p pþ2 cosh s cosh s qt ¼
ð6Þ ð7Þ ð8Þ ð9Þ
Substituting (6) to (9) into (1) yields
dA 1 dv tanh s þ aðtÞA bðtÞp2 B2 p þ pAB v þ t p dt dt cosh s cosh s pA dB s tanh s tanh s tanh s pAnþ1 B þ pðp þ 1Þðp þ 2ÞbðtÞAB3 ¼0 pþ2 pðnþ1Þ B dt coshp s cosh s cosh s
ð10Þ
From (10), one can say that the last two terms match up, provided the exponent of the cosh functions are the same. This gives
pðn þ 1Þ ¼ p þ 2
ð11Þ
which yields
p¼
2 n
ð12Þ
Also, from (10), one can see that the functions 1/coshps, tanhs/coshps, stanhs/coshps are linearly independent and therefore their coefficients must, respectively vanish. This leads to the following relations
dA þ aðtÞA ¼ 0 dt dv v þ t ¼ bp2 B2 dt
ð13Þ ð14Þ
A. Biswas / Commun Nonlinear Sci Numer Simulat 14 (2009) 3503–3506
3505
and
dB ¼0 dt
ð15Þ
From (13)–(15), one can, respectively conclude
AðtÞ ¼ A0 e
v ðtÞ ¼
2 2
p B t
R
aðtÞdt
Z
ð16Þ
bðtÞdt
ð17Þ
and
BðtÞ ¼ constant
ð18Þ
where A0 is the initial amplitude of the soliton. In order to determine the constant width B, one needs to set the sum of the coefficients of the last two terms, in (10), to zero. This gives
An ¼ bðp þ 1Þðp þ 2ÞB2
ð19Þ
which yields
R #12 An0 en aðtÞdt BðtÞ ¼ n 2bðtÞðn þ 1Þðn þ 2Þ "
ð20Þ
so that from (20), one needs to have b(t) > 0. In order to conform to the fact that B(t) must be a constant, one needs to have the constraint of the time-dependent coefficients a(t) and b(t) from (20) related as
bðtÞ ¼ ke
n
R
aðtÞdt
ð21Þ
for some positive constant k. Thus, the solitary wave solution to (1) is finally given by
qðx; tÞ ¼
AðtÞ 2 n
cosh ½BðtÞðx v ðtÞtÞ
ð22Þ
where the soliton parameters A(t), B(t) and t(t) are, respectively given by (16), (20) and (17) while the coefficients a(t) and b(t) are related as given in (21).
3. Example In this section, the same example, as considered before [3] will be studied, to illustrate the above technique. The gKdV equation that is considered here is
qt þ qn qx þ aq þ cenat qxxx ¼ 0
ð23Þ
where a and c are constants here. Starting with the same ansatze as in (2), Eq. (10) modifies to
dA 1 pA dB s tanh s tanh s pAnþ1 B þ aA p p pðnþ1Þ dt cosh s B dt cosh s cosh s dv tanh s 3 nat tanh s 2 2 nat þ pAB v þ t ¼0 cp B e p þ pðp þ 1Þðp þ 2ÞcAB e pþ2 dt cosh s cosh s
ð24Þ
which leads to the solitary wave solution
qðx; tÞ ¼
A 2 n
cosh ½Bðx v tÞ
ð25Þ
where
AðtÞ ¼ A0 eat
ð26Þ
n B ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2cðn þ 1Þðn þ 2Þ
ð27Þ
and
v ðtÞ ¼
2enat atnðn þ 1Þðn þ 2Þ
ð28Þ
3506
A. Biswas / Commun Nonlinear Sci Numer Simulat 14 (2009) 3503–3506
The constraint relation (21) is meaningful here as c is a constant as seen from the hypothesis. From (26), it can be seen that
limt!1 AðtÞ ¼
0;
: a>0
1; : a < 0
ð29Þ
while from (28), one can conclude that
limt!1 v ðtÞ ¼
0;
: a>0
1; : a < 0
ð30Þ
which shows that the amplitude and velocity of the soliton gradually die down, provided a > 0. Thus, the solitary wave solution of (23) is given by (25) with the respective parameters defined in (26)–(28). 4. Conclusions In this paper, solitary wave ansatze is used to integrate the gKdV equation with linear, time-dependent damping and time-dependent dispersion. The 1-soliton solution is obtained using this method. In addition, a concrete example is taken to illustrate the method with particular form of time-dependent coefficients that conform to the constraints of these coefficients. Acknowledgements This research work 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. References [1] Antonova M, Biswas A. Adiabatic parameter dynamics of perturbed solitary waves. Commun Nonlin Sci Numer Simulat 2009;14(3):734–48. [2] Biswas A, Konar S. Soliton perturbation theory for the compound KdV equation. Int J Theoret Phys 2007;46(2):237–43. [3] Mayil Vaganan B, Kumaran MS. Exact linearization and invariant solutions of the generalized Burger’s equation with linear damping and variable viscosity. Stud Appl Math 2006;117:95–108. [4] Senthilkumaran M, Pandiaraja D, Mayil Vaganan B. New explicit solutions of the generalized KdV equations. Appl Math Comput 2008;202(2):693–9. [5] Krishnan EV, Peng Y. Exact solutions to the combined KdV–mKdV equation by the extended mapping method. Phys Scripta 2006;73:405–9. [6] Krishnan EV, Khan QJA. Higher order KdV-type equations and their stability. Int J Math Math Sci 2001;27(4):215–20. [7] Li J, Xu T, Meng X-H, Zhang Y-X, Zhang H-Q, Tian B, et al. Bäcklund transformation and N-soliton like solution for a variable coefficient Gardner equation from nonlinear lattice, plasma physics and ocean dynamics with symbolic computation. J Math Anal Appl 2007;336(2):1443–55. [8] Wazwaz A-M. New sets of solitary wave solutions to the KdV, mKdV and generalized KdV equations. Commun Nonlin Sci Numer Simulat 2008;13(2):331–9. [9] Xiao-Yan T, Fei H, Sen-Yue L. Variable coefficient KdV equation and the analytical diagnosis of a dipole blocking life cycle. Chinese Phys Lett 2006;23:887–90. [10] Zhidkov PE. Korteweg–de Vries and nonlinear Schrödinger’s equations: qualitative theory. New York, NY: Springer-Verlag; 2001.
Contents lists available at ScienceDirect
Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns
Short communication
Solitary wave solution for the generalized KdV equation with time-dependent damping and dispersion Anjan Biswas * 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
a r t i c l e
i n f o
Article history: Received 3 September 2008 Received in revised form 19 September 2008 Accepted 19 September 2008 Available online 10 October 2008
a b s t r a c t The solitary wave solution of the generalized KdV equation is obtained in this paper in presence of time-dependent damping and dispersion. The approach is from a solitary wave ansatze that leads to the exact solution. A particular example is also considered to complete the analysis. Ó 2008 Elsevier B.V. All rights reserved.
MSC: 35Q51 35Q53 37K10 PACS codes: 02.30.Jr 02.30.Ik Keywords: Solitary wave Integrability Dispersion Damping
1. Introduction The study of nonlinear evolution equation is going on for the past few decades. It is an important area of study in the fields of Physics and Mathematics. In particular, in the last couple of decades there has been a large number of papers published that talks about the integrability aspects of various nonlinear evolution equations that proved to be a failure by the Painleve test of integrability [1–10]. Thus, those equations failed to be integrable by the classical method of integration, namely the inverse scattering transform (IST) which is the nonlinear analog of Fourier transform that is used to integrate linear partial differential equations. In the present times, there are a bunch of new techniques for carrying out the integration of various nonlinear evolution equations. These new methods include the Wadati trace method, pseudo-spectral method, tanh-sech method, sine-cosine method and the Riccati equation expansion method, exponential function method and many more. Without these modern methods of integrability, many such equations would not have been solved.
* Tel.: +1 302 659 0169; fax: +1 302 857 7517. E-mail address: [email protected] 1007-5704/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2008.09.026
3504
A. Biswas / Commun Nonlinear Sci Numer Simulat 14 (2009) 3503–3506
In this paper, the solitary wave ansatze will be exploited to find the 1-soliton solution of the generalized Korteweg–de Vries equation (gKdV), in presence of linear, time-dependent damping and time-dependent dispersion.
2. Mathematical analysis The gKdV equation with time-dependent damping and dispersion that is going to be studied in this paper is given by
qt þ qn qx þ aðtÞq þ bðtÞqxxx ¼ 0
ð1Þ
Here the first term is the evolution term, while the second term represents the nonlinear term. The third term is the linear damping with a time-dependent coefficient a(t) while the fourth term is the dispersion term with time-dependent coefficient b(t). In (1), a and b 2 R while n 2 Z+. This equation arises in various physical situations including the study of coastal waves in ocean, liquid drops and bubbles [3]. It also arises in the context of atmospheric blocking phenomenon and in particular in the issue of dipole blocking [3,9]. It is not possible to integrate this equation by the IST as the Painleve test will fail. However, a particular form of (1) was recently studied by the aid of Lie transform [3]. In this paper, the solitary wave ansatze will enable to find the solution to (1) for a general connection between the functions a(t) and b(t) as long as they are Riemann integrable. Without any loss of generality it is assumed that the solitary wave solution to (1) is given by
qðx; tÞ ¼
AðtÞ p cosh ½BðtÞðx v ðtÞtÞ
ð2Þ
where A represents the amplitude of the soliton, while B is the inverse width of the soliton and t represents the velocity of the soliton. It needs to be noted that since damping and dispersion have time-dependent coefficients, one needs to have, in general,
A ¼ AðtÞ
ð3Þ
B ¼ BðtÞ
ð4Þ
v ¼ v ðtÞ
ð5Þ
and
Thus from (2), one gets,
dA 1 dv tanh s pA dB s tanh s þ pAB v þ t dt coshp s dt coshp s B dt coshp s tanh s qn qx ¼ pAnþ1 B pðnþ1Þ cosh s aðtÞA aðtÞq ¼ p cosh s tanh s tanh s bðtÞqxxx ¼ pðp þ 1Þðp þ 2ÞbðtÞAB3 p3 bðtÞAB3 p pþ2 cosh s cosh s qt ¼
ð6Þ ð7Þ ð8Þ ð9Þ
Substituting (6) to (9) into (1) yields
dA 1 dv tanh s þ aðtÞA bðtÞp2 B2 p þ pAB v þ t p dt dt cosh s cosh s pA dB s tanh s tanh s tanh s pAnþ1 B þ pðp þ 1Þðp þ 2ÞbðtÞAB3 ¼0 pþ2 pðnþ1Þ B dt coshp s cosh s cosh s
ð10Þ
From (10), one can say that the last two terms match up, provided the exponent of the cosh functions are the same. This gives
pðn þ 1Þ ¼ p þ 2
ð11Þ
which yields
p¼
2 n
ð12Þ
Also, from (10), one can see that the functions 1/coshps, tanhs/coshps, stanhs/coshps are linearly independent and therefore their coefficients must, respectively vanish. This leads to the following relations
dA þ aðtÞA ¼ 0 dt dv v þ t ¼ bp2 B2 dt
ð13Þ ð14Þ
A. Biswas / Commun Nonlinear Sci Numer Simulat 14 (2009) 3503–3506
3505
and
dB ¼0 dt
ð15Þ
From (13)–(15), one can, respectively conclude
AðtÞ ¼ A0 e
v ðtÞ ¼
2 2
p B t
R
aðtÞdt
Z
ð16Þ
bðtÞdt
ð17Þ
and
BðtÞ ¼ constant
ð18Þ
where A0 is the initial amplitude of the soliton. In order to determine the constant width B, one needs to set the sum of the coefficients of the last two terms, in (10), to zero. This gives
An ¼ bðp þ 1Þðp þ 2ÞB2
ð19Þ
which yields
R #12 An0 en aðtÞdt BðtÞ ¼ n 2bðtÞðn þ 1Þðn þ 2Þ "
ð20Þ
so that from (20), one needs to have b(t) > 0. In order to conform to the fact that B(t) must be a constant, one needs to have the constraint of the time-dependent coefficients a(t) and b(t) from (20) related as
bðtÞ ¼ ke
n
R
aðtÞdt
ð21Þ
for some positive constant k. Thus, the solitary wave solution to (1) is finally given by
qðx; tÞ ¼
AðtÞ 2 n
cosh ½BðtÞðx v ðtÞtÞ
ð22Þ
where the soliton parameters A(t), B(t) and t(t) are, respectively given by (16), (20) and (17) while the coefficients a(t) and b(t) are related as given in (21).
3. Example In this section, the same example, as considered before [3] will be studied, to illustrate the above technique. The gKdV equation that is considered here is
qt þ qn qx þ aq þ cenat qxxx ¼ 0
ð23Þ
where a and c are constants here. Starting with the same ansatze as in (2), Eq. (10) modifies to
dA 1 pA dB s tanh s tanh s pAnþ1 B þ aA p p pðnþ1Þ dt cosh s B dt cosh s cosh s dv tanh s 3 nat tanh s 2 2 nat þ pAB v þ t ¼0 cp B e p þ pðp þ 1Þðp þ 2ÞcAB e pþ2 dt cosh s cosh s
ð24Þ
which leads to the solitary wave solution
qðx; tÞ ¼
A 2 n
cosh ½Bðx v tÞ
ð25Þ
where
AðtÞ ¼ A0 eat
ð26Þ
n B ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2cðn þ 1Þðn þ 2Þ
ð27Þ
and
v ðtÞ ¼
2enat atnðn þ 1Þðn þ 2Þ
ð28Þ
3506
A. Biswas / Commun Nonlinear Sci Numer Simulat 14 (2009) 3503–3506
The constraint relation (21) is meaningful here as c is a constant as seen from the hypothesis. From (26), it can be seen that
limt!1 AðtÞ ¼
0;
: a>0
1; : a < 0
ð29Þ
while from (28), one can conclude that
limt!1 v ðtÞ ¼
0;
: a>0
1; : a < 0
ð30Þ
which shows that the amplitude and velocity of the soliton gradually die down, provided a > 0. Thus, the solitary wave solution of (23) is given by (25) with the respective parameters defined in (26)–(28). 4. Conclusions In this paper, solitary wave ansatze is used to integrate the gKdV equation with linear, time-dependent damping and time-dependent dispersion. The 1-soliton solution is obtained using this method. In addition, a concrete example is taken to illustrate the method with particular form of time-dependent coefficients that conform to the constraints of these coefficients. Acknowledgements This research work 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. References [1] Antonova M, Biswas A. Adiabatic parameter dynamics of perturbed solitary waves. Commun Nonlin Sci Numer Simulat 2009;14(3):734–48. [2] Biswas A, Konar S. Soliton perturbation theory for the compound KdV equation. Int J Theoret Phys 2007;46(2):237–43. [3] Mayil Vaganan B, Kumaran MS. Exact linearization and invariant solutions of the generalized Burger’s equation with linear damping and variable viscosity. Stud Appl Math 2006;117:95–108. [4] Senthilkumaran M, Pandiaraja D, Mayil Vaganan B. New explicit solutions of the generalized KdV equations. Appl Math Comput 2008;202(2):693–9. [5] Krishnan EV, Peng Y. Exact solutions to the combined KdV–mKdV equation by the extended mapping method. Phys Scripta 2006;73:405–9. [6] Krishnan EV, Khan QJA. Higher order KdV-type equations and their stability. Int J Math Math Sci 2001;27(4):215–20. [7] Li J, Xu T, Meng X-H, Zhang Y-X, Zhang H-Q, Tian B, et al. Bäcklund transformation and N-soliton like solution for a variable coefficient Gardner equation from nonlinear lattice, plasma physics and ocean dynamics with symbolic computation. J Math Anal Appl 2007;336(2):1443–55. [8] Wazwaz A-M. New sets of solitary wave solutions to the KdV, mKdV and generalized KdV equations. Commun Nonlin Sci Numer Simulat 2008;13(2):331–9. [9] Xiao-Yan T, Fei H, Sen-Yue L. Variable coefficient KdV equation and the analytical diagnosis of a dipole blocking life cycle. Chinese Phys Lett 2006;23:887–90. [10] Zhidkov PE. Korteweg–de Vries and nonlinear Schrödinger’s equations: qualitative theory. New York, NY: Springer-Verlag; 2001.