- Email: [email protected]
Exact 1-soliton solution of the Zakharov equation in plasmas withpower law nonlinearity
Applied Mathematics and Computation 217 (2011) 7372–7375
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Exact 1-soliton solution of the Zakharov equation in plasmas with power law nonlinearity Pablo Suarez a, Anjan Biswas b,⇑ a
Department of Mathematical Sciences, Delaware State University, Dover, DE 19901-2277, USA Department of Mathematical Sciences, Center for Research and Education in Optical Sciences and Applications, 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 studies the Zakharov equation with power law nonlinearity. An exact 1-soliton solution is obtained by the ansatz method. The parameter regimes are identified in the process. The numerical simulation is also given to complete the study. Ó 2011 Elsevier Inc. All rights reserved.
Keywords: Solitons Integrability Exact solution
1. Introduction The study of nonlinear evolution equations (NLEES) appear in a lot of places in Applied Mathematics and Theoretical Physics [1–10]. They also appear in Mathematical Biology, Mathematical Chemistry, Nuclear physics, Fluid Dynamics, Plasma Physics, Nonlinear Optics and many other places. In this paper, the Zakharov equation (ZE), that appears in Plasma Physics, will be studied both analytically and numerically. There are various mathematical tricks and techniques to solve these NLEEs analytically. Besides the age-old method of the then popular Inverse Scattering Transform method, there are various modern methods of integrability that has been developed and hence popular, since the last decade or so. Some of these popular methods are Adomian decomposition method, G0 /G method, exponential function method, Riccati’s method, variational iteration method, homotopy perturbation method, just to name a few. However, one has to be very careful in applying these techniques. These methods could lead to incorrect solutions as pointed out by Kudryashov [5]. In this paper, however, there will be one such method of integrability that will be used to integrate the ZE. It is the solitary wave ansatz method. Subsequently, the numerical simulations will be given in the next section. It needs to be noted that this ZE was already studied earlier in 2010, where an analytical solution was obtained by He’s semi-inverse variational principle, although the solution obtained there was not exact. 2. Mathematical analysis The dimensionless form of the Zakharov equation (ZE) that is going to be studied in this paper is given by Abbasbandy et al. [1], Javidi and Golbabai [4]
iqt þ aqxx þ bF jqj2 q ¼ qr; 2
rtt k r xx ¼ ðjqj2m Þxx :
ð1Þ ð2Þ
In (1) and (2), x and t represents the spatial and temporal variables respectively. In (1), a and b represent the coefficients of dispersion and nonlinearity. Also, the two dependent variables are q and r where q is complex valued dependent variable ⇑ Corresponding author. E-mail address: [email protected] (A. Biswas). 0096-3003/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2011.02.036
P. Suarez, A. Biswas / Applied Mathematics and Computation 217 (2011) 7372–7375
7373
while r is a real-valued dependent variable. In this paper, the focus is on obtaining the exact 1-soliton solution to (1) and (2) and then solving the system numerically. Solitons are the result of a delicate balance between dispersion and nonlinearity. Eqs. (1) and (2) thus form a coupled system. Here, F is a real-valued algebraic nonlinear function and it is necessary to have the smoothness of the complex function F(jqj2)q : C ´ C. Considering the complex plane C as a two-dimensional linear space R2, the function F(jqj2)q is k times continuously differentiable, so that [1] 1 [
Fðjqj2 Þq 2
C k ððn; nÞ ðm; mÞ; R2 Þ:
ð3Þ
m;n¼1
In order to seek a 1-soliton solution to (1) and (2) with F(s) = s, the starting hypothesis is given by
qðx; tÞ ¼
A1 ei/ p cosh 1 s
ð4Þ
rðx; tÞ ¼
A2 ; p cosh 2 s
ð5Þ
and
where
s ¼ Bðx v tÞ
ð6Þ
/ ¼ jx þ xt þ h:
ð7Þ
and
Here, in (4)–(7), A1 and A2 represent the respective amplitudes of the solitons and v is the velocity of the soliton while B is the inverse widths of the solitons. Then, j is the frequency of the soliton, while x is the wave number of the soliton and h is the phase constant. The unknown exponents p1 and p2 will be determined by the application of the balancing principle [8]. Now, from (4) and (5),
iqt ¼
ip1 v A1 B tanh s xA1 ei/ ; p p cosh 1 s cosh 1 s
ð8Þ
(
) p21 A1 B2 p1 ðp1 þ 1ÞA1 B2 2ijp1 A1 B tanh s j2 A1 ei/ ; þ p p p p þ2 cosh 1 s cosh 1 s cosh 1 s cosh 1 s
qxx ¼
ð9Þ
qr ¼
A1 A2 ei/ ; p þp cosh 1 2 s
ð10Þ
rtt ¼
p22 v 2 A2 B2 p2 ðp2 þ 1ÞA2 B2 ; p p þ2 cosh 2 s cosh 2 s
ð11Þ
rxx ¼
p22 A2 B2 p2 ðp2 þ 1ÞA2 B2 ; p p þ2 cosh 2 s cosh 2 s
ð12Þ
ðjqj2m Þxx ¼
2 4m2 p21 A2m 1 B
cosh
2mp1
s
2 2mp1 ð2mp1 þ 1ÞA2m 1 B
cosh
2mp1 þ2
s
:
ð13Þ
Substituting (8)–(13) into (1) and (2) yields 2mþ1
ip1 v A1 B tanh s xA1 bA1 ap2 A1 B2 p ðp þ 1ÞA1 B2 2iajp1 A1 B tanh s aj2 A1 A1 A2 þ 1 p1 1 1 p þ2 þ ¼ p1 p1 þ p1 p p þp ð2mþ1Þp 1 cosh s cosh s cosh cosh s cosh 1 s cosh 1 2 s cosh s s cosh s
ð14Þ
and 2
2
2 2 p22 A2 B2 ðv 2 k Þ p2 ðp2 þ 1ÞA2 B2 ðv 2 k Þ 4m2 p21 A2m 2mp1 ð2mp1 þ 1ÞA2m 1 B 1 B ¼ : p2 p þ2 2mp 2mp þ2 cosh s cosh 1 s cosh 1 s cosh 2 s
ð15Þ
Now, splitting (14) into real and imaginary parts, yield
xA1 p1
cosh
2mþ1
þ
bA1
s coshð2mþ1Þp s
þ
ap21 A1 B2 p1 ðp1 þ 1ÞA1 B2 aj2 A1 A1 A2 p p1 ¼ p1 þp2 p1 þ2 cosh 1 s cosh s cosh s cosh s
ð16Þ
7374
P. Suarez, A. Biswas / Applied Mathematics and Computation 217 (2011) 7372–7375
and
ðp1 v A1 B þ 2ajp1 A1 BÞ tanh s ¼ 0: p cosh 1 s
ð17Þ
Eq. (17) gives the velocity of the soliton as
v ¼ 2aj:
ð18Þ
From (15), by balancing principle [8], equating the exponents 2mp1 and p2 gives
p2 ¼ 2mp1 ;
ð19Þ
while from (16), equating the exponents p1 + 2 and p1 + p2 yields
p1 þ 2 ¼ p1 þ p2 ;
ð20Þ
which leads to
p2 ¼ 2
ð21Þ
and hence from (19)
p1 ¼
1 : m
ð22Þ p1 þj
Now from (16), setting the coefficients of the linearly independent functions 1=cosh
s to zero, for j = 1, 2 gives
a x ¼ 2 ðB2 m2 j2 Þ m
ð23Þ
and 2
aðm þ 1ÞB2 þ m2 A2 bm A2m 1 ¼ 0:
ð24Þ p2 þj
Similarly, from (15), the linearly independent functions 1=cosh 2
ðv k ÞA2 ¼ 2
s for j = 1, 2 both yield
A2m 1 :
ð25Þ
Thus finally from the coupled system of Eq. (24) and (25), it is possible to obtain
A1 ¼
1 " #2m 2 aðm þ 1Þðv 2 k ÞB2
ð26Þ
2
m2 fbðv 2 k Þ 1g
and
A2 ¼
aðm þ 1ÞB2 2
m2 fbðv 2 k Þ 1g
ð27Þ
:
From (26) and (27) it is necessary to observe that 2
bðv 2 k Þ – 1:
ð28Þ
Fig. 1. Soliton profiles for the Zakharov equation.
P. Suarez, A. Biswas / Applied Mathematics and Computation 217 (2011) 7372–7375
7375
Thus, finally, the 1-soliton solution to (1) and (2) are respectively given by
qðx; tÞ ¼
A1 1
eiðjxþxtþhÞ
ð29Þ
;
ð30Þ
coshm s
and
rðx; tÞ ¼
A2 2
cosh
s
where the amplitudes A1 and A2 are given by (26) and (27), the velocity of the solitons is given in (18) and the wave number x is given by (23). All of these pose a restriction that is given by (23). The solutions of jq(x, t)j2 and r(x, t) are plotted in Fig. 1(a) and (b), respectively. For this example, the parameter values that are chosen are a = b = 1, n = 1, v = 4, k = 1, B = 1, h = 0, x = p and j = 2. Although these values are chosen arbitarily, the restriction given by (28) is valid for these set of parameter values. 3. Conclusions In this paper, the ZE is solved analytically and 1-soliton solution is obtained. The solitary wave ansatz is used to carry out the integration of the ZE. The numerical simulations are also obtained that support the analytical results. In future, these results will be extended to time-dependent coefficients or with perturbation terms as well as with stochastic perturbation terms. Those results will be published in future. Acknowledgments The research work of the second author (AB) is fully supported by NSF-CREST Grant No: HRD-0630388 and this support is also thankfully appreciated. References [1] S. Abbasbandy, E. Babolian, M. Ashtani, Numerical solution of the generalized Zakharov equation by homotopy analysis method, Communications in Nonlinear Science and Numerical Simulation 14 (12) (2009) 4114–4121. [2] A. Biswas, E. Zerrad, J. Gwanmesia, R. Khouri, 1-soliton solution of the generalized Zakharov equation in plasmas by He’s semi-inverse variational principle, Applied Mathematics and Computation 215 (12) (2010) 4462–4466. [3] B. Guo, J. Zhang, X. Pu, On the existence and uniqueness of smooth solution for a generalized Zakharov equation, Journal of Mathematical Analysis and its Application 365 (1) (2010) 238–253. [4] M. Javidi, M. Golbabai, Construction of a solitary wave solution for the generalized Zakharov equation by a variational iteration method, Computers and Mathematics with Applications 54 (7–8) (2009) 3571–3584. [5] 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) 3507–3529. [6] Y-Z. Li, K-M. Li, C. Lin, Exp-function method for solving the generalized Zakharov equations, Applied Mathematics and Computation 205 (1) (2008) 197–201. [7] S.A.E. Wakil, A.R. Degheidy, E.M. Abulwafa, M.A. Madkour, M.A. Abdou, Exact travelling wave solutions of generalized Zakharov equations with arbitrary power nonlinearities, International Journal of Nonlinear Science 7 (4) (2009) 455–461. [8] A.M. Wazwaz, The extended tanh method for abundant solitary wave solutions of nonlinear wave equations, Applied Mathematics and Computation 187 (2) (2007) 1131–1142. [9] Y. X-Lin, T. J-Shi, Explicit exact solutions for the generalized Zakharov equations with nonlinear terms of any order, Computers and Mathematics with Applications 57 (2009) 1622–1629. [10] J. Zhang, Variational approach to solitary wave solution of the generalized Zakharov equation, Computers and Mathematics with Applications. 54 (2007) 1043–1046.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Exact 1-soliton solution of the Zakharov equation in plasmas with power law nonlinearity Pablo Suarez a, Anjan Biswas b,⇑ a
Department of Mathematical Sciences, Delaware State University, Dover, DE 19901-2277, USA Department of Mathematical Sciences, Center for Research and Education in Optical Sciences and Applications, 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 studies the Zakharov equation with power law nonlinearity. An exact 1-soliton solution is obtained by the ansatz method. The parameter regimes are identified in the process. The numerical simulation is also given to complete the study. Ó 2011 Elsevier Inc. All rights reserved.
Keywords: Solitons Integrability Exact solution
1. Introduction The study of nonlinear evolution equations (NLEES) appear in a lot of places in Applied Mathematics and Theoretical Physics [1–10]. They also appear in Mathematical Biology, Mathematical Chemistry, Nuclear physics, Fluid Dynamics, Plasma Physics, Nonlinear Optics and many other places. In this paper, the Zakharov equation (ZE), that appears in Plasma Physics, will be studied both analytically and numerically. There are various mathematical tricks and techniques to solve these NLEEs analytically. Besides the age-old method of the then popular Inverse Scattering Transform method, there are various modern methods of integrability that has been developed and hence popular, since the last decade or so. Some of these popular methods are Adomian decomposition method, G0 /G method, exponential function method, Riccati’s method, variational iteration method, homotopy perturbation method, just to name a few. However, one has to be very careful in applying these techniques. These methods could lead to incorrect solutions as pointed out by Kudryashov [5]. In this paper, however, there will be one such method of integrability that will be used to integrate the ZE. It is the solitary wave ansatz method. Subsequently, the numerical simulations will be given in the next section. It needs to be noted that this ZE was already studied earlier in 2010, where an analytical solution was obtained by He’s semi-inverse variational principle, although the solution obtained there was not exact. 2. Mathematical analysis The dimensionless form of the Zakharov equation (ZE) that is going to be studied in this paper is given by Abbasbandy et al. [1], Javidi and Golbabai [4]
iqt þ aqxx þ bF jqj2 q ¼ qr; 2
rtt k r xx ¼ ðjqj2m Þxx :
ð1Þ ð2Þ
In (1) and (2), x and t represents the spatial and temporal variables respectively. In (1), a and b represent the coefficients of dispersion and nonlinearity. Also, the two dependent variables are q and r where q is complex valued dependent variable ⇑ Corresponding author. E-mail address: [email protected] (A. Biswas). 0096-3003/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2011.02.036
P. Suarez, A. Biswas / Applied Mathematics and Computation 217 (2011) 7372–7375
7373
while r is a real-valued dependent variable. In this paper, the focus is on obtaining the exact 1-soliton solution to (1) and (2) and then solving the system numerically. Solitons are the result of a delicate balance between dispersion and nonlinearity. Eqs. (1) and (2) thus form a coupled system. Here, F is a real-valued algebraic nonlinear function and it is necessary to have the smoothness of the complex function F(jqj2)q : C ´ C. Considering the complex plane C as a two-dimensional linear space R2, the function F(jqj2)q is k times continuously differentiable, so that [1] 1 [
Fðjqj2 Þq 2
C k ððn; nÞ ðm; mÞ; R2 Þ:
ð3Þ
m;n¼1
In order to seek a 1-soliton solution to (1) and (2) with F(s) = s, the starting hypothesis is given by
qðx; tÞ ¼
A1 ei/ p cosh 1 s
ð4Þ
rðx; tÞ ¼
A2 ; p cosh 2 s
ð5Þ
and
where
s ¼ Bðx v tÞ
ð6Þ
/ ¼ jx þ xt þ h:
ð7Þ
and
Here, in (4)–(7), A1 and A2 represent the respective amplitudes of the solitons and v is the velocity of the soliton while B is the inverse widths of the solitons. Then, j is the frequency of the soliton, while x is the wave number of the soliton and h is the phase constant. The unknown exponents p1 and p2 will be determined by the application of the balancing principle [8]. Now, from (4) and (5),
iqt ¼
ip1 v A1 B tanh s xA1 ei/ ; p p cosh 1 s cosh 1 s
ð8Þ
(
) p21 A1 B2 p1 ðp1 þ 1ÞA1 B2 2ijp1 A1 B tanh s j2 A1 ei/ ; þ p p p p þ2 cosh 1 s cosh 1 s cosh 1 s cosh 1 s
qxx ¼
ð9Þ
qr ¼
A1 A2 ei/ ; p þp cosh 1 2 s
ð10Þ
rtt ¼
p22 v 2 A2 B2 p2 ðp2 þ 1ÞA2 B2 ; p p þ2 cosh 2 s cosh 2 s
ð11Þ
rxx ¼
p22 A2 B2 p2 ðp2 þ 1ÞA2 B2 ; p p þ2 cosh 2 s cosh 2 s
ð12Þ
ðjqj2m Þxx ¼
2 4m2 p21 A2m 1 B
cosh
2mp1
s
2 2mp1 ð2mp1 þ 1ÞA2m 1 B
cosh
2mp1 þ2
s
:
ð13Þ
Substituting (8)–(13) into (1) and (2) yields 2mþ1
ip1 v A1 B tanh s xA1 bA1 ap2 A1 B2 p ðp þ 1ÞA1 B2 2iajp1 A1 B tanh s aj2 A1 A1 A2 þ 1 p1 1 1 p þ2 þ ¼ p1 p1 þ p1 p p þp ð2mþ1Þp 1 cosh s cosh s cosh cosh s cosh 1 s cosh 1 2 s cosh s s cosh s
ð14Þ
and 2
2
2 2 p22 A2 B2 ðv 2 k Þ p2 ðp2 þ 1ÞA2 B2 ðv 2 k Þ 4m2 p21 A2m 2mp1 ð2mp1 þ 1ÞA2m 1 B 1 B ¼ : p2 p þ2 2mp 2mp þ2 cosh s cosh 1 s cosh 1 s cosh 2 s
ð15Þ
Now, splitting (14) into real and imaginary parts, yield
xA1 p1
cosh
2mþ1
þ
bA1
s coshð2mþ1Þp s
þ
ap21 A1 B2 p1 ðp1 þ 1ÞA1 B2 aj2 A1 A1 A2 p p1 ¼ p1 þp2 p1 þ2 cosh 1 s cosh s cosh s cosh s
ð16Þ
7374
P. Suarez, A. Biswas / Applied Mathematics and Computation 217 (2011) 7372–7375
and
ðp1 v A1 B þ 2ajp1 A1 BÞ tanh s ¼ 0: p cosh 1 s
ð17Þ
Eq. (17) gives the velocity of the soliton as
v ¼ 2aj:
ð18Þ
From (15), by balancing principle [8], equating the exponents 2mp1 and p2 gives
p2 ¼ 2mp1 ;
ð19Þ
while from (16), equating the exponents p1 + 2 and p1 + p2 yields
p1 þ 2 ¼ p1 þ p2 ;
ð20Þ
which leads to
p2 ¼ 2
ð21Þ
and hence from (19)
p1 ¼
1 : m
ð22Þ p1 þj
Now from (16), setting the coefficients of the linearly independent functions 1=cosh
s to zero, for j = 1, 2 gives
a x ¼ 2 ðB2 m2 j2 Þ m
ð23Þ
and 2
aðm þ 1ÞB2 þ m2 A2 bm A2m 1 ¼ 0:
ð24Þ p2 þj
Similarly, from (15), the linearly independent functions 1=cosh 2
ðv k ÞA2 ¼ 2
s for j = 1, 2 both yield
A2m 1 :
ð25Þ
Thus finally from the coupled system of Eq. (24) and (25), it is possible to obtain
A1 ¼
1 " #2m 2 aðm þ 1Þðv 2 k ÞB2
ð26Þ
2
m2 fbðv 2 k Þ 1g
and
A2 ¼
aðm þ 1ÞB2 2
m2 fbðv 2 k Þ 1g
ð27Þ
:
From (26) and (27) it is necessary to observe that 2
bðv 2 k Þ – 1:
ð28Þ
Fig. 1. Soliton profiles for the Zakharov equation.
P. Suarez, A. Biswas / Applied Mathematics and Computation 217 (2011) 7372–7375
7375
Thus, finally, the 1-soliton solution to (1) and (2) are respectively given by
qðx; tÞ ¼
A1 1
eiðjxþxtþhÞ
ð29Þ
;
ð30Þ
coshm s
and
rðx; tÞ ¼
A2 2
cosh
s
where the amplitudes A1 and A2 are given by (26) and (27), the velocity of the solitons is given in (18) and the wave number x is given by (23). All of these pose a restriction that is given by (23). The solutions of jq(x, t)j2 and r(x, t) are plotted in Fig. 1(a) and (b), respectively. For this example, the parameter values that are chosen are a = b = 1, n = 1, v = 4, k = 1, B = 1, h = 0, x = p and j = 2. Although these values are chosen arbitarily, the restriction given by (28) is valid for these set of parameter values. 3. Conclusions In this paper, the ZE is solved analytically and 1-soliton solution is obtained. The solitary wave ansatz is used to carry out the integration of the ZE. The numerical simulations are also obtained that support the analytical results. In future, these results will be extended to time-dependent coefficients or with perturbation terms as well as with stochastic perturbation terms. Those results will be published in future. Acknowledgments The research work of the second author (AB) is fully supported by NSF-CREST Grant No: HRD-0630388 and this support is also thankfully appreciated. References [1] S. Abbasbandy, E. Babolian, M. Ashtani, Numerical solution of the generalized Zakharov equation by homotopy analysis method, Communications in Nonlinear Science and Numerical Simulation 14 (12) (2009) 4114–4121. [2] A. Biswas, E. Zerrad, J. Gwanmesia, R. Khouri, 1-soliton solution of the generalized Zakharov equation in plasmas by He’s semi-inverse variational principle, Applied Mathematics and Computation 215 (12) (2010) 4462–4466. [3] B. Guo, J. Zhang, X. Pu, On the existence and uniqueness of smooth solution for a generalized Zakharov equation, Journal of Mathematical Analysis and its Application 365 (1) (2010) 238–253. [4] M. Javidi, M. Golbabai, Construction of a solitary wave solution for the generalized Zakharov equation by a variational iteration method, Computers and Mathematics with Applications 54 (7–8) (2009) 3571–3584. [5] 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) 3507–3529. [6] Y-Z. Li, K-M. Li, C. Lin, Exp-function method for solving the generalized Zakharov equations, Applied Mathematics and Computation 205 (1) (2008) 197–201. [7] S.A.E. Wakil, A.R. Degheidy, E.M. Abulwafa, M.A. Madkour, M.A. Abdou, Exact travelling wave solutions of generalized Zakharov equations with arbitrary power nonlinearities, International Journal of Nonlinear Science 7 (4) (2009) 455–461. [8] A.M. Wazwaz, The extended tanh method for abundant solitary wave solutions of nonlinear wave equations, Applied Mathematics and Computation 187 (2) (2007) 1131–1142. [9] Y. X-Lin, T. J-Shi, Explicit exact solutions for the generalized Zakharov equations with nonlinear terms of any order, Computers and Mathematics with Applications 57 (2009) 1622–1629. [10] J. Zhang, Variational approach to solitary wave solution of the generalized Zakharov equation, Computers and Mathematics with Applications. 54 (2007) 1043–1046.