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Exact explicit travelling wave solutions for (n + 1)-dimensional Klein–Gordon–Zakharov equations
Chaos, Solitons and Fractals 34 (2007) 867–871 www.elsevier.com/locate/chaos
Exact explicit travelling wave solutions for (n + 1)dimensional Klein–Gordon–Zakharov equations q Jibin Li Department of Mathematics, Zhejiang Normal University, Jinhua, Zhejiang 321004, PR China Center for Nonlinear Science Studies, Kunming University of Science and Technology, Kunming, Yunnan 650093, PR China Accepted 23 March 2006
Communicated by Prof. M. Wadati
Abstract Using the methods of dynamical systems for the (n + 1)-dimensional KGS nonlinear wave equations, five classes of exact explicit parametric representations of the bounded travelling solutions are obtained. To guarantee the existence of the above solutions, all parameter conditions are given. Ó 2006 Elsevier Ltd. All rights reserved.
1. Introduction It is known that the following Klein–Gordon–Zakharov equations (KGZ-equations in short) are governed by wtt c2 Dw ¼ Dj/j2 ;
/tt D/ þ / ¼ /w;
ð1Þ
where D ¼ oxo2 þ oxo2 þ þ oxo2 is the Laplacian operator, x 2 Rn, c is the propagation speed of a wave. Eqs. (1) are a cou1
2
n
pled nonlinear wave model which describes the interaction of the Langmuir wave and the ion acoustic wave in a plasma and so on (see [1,2,9]). The real value function / is the fast time scale component of electric field raised by electrons and the real value function w is the deviation of ion density from its equilibrium. For the case n = 3, Ozawa et al. [6–8] studied the local well-posedness for the Cauchy problem in the energy space and the global existence of small amplitude solutions of (1). Gan and Zhang [3] derived a sharp condition of global existence, stability and existence of standing wave of (1). To our knowledge, a general integer n, exact explicit travelling wave solutions of KGZ equation (1) have not been considered before. In this paper, we shall consider the existence of exact explicit bounded travelling wave solutions of (1), by using the methods of dynamical systems (see [5]). In different regions of the parametric space, explicit parameter conditions to guarantee the above solutions are given.
q This research was supported by the National Natural Science Foundation of China (10231020) and the Natural Science Foundation of Yunnan Province (2005A0092M). E-mail addresses: [email protected], [email protected]
0960-0779/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2006.03.088
868
J. Li / Chaos, Solitons and Fractals 34 (2007) 867–871
2. The reduced travelling wave system of (1) To find travelling wave solutions of KGZ equations, we assume that /ðx; tÞ ¼ eig uðnÞ;
x 2 Rn ; t 2 R;
wðx; tÞ ¼ vðnÞ;
ð2Þ
where g¼
n X
aj xj þ bt;
n¼
j¼1
n X
cj xj rt:
ð3Þ
j¼1
Substituting (2) into (1), cancelling out eig and separating the real and imaginary parts, we have 8 ! ! ! n n n > P P P > 2 2 2 2 > > c r unn aj þ 1 b u uv ¼ 0; aj cj þ br un ¼ 0; > < j¼1 j j¼1 j¼1 ! ! > n n > P P > 2 2 2 2 2 > > : c j¼1 cj r vnn þ 2 j¼1 cj ðun þ uunn Þ ¼ 0:
ð4Þ
We see from (4) that to have non-constant solutions of (1) depending on n, we must suppose that n X
aj cj þ br ¼ 0:
ð5Þ
j¼1
Denote that a¼
n X
a2j þ 1 b2 ;
l¼
j¼1
n X
c2j r2 ;
j¼1
c¼
n X
c2j ;
j¼1
d ¼ c2
n X
c2j r2 :
j¼1
Thus, we obtain the following ordinary differential system: ( lunn au uv ¼ 0; dvnn þ 2cðu2n þ uunn Þ ¼ 0:
ð6Þ
We next consider the dynamical behavior of (6) on the super-surface v Au2 = 0 in the four-dimensional phase space _ v_ Þ, where ‘‘Æ’’ stand for the derivative with respect to n. ðu; v; u; Substituting v = Au2 to (6) and taking Pn 2 c j¼1 cj P ; ð7Þ A¼ ¼ 2 d r c2 nj¼1 c2j then the second equation of system (6) vanishes and (6) becomes lunn au Au3 ¼ 0: Suppose that l 5 0 and write that a¼
a ¼ l
Pn 2 2 j¼1 aj þ 1 b P ; n 2 2 j¼1 cj r
b¼
Pn 2 A j¼1 cj P : ¼ Pn 2 2 l ð j¼1 cj r Þðr2 c2 nj¼1 c2j Þ
ð8Þ
Thus, we have the following equation: unn au bu3 ¼ 0; which corresponds to the two-dimensional Hamiltonian system du ¼ y; dn
dy ¼ au þ bu3 dn
ð9Þ
J. Li / Chaos, Solitons and Fractals 34 (2007) 867–871
869
with the Hamiltonian H ðu; yÞ ¼ 12au2 þ 12y 2 14bu4 ¼ h:
ð10Þ
Clearly, (9) is well-known Duffing oscillator system (see [4]). 3. The parametric representations of bounded solutions of system (9) In this section, we consider the dynamics of the phase orbits of (9) in its parameter space. By qualitative analysis and using the Jacobian elliptic functions (see [1]), we have the following results. (1) a > 0, b > 0. The origin O(0, 0) of (9) is an unique equilibrium which is a saddle point. There is no bounded orbit of u(n) for n 2 (1, 1). (2) a < 0, b < 0. The origin O(0, 0) of (9) is an unique equilibrium point which is a center. There exists a family of peri½ 4h 2ab u2 u4 ¼ ðbÞ ½ðz21 þ u2 Þðz22 u2 Þ, odic orbits of (9) enclosing the origin. We see from (10) that y 2 ¼ ðbÞ 2 ðbÞ 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 where z21 ¼ ðbÞ ½ðaÞ þ a2 4hb, z22 ¼ ðbÞ ½a þ a2 4hb. The family of periodic orbits defined by H(u, y) = h, h 2 (0, 1) of (10) has the parametric representation ! rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðbÞðz21 þ z22 Þ z2 ffi : uðnÞ ¼ z2 cn n; pffiffiffiffiffiffiffiffiffiffiffiffiffi 2 z21 þ z22
ð11Þ qffiffiffiffiffiffiffi (3) a < 0, b > 0. There exist three equilibrium points of (9) A ðaÞ ; 0 , O(0, 0). O is a center, A± are saddle b 2
a points. The heteroclinic orbits defined by H ðu; yÞ ¼ 4b ¼ h1 have the parametric representations rffiffiffi rffiffiffiffiffiffiffi a b uðnÞ ¼ tanh n: b 2
ð12Þ
2
a Þ, it can be written as y 2 ¼ b2 ½4hb þ 2ab u2 þ u4 ¼ We see from (10) that y 2 ¼ 2h þ au2 þ b2 u4 . For h 2 ð0; 4b pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b 2 2 2 2 2 1 2 1 2 ½ðr1 u Þðr2 u Þ, where r1 ¼ b ½ðaÞ þ a 4hb, r2 ¼ b ½ðaÞ a2 4hb. Thus, the family of periodic 2
orbits defined by H(u, y) = h has the parametric representation ! rffiffiffi b r2 uðnÞ ¼ r2 sn : ð13Þ r1 n; 2 r1 qffiffiffiffiffiffiffi ; 0 , O(0, 0). O is a saddle point, A± are center (4) a > 0, b < 0. There exist three equilibrium points of (9) A ðaÞ b points. Two homoclinic orbits defined by H(u, y) = 0 have the parametric representations rffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffi 2a aðbÞ uðnÞ ¼ n: sech b 2
ð14Þ
2a ½ 4h þ ðbÞ u2 u4 ¼ ðbÞ ½ðz22 u2 Þðu2 ðz21 ÞÞ, where z21 , z22 are For h 2 ð4b ; 0Þ, we have from (10) that y 2 ¼ ðbÞ 2 ðbÞ 2 a2
the same as the case (2). Thus, two families of periodic orbits defined by H(u, y) = h have the parametric representations rffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi! z22 þ z21 ðbÞ uðnÞ ¼ z2 dn : ð15Þ z2 n; z2 2 The family of periodic orbits of (9) enclosing three equilibrium points A± and O defined by H(u, y) = h, h 2 (0, 1) has the same parametric representations as (11).
4. Conclusion By using the results of Sections 2 and 3 and noting w(x, t) = v(n) = Au2(n), we obtain the following conclusion of this paper. Theorem 1. Suppose that the parameters A, a, b are given by (7), (8) and z21 , z22 , r21 , r22 are defined by Section 3. In addition, the condition (5) holds. Then, the (n + 1)-dimensional KGZ equation (1) have the following five formulas of exact explicit bounded travelling wave solutions.
870
J. Li / Chaos, Solitons and Fractals 34 (2007) 867–871
(1) When a < 0, b < 0, 8 n P ! ! > qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > i a x þbt j j n > > ðbÞðz21 þz22 Þ P z2 > j¼1 > cj xj rt ; pffiffiffiffiffiffiffiffi z2 cn ; < /ðx; tÞ ¼ e 2 2 2 z1 þz2
j¼1
! ! > qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > n > ðbÞðz21 þz22 Þ P z2 2 > 2 > cj xj rt ; pffiffiffiffiffiffiffiffi : : wðx; tÞ ¼ Az2 cn 2 2 2
ð16Þ
z1 þz2
j¼1
(2) When a < 0, b > 0, 8 n P ! > > qffiffi P i aj xj þbt pffiffiffiffi n > > a b > j¼1 > tanh 2 cj xj rt ; < /ðx; tÞ ¼ e b j¼1
! > > qffiffi P > n > 2 > a b > cj xj rt : wðx; tÞ ¼ Að b Þtanh 2
ð17Þ
j¼1
and
8 n P ! ! > > qffiffi i aj xj þbt n > P > b > j¼1 > r cj xj rt ; rr21 ; sn < /ðx; tÞ ¼ r2 e 2 1 j¼1
! ! > > qffiffi > n P > 2 > 2 b > cj xj rt ; rr21 : r : wðx; tÞ ¼ Ar2 sn 2 1
ð18Þ
j¼1
(3) When a > 0, b < 0, 8 n P ! > qffiffiffiffiffiffiffiffi P > i aj xj þbt qffiffiffiffi n > > aðbÞ 2a > j¼1 > cj xj rt ; sech < /ðx; tÞ ¼ e 2 b j¼1
! > > qffiffiffiffiffiffiffiffi P > n > aðbÞ 2 > 2a > cj xj rt : wðx; tÞ ¼ AðbÞsech 2
ð19Þ
j¼1
and
8 n P ! pffiffiffiffiffiffiffiffi! > qffiffiffiffiffiffiffi > i aj xj þbt n > P z2 þz2 > ðbÞ > j¼1 > cj xj rt ; z22 1 ; z2 dn z < /ðx; tÞ ¼ e 2 2 j¼1
! pffiffiffiffiffiffiffiffi! > > qffiffiffiffiffiffiffi > n P > z2 þz2 ðbÞ 2 2 > > cj xj rt ; z22 1 : z : wðx; tÞ ¼ Az2 dn 2 2
ð20Þ
j¼1
References [1] Byrd PF, Fridman MD. Handbook of elliptic integrals for engineers and scientists. Berlin: Springer; 1971. [2] Dendy RO. Plasma dynamics. Oxford: Oxford University Press; 1990. [3] Gan Zaihui, Zhang Jian. Instability of standing wave for Klein–Gordon–Zakharov equations with different propagation speeds in three space dimensions. J Math Anal Appl 2005;307(1):219–31. [4] Guckenheimer J, Holmes PJ. Nonlinear oscillations, dynamical systems and bifurcations of vector fields. New York: SpringerVerlag; 1983. [5] Li Jibin. Solitary and periodic traveling wave solutions in Klein–Gordon–Schrodinger equations. J Yunnan Univ 2003;25(3): 176–80. [6] Ozawa T, Tsutaya K, Tsutsumi Y. Normal form and global solutions for the Klein–Gordon–Zakharov equations. Ann Inst H Poincare Anal Non Lineaire 1995;12:459–503.
J. Li / Chaos, Solitons and Fractals 34 (2007) 867–871
871
[7] Ozawa T, Tsutaya K, Tsutsumi Y. Well-posedness in energy space for Cauchy problem of the Klein–Gordon–Zakharov equations with different propagation speeds in three space dimensions. Math Ann 1999;313:127–207. [8] Tsutaya K. Global existence of small amplitude solutions for the Klein–Gordon–Zakharov equations. Non-Linear Anal 1996;27:1373–80. [9] Zakharov VE. Collapse of Langmuir wave. Sov Phys JETP 1972;35:908–14.
Exact explicit travelling wave solutions for (n + 1)dimensional Klein–Gordon–Zakharov equations q Jibin Li Department of Mathematics, Zhejiang Normal University, Jinhua, Zhejiang 321004, PR China Center for Nonlinear Science Studies, Kunming University of Science and Technology, Kunming, Yunnan 650093, PR China Accepted 23 March 2006
Communicated by Prof. M. Wadati
Abstract Using the methods of dynamical systems for the (n + 1)-dimensional KGS nonlinear wave equations, five classes of exact explicit parametric representations of the bounded travelling solutions are obtained. To guarantee the existence of the above solutions, all parameter conditions are given. Ó 2006 Elsevier Ltd. All rights reserved.
1. Introduction It is known that the following Klein–Gordon–Zakharov equations (KGZ-equations in short) are governed by wtt c2 Dw ¼ Dj/j2 ;
/tt D/ þ / ¼ /w;
ð1Þ
where D ¼ oxo2 þ oxo2 þ þ oxo2 is the Laplacian operator, x 2 Rn, c is the propagation speed of a wave. Eqs. (1) are a cou1
2
n
pled nonlinear wave model which describes the interaction of the Langmuir wave and the ion acoustic wave in a plasma and so on (see [1,2,9]). The real value function / is the fast time scale component of electric field raised by electrons and the real value function w is the deviation of ion density from its equilibrium. For the case n = 3, Ozawa et al. [6–8] studied the local well-posedness for the Cauchy problem in the energy space and the global existence of small amplitude solutions of (1). Gan and Zhang [3] derived a sharp condition of global existence, stability and existence of standing wave of (1). To our knowledge, a general integer n, exact explicit travelling wave solutions of KGZ equation (1) have not been considered before. In this paper, we shall consider the existence of exact explicit bounded travelling wave solutions of (1), by using the methods of dynamical systems (see [5]). In different regions of the parametric space, explicit parameter conditions to guarantee the above solutions are given.
q This research was supported by the National Natural Science Foundation of China (10231020) and the Natural Science Foundation of Yunnan Province (2005A0092M). E-mail addresses: [email protected], [email protected]
0960-0779/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2006.03.088
868
J. Li / Chaos, Solitons and Fractals 34 (2007) 867–871
2. The reduced travelling wave system of (1) To find travelling wave solutions of KGZ equations, we assume that /ðx; tÞ ¼ eig uðnÞ;
x 2 Rn ; t 2 R;
wðx; tÞ ¼ vðnÞ;
ð2Þ
where g¼
n X
aj xj þ bt;
n¼
j¼1
n X
cj xj rt:
ð3Þ
j¼1
Substituting (2) into (1), cancelling out eig and separating the real and imaginary parts, we have 8 ! ! ! n n n > P P P > 2 2 2 2 > > c r unn aj þ 1 b u uv ¼ 0; aj cj þ br un ¼ 0; > < j¼1 j j¼1 j¼1 ! ! > n n > P P > 2 2 2 2 2 > > : c j¼1 cj r vnn þ 2 j¼1 cj ðun þ uunn Þ ¼ 0:
ð4Þ
We see from (4) that to have non-constant solutions of (1) depending on n, we must suppose that n X
aj cj þ br ¼ 0:
ð5Þ
j¼1
Denote that a¼
n X
a2j þ 1 b2 ;
l¼
j¼1
n X
c2j r2 ;
j¼1
c¼
n X
c2j ;
j¼1
d ¼ c2
n X
c2j r2 :
j¼1
Thus, we obtain the following ordinary differential system: ( lunn au uv ¼ 0; dvnn þ 2cðu2n þ uunn Þ ¼ 0:
ð6Þ
We next consider the dynamical behavior of (6) on the super-surface v Au2 = 0 in the four-dimensional phase space _ v_ Þ, where ‘‘Æ’’ stand for the derivative with respect to n. ðu; v; u; Substituting v = Au2 to (6) and taking Pn 2 c j¼1 cj P ; ð7Þ A¼ ¼ 2 d r c2 nj¼1 c2j then the second equation of system (6) vanishes and (6) becomes lunn au Au3 ¼ 0: Suppose that l 5 0 and write that a¼
a ¼ l
Pn 2 2 j¼1 aj þ 1 b P ; n 2 2 j¼1 cj r
b¼
Pn 2 A j¼1 cj P : ¼ Pn 2 2 l ð j¼1 cj r Þðr2 c2 nj¼1 c2j Þ
ð8Þ
Thus, we have the following equation: unn au bu3 ¼ 0; which corresponds to the two-dimensional Hamiltonian system du ¼ y; dn
dy ¼ au þ bu3 dn
ð9Þ
J. Li / Chaos, Solitons and Fractals 34 (2007) 867–871
869
with the Hamiltonian H ðu; yÞ ¼ 12au2 þ 12y 2 14bu4 ¼ h:
ð10Þ
Clearly, (9) is well-known Duffing oscillator system (see [4]). 3. The parametric representations of bounded solutions of system (9) In this section, we consider the dynamics of the phase orbits of (9) in its parameter space. By qualitative analysis and using the Jacobian elliptic functions (see [1]), we have the following results. (1) a > 0, b > 0. The origin O(0, 0) of (9) is an unique equilibrium which is a saddle point. There is no bounded orbit of u(n) for n 2 (1, 1). (2) a < 0, b < 0. The origin O(0, 0) of (9) is an unique equilibrium point which is a center. There exists a family of peri½ 4h 2ab u2 u4 ¼ ðbÞ ½ðz21 þ u2 Þðz22 u2 Þ, odic orbits of (9) enclosing the origin. We see from (10) that y 2 ¼ ðbÞ 2 ðbÞ 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 where z21 ¼ ðbÞ ½ðaÞ þ a2 4hb, z22 ¼ ðbÞ ½a þ a2 4hb. The family of periodic orbits defined by H(u, y) = h, h 2 (0, 1) of (10) has the parametric representation ! rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðbÞðz21 þ z22 Þ z2 ffi : uðnÞ ¼ z2 cn n; pffiffiffiffiffiffiffiffiffiffiffiffiffi 2 z21 þ z22
ð11Þ qffiffiffiffiffiffiffi (3) a < 0, b > 0. There exist three equilibrium points of (9) A ðaÞ ; 0 , O(0, 0). O is a center, A± are saddle b 2
a points. The heteroclinic orbits defined by H ðu; yÞ ¼ 4b ¼ h1 have the parametric representations rffiffiffi rffiffiffiffiffiffiffi a b uðnÞ ¼ tanh n: b 2
ð12Þ
2
a Þ, it can be written as y 2 ¼ b2 ½4hb þ 2ab u2 þ u4 ¼ We see from (10) that y 2 ¼ 2h þ au2 þ b2 u4 . For h 2 ð0; 4b pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b 2 2 2 2 2 1 2 1 2 ½ðr1 u Þðr2 u Þ, where r1 ¼ b ½ðaÞ þ a 4hb, r2 ¼ b ½ðaÞ a2 4hb. Thus, the family of periodic 2
orbits defined by H(u, y) = h has the parametric representation ! rffiffiffi b r2 uðnÞ ¼ r2 sn : ð13Þ r1 n; 2 r1 qffiffiffiffiffiffiffi ; 0 , O(0, 0). O is a saddle point, A± are center (4) a > 0, b < 0. There exist three equilibrium points of (9) A ðaÞ b points. Two homoclinic orbits defined by H(u, y) = 0 have the parametric representations rffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffi 2a aðbÞ uðnÞ ¼ n: sech b 2
ð14Þ
2a ½ 4h þ ðbÞ u2 u4 ¼ ðbÞ ½ðz22 u2 Þðu2 ðz21 ÞÞ, where z21 , z22 are For h 2 ð4b ; 0Þ, we have from (10) that y 2 ¼ ðbÞ 2 ðbÞ 2 a2
the same as the case (2). Thus, two families of periodic orbits defined by H(u, y) = h have the parametric representations rffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi! z22 þ z21 ðbÞ uðnÞ ¼ z2 dn : ð15Þ z2 n; z2 2 The family of periodic orbits of (9) enclosing three equilibrium points A± and O defined by H(u, y) = h, h 2 (0, 1) has the same parametric representations as (11).
4. Conclusion By using the results of Sections 2 and 3 and noting w(x, t) = v(n) = Au2(n), we obtain the following conclusion of this paper. Theorem 1. Suppose that the parameters A, a, b are given by (7), (8) and z21 , z22 , r21 , r22 are defined by Section 3. In addition, the condition (5) holds. Then, the (n + 1)-dimensional KGZ equation (1) have the following five formulas of exact explicit bounded travelling wave solutions.
870
J. Li / Chaos, Solitons and Fractals 34 (2007) 867–871
(1) When a < 0, b < 0, 8 n P ! ! > qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > i a x þbt j j n > > ðbÞðz21 þz22 Þ P z2 > j¼1 > cj xj rt ; pffiffiffiffiffiffiffiffi z2 cn ; < /ðx; tÞ ¼ e 2 2 2 z1 þz2
j¼1
! ! > qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > n > ðbÞðz21 þz22 Þ P z2 2 > 2 > cj xj rt ; pffiffiffiffiffiffiffiffi : : wðx; tÞ ¼ Az2 cn 2 2 2
ð16Þ
z1 þz2
j¼1
(2) When a < 0, b > 0, 8 n P ! > > qffiffi P i aj xj þbt pffiffiffiffi n > > a b > j¼1 > tanh 2 cj xj rt ; < /ðx; tÞ ¼ e b j¼1
! > > qffiffi P > n > 2 > a b > cj xj rt : wðx; tÞ ¼ Að b Þtanh 2
ð17Þ
j¼1
and
8 n P ! ! > > qffiffi i aj xj þbt n > P > b > j¼1 > r cj xj rt ; rr21 ; sn < /ðx; tÞ ¼ r2 e 2 1 j¼1
! ! > > qffiffi > n P > 2 > 2 b > cj xj rt ; rr21 : r : wðx; tÞ ¼ Ar2 sn 2 1
ð18Þ
j¼1
(3) When a > 0, b < 0, 8 n P ! > qffiffiffiffiffiffiffiffi P > i aj xj þbt qffiffiffiffi n > > aðbÞ 2a > j¼1 > cj xj rt ; sech < /ðx; tÞ ¼ e 2 b j¼1
! > > qffiffiffiffiffiffiffiffi P > n > aðbÞ 2 > 2a > cj xj rt : wðx; tÞ ¼ AðbÞsech 2
ð19Þ
j¼1
and
8 n P ! pffiffiffiffiffiffiffiffi! > qffiffiffiffiffiffiffi > i aj xj þbt n > P z2 þz2 > ðbÞ > j¼1 > cj xj rt ; z22 1 ; z2 dn z < /ðx; tÞ ¼ e 2 2 j¼1
! pffiffiffiffiffiffiffiffi! > > qffiffiffiffiffiffiffi > n P > z2 þz2 ðbÞ 2 2 > > cj xj rt ; z22 1 : z : wðx; tÞ ¼ Az2 dn 2 2
ð20Þ
j¼1
References [1] Byrd PF, Fridman MD. Handbook of elliptic integrals for engineers and scientists. Berlin: Springer; 1971. [2] Dendy RO. Plasma dynamics. Oxford: Oxford University Press; 1990. [3] Gan Zaihui, Zhang Jian. Instability of standing wave for Klein–Gordon–Zakharov equations with different propagation speeds in three space dimensions. J Math Anal Appl 2005;307(1):219–31. [4] Guckenheimer J, Holmes PJ. Nonlinear oscillations, dynamical systems and bifurcations of vector fields. New York: SpringerVerlag; 1983. [5] Li Jibin. Solitary and periodic traveling wave solutions in Klein–Gordon–Schrodinger equations. J Yunnan Univ 2003;25(3): 176–80. [6] Ozawa T, Tsutaya K, Tsutsumi Y. Normal form and global solutions for the Klein–Gordon–Zakharov equations. Ann Inst H Poincare Anal Non Lineaire 1995;12:459–503.
J. Li / Chaos, Solitons and Fractals 34 (2007) 867–871
871
[7] Ozawa T, Tsutaya K, Tsutsumi Y. Well-posedness in energy space for Cauchy problem of the Klein–Gordon–Zakharov equations with different propagation speeds in three space dimensions. Math Ann 1999;313:127–207. [8] Tsutaya K. Global existence of small amplitude solutions for the Klein–Gordon–Zakharov equations. Non-Linear Anal 1996;27:1373–80. [9] Zakharov VE. Collapse of Langmuir wave. Sov Phys JETP 1972;35:908–14.