- Email: [email protected]
Explicit and exact solutions for the generalized reaction duffing equation
224
Communications
in Nonlinear
Science & Numerical
Vo1.4, No.3 (Sep. 1999)
Simulation
Explicit and Exact Solutions for the Generalized Reaction Duffing Equation
l1
Zhenya YAN and Hongqing ZHANG (Institute of Mathematical Science, Dalian University of Technology, Dalian 116024, China) E-mail:[email protected] Abstract: With the aid of Mathematics, several kinds for the generalized reaction duffing equation are obtained solutions contain new solitary wave solutions and period also be applied to other nonlinear evolution equations. Keywords: nonlinear evolution equation, reaction duffing tary wave solution, period solution
of explicit and exact solutions by using a new ansatz. These solutions. This approach can equation,
exact solution,
soli-
Introduction In the soliton theory, finding more efficient method plays an important role in solving exact solutions of nonlinear evolution equations11-41. The following reaction duffing equation utt + au,, + bu + du3 = 0
(1)
where a, b, d are all constants, can be reduced to many famous equations, such as 44 equation, Klein-Gordon equation, Landau-ginburg-Higgs equation and so on[1-3]. We have derived three soliton solutions of Eq.( 1) by using sine-cosine method131. In this paper, we would like to consider the generalized form of Eq.(l), namely utt + auzz + bu + cu2 + du3 = 0
(2)
By the use of a new ansatz, several kinds of explicit and exact solutions for Eq.(2) are obtained. These solutions contain new solitary wave solutions and period solutions. This approach can also be applied to other nonlinear evolution equations.
1 Approach
and Exact
Solutions
We consider the following formal travelling u(z,t)=qb(<),
wave solutions of Eq.(2) <=s-At+%
Where X a constant to be determined later and ~0 is an arbitrary into Eq.(2), the Eq.(2) is reduced to (u f X2)&
(3) constant.
Substituting
+ bq5+ CC$~+ d$3 = 0
(3)
(4)
In order to solve Eq.(4), we suppose that Eq.(4) has the following formal solutions qb(<) = 2 ~~-~[Aiw + Bid-1 i=l
w’(C)= q1+ Pm21 “The
paper was received on July 8, 1999
+ Ao
(5) (6)
YAN et al.: Explicit
No.3
225
and Exact . . .
whereS=fl,p=fl;R,As,Ai,&(i=1,2 ,... , n) are contants to be determined later. In fact, we can determine the value of m in Eq.(5). By balancing the highest order nonlinear term $3 and linear term C&Cin (4), it is easy to find that m = 1. Thus (5) becomes (7) and then = 2PA1R2w(l f CLW~)+ B~R~(~ + 2w2)dm
&
(8) With the aid of Mathematics, substituting (7) and (8) into Eq.(4) and collecting all terms with the same power in wi[dm]j(i = 0, 1,2,3, ; j = 0,l). And then setting the coefficients of all these terms to zero yields a set of overdetermined algebraic polynomials with respect to the unknowns R, X, AC,,AI, BI.
bAo + c(A; + 6B;) + d(A; + 36AoB;) = 0 2p(a + X2)A1R2 + bAl + 2cAoAl + d(3A;A1
p(a + X2)&R2
+ 3SA&)
(94 = 0
(9b)
+ bB1 + 2cAoBI + d(3A;B1 + SB;) = 0
PC)
2cA1B1 + 6dAoA1B1 = 0
(94
c(A; + 3S&)
+ d(A; + 3SpAoB,2) = 0
(94
2(u + X2)A1R2 + d(A; + 3bpA&)
= 0
w
2(a + X2)B1 R2 + d(G&
= 0
(W
+ 3A:&)
TO solve Eqs.(Sa)-(Sg), by using Wu-elimination method which is a sufficient method to solve the systems of algebraic polynomial equations with more unknowns[51, we get the following conclusions from system(g) Case 1 when 6 = fl; p = -1; 9bd - 2c2 = 0; d(a + X2) < 0
AO=-5, Case
2 when 6 = 1; p = -1;
Ao=--
Case
R=f
3;,
R=iz
AI = 0, 13~ = zt
3 when 6 = p = 1; 9bd - 22 = 0;
Ao=-5, Case
9bd - 2c2 = 0; d(a + X2) > 0
d(u + X2) < 0, c2 - 3bd > 0
R=zt
4 when 6 = p = -1;
Ao=-2,
Al =O,
B1 = f
9bd - 2s = 0; d(u + X2) < 0 R=f
d -
C2
3d(a + X2) ’
Al = B,2 = $
Nowadays, we consider the Biccati Eq.(6). It is easy to show that Eq.(6) has the following general solutions WI(<) = t=Mx3, ~3 (5) = tan(%),
ws(5) = coth(R[) wq(5)
= cot(R[)
(p = -1)
(10)
(p = 1)
(11)
226
Communications
in Nonlinear
Science & Numerical
Simulation
Vo1.4, No.3 (Sep. 1999)
Therefore, combining (3),(7),(10),(11) along with Case 1-4, we find the following many types of exact solutions for Eq.(2), which contain new solitary wave solutions and period solutions
At+ Q)] -
w(x, t) = t&$tanhi/G-
(x -
us,e(x,t)
(x - At + co)] - J,
= */$coth,/~
5, (9bd - 2c2 = 0; d(a + A”) < 0)
(9bd - 2c2 = 0; d(a + X2) < 0)
.,r(,_t)=*~sech~~~(x-ht+Co)j-~,(gM-2c2=O;d(U+i2)>O)
.,,(.,t)=*~~eci~~(x-~t+~)~-~,(9~~-2~2=O;~(~+~2)
u9,10(x,t)
m-14(x,
=
(x - At + co)] - 5, (9bd - 2c2 = 0; d(” + A”) < 0)
*~csc~~~
t) = /${a
coth~~~(x-~t+o)j+gcschlJ-~(x-ht+co)i) -5,
(9bd - 2c2 = 0; d(u + X2) < 0; Q = fl;
p = H)
If we consider Eq.(2) in the complex field, according to the same steps as the above, we can also derive other formal solitary wave solutions as follows
%5,16(&t)
=
fl .~csch~~~(x-,i,,,1-g.
2 Conclusion In summary, we find many exact solutions for the generalized reaction duffing equation by virtue of a new ansatz. These solutions may be useful to explain some physical phenomena. The method is general, and can be also applied to other nonlinear evolution equations, such as KdV equation, KP equation, WKB equation, Boussinesq equation and so on [1,2]. In addition, the method can be carried out in computer with the aid of Mathematics and Wu-elimination method. Acknowledgment The work is supported by the National Natural Science Foundation of China under the Grant No.19572022 and the National Climbing Project Foundation of China.
227
FANG: A Pseudo-spectralApproximation 1. .
No.3
References
[l] Ablowitz, M. J., et al., Solitons, Nonlinear Evolution Equations and Inverse Scatting, New York: Cambridge University Press, 1991 [2] Gu, C. H., et al., Soliton theory and its application, Zhejiang: Zhejiang Science and Technology Press. 1990 [3] Yan, Z. Y., et al., Acta Physica Sinica, 1999, 49(l): l-5 [4] Zhang, H. Q. and Yan, Z. Y., Mathematics Application, 1999, 12(l): 77-81 [5] Wu, W., Kexue Tongbao, 1986, 35: 1-7
A Pseudo-spectral Approximation System of Generalized Zakharov Shaomei FANG (Department of Mathematics, E-mail: [email protected]
Shaoguan University,
for the Equations
l2
Shaoguan 512005, China)
In this paper, a class of system of generalized Zakharov equations with periodic initial conditions is considered. Fully discrete pseudo-spectral approximation for the system are proposed. Moreover, convergence with spectral accuracy is proved for the approximation by means of a priori estimates and Sobolev’s imbedding theorem and unconditional stability is obtained. Finally, solvability of the scheme is developed and numerical examples are described. Keywords: Zakharov equation, pseudo-spectral approximation Abstract:
Introduction In this paper, we are concerned with the theoretical analysis of the fully discrete pseudospectral approximation of the system of generalized Zakharov equations with periodic initial conditions by ’ ict + csz - cync + /?q(]c12)c = 0; nt + ‘u, = 0; E(x, 0) = co(x); 2)(x, 0) = v~(x); n(x, 0) = no(z);
(x,t) E Rx J (x,t)~RxJ (x, t) E R x J XER
(14 (lb) UC) (14
E(X + 27r, t) = c(x, t); v(x + 27r, t) = 21(x, q; n(x + 27r, t) = n(x, t);
(x, t) E R x J
(14
ut + f(v),
- bxt
+ nx -t 44
= 9(v);
where i = &i, q(.) denotes a continuous real-valued function of real argument, co(x), vo(x) and no(z) are 2n-periodic functions. J = [0, T](T > 0); Q, p and 6 are real constants; and 6(x, t) is a 2n-periodic complex function with respect to x. The functions o(x, t) and n(x, t) are 2r-periodic real valued function. Zakharov’s equations (1) have been used by plasma physicists as model equations, who described the interaction between ion sound (density n) and Langmuir waves (envelop amplitude c)lll , when (lb) and (lc) take the form of wave equations which are driven by 12The
paper was receivedon July.
2, 1999
Communications
in Nonlinear
Science & Numerical
Vo1.4, No.3 (Sep. 1999)
Simulation
Explicit and Exact Solutions for the Generalized Reaction Duffing Equation
l1
Zhenya YAN and Hongqing ZHANG (Institute of Mathematical Science, Dalian University of Technology, Dalian 116024, China) E-mail:[email protected] Abstract: With the aid of Mathematics, several kinds for the generalized reaction duffing equation are obtained solutions contain new solitary wave solutions and period also be applied to other nonlinear evolution equations. Keywords: nonlinear evolution equation, reaction duffing tary wave solution, period solution
of explicit and exact solutions by using a new ansatz. These solutions. This approach can equation,
exact solution,
soli-
Introduction In the soliton theory, finding more efficient method plays an important role in solving exact solutions of nonlinear evolution equations11-41. The following reaction duffing equation utt + au,, + bu + du3 = 0
(1)
where a, b, d are all constants, can be reduced to many famous equations, such as 44 equation, Klein-Gordon equation, Landau-ginburg-Higgs equation and so on[1-3]. We have derived three soliton solutions of Eq.( 1) by using sine-cosine method131. In this paper, we would like to consider the generalized form of Eq.(l), namely utt + auzz + bu + cu2 + du3 = 0
(2)
By the use of a new ansatz, several kinds of explicit and exact solutions for Eq.(2) are obtained. These solutions contain new solitary wave solutions and period solutions. This approach can also be applied to other nonlinear evolution equations.
1 Approach
and Exact
Solutions
We consider the following formal travelling u(z,t)=qb(<),
wave solutions of Eq.(2) <=s-At+%
Where X a constant to be determined later and ~0 is an arbitrary into Eq.(2), the Eq.(2) is reduced to (u f X2)&
(3) constant.
Substituting
+ bq5+ CC$~+ d$3 = 0
(3)
(4)
In order to solve Eq.(4), we suppose that Eq.(4) has the following formal solutions qb(<) = 2 ~~-~[Aiw + Bid-1 i=l
w’(C)= q1+ Pm21 “The
paper was received on July 8, 1999
+ Ao
(5) (6)
YAN et al.: Explicit
No.3
225
and Exact . . .
whereS=fl,p=fl;R,As,Ai,&(i=1,2 ,... , n) are contants to be determined later. In fact, we can determine the value of m in Eq.(5). By balancing the highest order nonlinear term $3 and linear term C&Cin (4), it is easy to find that m = 1. Thus (5) becomes (7) and then = 2PA1R2w(l f CLW~)+ B~R~(~ + 2w2)dm
&
(8) With the aid of Mathematics, substituting (7) and (8) into Eq.(4) and collecting all terms with the same power in wi[dm]j(i = 0, 1,2,3, ; j = 0,l). And then setting the coefficients of all these terms to zero yields a set of overdetermined algebraic polynomials with respect to the unknowns R, X, AC,,AI, BI.
bAo + c(A; + 6B;) + d(A; + 36AoB;) = 0 2p(a + X2)A1R2 + bAl + 2cAoAl + d(3A;A1
p(a + X2)&R2
+ 3SA&)
(94 = 0
(9b)
+ bB1 + 2cAoBI + d(3A;B1 + SB;) = 0
PC)
2cA1B1 + 6dAoA1B1 = 0
(94
c(A; + 3S&)
+ d(A; + 3SpAoB,2) = 0
(94
2(u + X2)A1R2 + d(A; + 3bpA&)
= 0
w
2(a + X2)B1 R2 + d(G&
= 0
(W
+ 3A:&)
TO solve Eqs.(Sa)-(Sg), by using Wu-elimination method which is a sufficient method to solve the systems of algebraic polynomial equations with more unknowns[51, we get the following conclusions from system(g) Case 1 when 6 = fl; p = -1; 9bd - 2c2 = 0; d(a + X2) < 0
AO=-5, Case
2 when 6 = 1; p = -1;
Ao=--
Case
R=f
3;,
R=iz
AI = 0, 13~ = zt
3 when 6 = p = 1; 9bd - 22 = 0;
Ao=-5, Case
9bd - 2c2 = 0; d(a + X2) > 0
d(u + X2) < 0, c2 - 3bd > 0
R=zt
4 when 6 = p = -1;
Ao=-2,
Al =O,
B1 = f
9bd - 2s = 0; d(u + X2) < 0 R=f
d -
C2
3d(a + X2) ’
Al = B,2 = $
Nowadays, we consider the Biccati Eq.(6). It is easy to show that Eq.(6) has the following general solutions WI(<) = t=Mx3, ~3 (5) = tan(%),
ws(5) = coth(R[) wq(5)
= cot(R[)
(p = -1)
(10)
(p = 1)
(11)
226
Communications
in Nonlinear
Science & Numerical
Simulation
Vo1.4, No.3 (Sep. 1999)
Therefore, combining (3),(7),(10),(11) along with Case 1-4, we find the following many types of exact solutions for Eq.(2), which contain new solitary wave solutions and period solutions
At+ Q)] -
w(x, t) = t&$tanhi/G-
(x -
us,e(x,t)
(x - At + co)] - J,
= */$coth,/~
5, (9bd - 2c2 = 0; d(a + A”) < 0)
(9bd - 2c2 = 0; d(a + X2) < 0)
.,r(,_t)=*~sech~~~(x-ht+Co)j-~,(gM-2c2=O;d(U+i2)>O)
.,,(.,t)=*~~eci~~(x-~t+~)~-~,(9~~-2~2=O;~(~+~2)
u9,10(x,t)
m-14(x,
=
(x - At + co)] - 5, (9bd - 2c2 = 0; d(” + A”) < 0)
*~csc~~~
t) = /${a
coth~~~(x-~t+o)j+gcschlJ-~(x-ht+co)i) -5,
(9bd - 2c2 = 0; d(u + X2) < 0; Q = fl;
p = H)
If we consider Eq.(2) in the complex field, according to the same steps as the above, we can also derive other formal solitary wave solutions as follows
%5,16(&t)
=
fl .~csch~~~(x-,i,,,1-g.
2 Conclusion In summary, we find many exact solutions for the generalized reaction duffing equation by virtue of a new ansatz. These solutions may be useful to explain some physical phenomena. The method is general, and can be also applied to other nonlinear evolution equations, such as KdV equation, KP equation, WKB equation, Boussinesq equation and so on [1,2]. In addition, the method can be carried out in computer with the aid of Mathematics and Wu-elimination method. Acknowledgment The work is supported by the National Natural Science Foundation of China under the Grant No.19572022 and the National Climbing Project Foundation of China.
227
FANG: A Pseudo-spectralApproximation 1. .
No.3
References
[l] Ablowitz, M. J., et al., Solitons, Nonlinear Evolution Equations and Inverse Scatting, New York: Cambridge University Press, 1991 [2] Gu, C. H., et al., Soliton theory and its application, Zhejiang: Zhejiang Science and Technology Press. 1990 [3] Yan, Z. Y., et al., Acta Physica Sinica, 1999, 49(l): l-5 [4] Zhang, H. Q. and Yan, Z. Y., Mathematics Application, 1999, 12(l): 77-81 [5] Wu, W., Kexue Tongbao, 1986, 35: 1-7
A Pseudo-spectral Approximation System of Generalized Zakharov Shaomei FANG (Department of Mathematics, E-mail: [email protected]
Shaoguan University,
for the Equations
l2
Shaoguan 512005, China)
In this paper, a class of system of generalized Zakharov equations with periodic initial conditions is considered. Fully discrete pseudo-spectral approximation for the system are proposed. Moreover, convergence with spectral accuracy is proved for the approximation by means of a priori estimates and Sobolev’s imbedding theorem and unconditional stability is obtained. Finally, solvability of the scheme is developed and numerical examples are described. Keywords: Zakharov equation, pseudo-spectral approximation Abstract:
Introduction In this paper, we are concerned with the theoretical analysis of the fully discrete pseudospectral approximation of the system of generalized Zakharov equations with periodic initial conditions by ’ ict + csz - cync + /?q(]c12)c = 0; nt + ‘u, = 0; E(x, 0) = co(x); 2)(x, 0) = v~(x); n(x, 0) = no(z);
(x,t) E Rx J (x,t)~RxJ (x, t) E R x J XER
(14 (lb) UC) (14
E(X + 27r, t) = c(x, t); v(x + 27r, t) = 21(x, q; n(x + 27r, t) = n(x, t);
(x, t) E R x J
(14
ut + f(v),
- bxt
+ nx -t 44
= 9(v);
where i = &i, q(.) denotes a continuous real-valued function of real argument, co(x), vo(x) and no(z) are 2n-periodic functions. J = [0, T](T > 0); Q, p and 6 are real constants; and 6(x, t) is a 2n-periodic complex function with respect to x. The functions o(x, t) and n(x, t) are 2r-periodic real valued function. Zakharov’s equations (1) have been used by plasma physicists as model equations, who described the interaction between ion sound (density n) and Langmuir waves (envelop amplitude c)lll , when (lb) and (lc) take the form of wave equations which are driven by 12The
paper was receivedon July.
2, 1999