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Non-perturbative solution of the Klein-Gordon-Zakharov equation
NORI~-HOLLAND Non-Perturbative Solution of the Klein-Gordon-Zakharov Equation G. Adomian G e n e r a l A n a l y t i c s Corporation 155 Clyde R o a d A t h e n s , Georgia 30605
ABSTRACT The initial-value problem for the Klein-Gordon-Zakharov equation is solved by decomposition for x E R 3. (~ Elsevier Science Inc., 1997
I N T R O D U C T I O N AND DISCUSSION The Cauchy problem for the K-G-Z equation is given by u u = V 2 u + u + n u + [u[2u nt+V.V=O ¼ + Vn + Vlul 2 = 0
u(t = o) = ~(z) u , ( t = 0) = ~ ( x ) n ( t = 0) = ~ ( x )
y ( t = 0) = ~ ( x ) ,
x e n3
where u ( x , t) is a complex vector function, V is a real vector function, and n ( x , t ) is a real function. Write Lu = 02/~t 2 and L -1 is a two-fold integration from 0 to t. L t t u = V2u - u - n u - [u[2u.
(1)
Operating with L~ 1 on both sides, the left side becomes u - u ( t = O) t u t ( t = 0). Thus, u = c~(x) + t ~ ( x ) + L - 1 V 2 u - L - l u
- L-lnu
- L-11u[2u.
APPLIED MATHEMATICS AND COMPUTATION 81"89-92 (1997)
(~) Elsevier Science Inc., 1997 655 Avenue of the Americas, New York, NY 10010
0096-3003/97/$17.00 SSDI 0096-3003(95)00311-8
90
G. ADOMIAN
oc U oo Decomposing u into }--~j=o J, and n = Y-:~j=o nj, [u[2u = ~ = o Aj{[u[2u}, and n u : )-~]=o A j { n u } , where the Aj are the Adomian Polynomials [1] (usually denoted by A~). Then
u = a ( x ) +t13(x) + L - I V 2 E u j j=O L-1EuJ j=o
-
- L -1EAj{nu} j=o
- L -1 E A j { l u l 2 u } "
(2)
j=o
Now consider the equation for V writing Lt = O/Ot and L [ 1 is a definite integration from 0 to t. L2rlLtV = V - V ( t = O) so that - Lg~VWI 2
V = 6(x) - L t l V n
¢2¢)
V
=
6(X)
- -
L;lv
Z
nj
- -
nt-lvlul 2
j=O = 6(x) - L t l V E n j
j=0
- LtlVEAj{[ul2}. j=o
(3)
Similarly, Ltn = -V • V Ltl Ltn = -LtlV
•V
n = r(z) - L~IV.
V oo
n = T(X) - L;-1V • E S " j=o
Solving (2), (3), and (4), we identify the triplet of known terms
{
~o = ~(~) + t~(~)
Vo= 6(~)
Then ul = L-1V2Uo - L - l u o - L - 1 A o { n u } - L - 1 A o { l u ] 2 u } V1 = - L [ l n o
- L [ 1 V A o { ] u ] 2}
n l -- - L t l V
. 1Io
(4)
Klein-Gordon-Zakharov Equation
91
which are also d e t e r m i n e d b e c a u s e t h e Ao t e r m s involve o n l y uo. Next, we calculate u2 = L [ 1 V 2 U l - L - l u l
- L-l(noul
V2 : - L t l n l
- L [ 1 V A I { l U l 2}
n2 = - L ~ I V
• V1
+ nluo) - n71Al{lu12u}
W e can continue for as m a n y t e r m s as necessary; t h e c o m p u t a t i o n is s t r a i g h t f o r w a r d once t h e An are c a l c u l a t e d (which we list for convenience). However, a few t e r m s g e n e r a l l y suffice [2]. T h e n we w r i t e rn--1
m[ul :
us
5=0 rn-1
.[vl = E vj j=O m--1
~m[n] = E ?%J j=O as a p p r o x i m a n t s to u, V, n for a sufficient m for convergence t o a r e a s o n a b l e accuracy. T h e A{lul 2} = A { u . u} are given by Ao = uo • uo
A1 = 2uo - Ul A2 -- Ul • Ul + 2uo • u2 A3:2uo-u3+2Ul'U2 A4 = u 2 . u 2 + 2 u l ' u 3 + 2 u o ' u 4 A5 = 2uo - u5 + 2u2 • u3 + 2Ul • u4. T h e A{lul2u} = A { ( u . u ) u } are given by Ao = (Uo . uo)uo AI = (2uo-ul)uo
+ (uo . uo)ul
A2 = ( 2 u o ' u 2 + u l " u J u o + (2uo" U J U l + (uo " uo)u2 A3 = (2uo" u3 + 2 u l -u2)uo + (2uo • u2 + Ul " Ul)Ul + (2uo" Ul)U2 + (uo" uo)u3
92
G. ADOMIAN
A4 = (2uo" u4 + u2" u2 + 2ul • u3)uo + (2ul • u2 + 2u0 + ( u l • Ul +
• u3)ul
2u0. u2)u2 + (2u0. ul)u3 + (uo. uo)u4
A5 = (2uo • u5 + 2ua • u4 + 2u2 • u3)uo + (2ul • u3 + u2. u2 + 2uo. u4)ul + (2ul • u2 + 2uo- u3)u2 + +
( U l * U l + 2U0 "
U2)U3
(2uo- ul)u4 + (uo. uo)us.
The A j { n u } are given by Ao
=
nouo
A1 = n o u l + n l u o A2 = nou2 • n l U l -~ n2uo A3 = nou3 -~ n l u 2 • n 2 u l -~ n3uo A4 = nou4 • n l u 3 ~- n2u2 • n 3 u l • n4uo A5 = nou5 + n l u 4 -~ n2u3 -~ n3u2 + n4Ul -{- n5uo
which completes the solution with no smallness assumptions. Convergence has been previously established by a number of authors and, because of the rapidity of the convergence, few terms are necessary for sufficient accuracy. REFERENCES 1 G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method, Kluwer Academic Publishers, 1994. 2 G. Boling and Y. Guangwei, Global Smooth Solution for the Klein-GordonZakharov Equations, J. Math. Phys. 36(8) (1995).
ABSTRACT The initial-value problem for the Klein-Gordon-Zakharov equation is solved by decomposition for x E R 3. (~ Elsevier Science Inc., 1997
I N T R O D U C T I O N AND DISCUSSION The Cauchy problem for the K-G-Z equation is given by u u = V 2 u + u + n u + [u[2u nt+V.V=O ¼ + Vn + Vlul 2 = 0
u(t = o) = ~(z) u , ( t = 0) = ~ ( x ) n ( t = 0) = ~ ( x )
y ( t = 0) = ~ ( x ) ,
x e n3
where u ( x , t) is a complex vector function, V is a real vector function, and n ( x , t ) is a real function. Write Lu = 02/~t 2 and L -1 is a two-fold integration from 0 to t. L t t u = V2u - u - n u - [u[2u.
(1)
Operating with L~ 1 on both sides, the left side becomes u - u ( t = O) t u t ( t = 0). Thus, u = c~(x) + t ~ ( x ) + L - 1 V 2 u - L - l u
- L-lnu
- L-11u[2u.
APPLIED MATHEMATICS AND COMPUTATION 81"89-92 (1997)
(~) Elsevier Science Inc., 1997 655 Avenue of the Americas, New York, NY 10010
0096-3003/97/$17.00 SSDI 0096-3003(95)00311-8
90
G. ADOMIAN
oc U oo Decomposing u into }--~j=o J, and n = Y-:~j=o nj, [u[2u = ~ = o Aj{[u[2u}, and n u : )-~]=o A j { n u } , where the Aj are the Adomian Polynomials [1] (usually denoted by A~). Then
u = a ( x ) +t13(x) + L - I V 2 E u j j=O L-1EuJ j=o
-
- L -1EAj{nu} j=o
- L -1 E A j { l u l 2 u } "
(2)
j=o
Now consider the equation for V writing Lt = O/Ot and L [ 1 is a definite integration from 0 to t. L2rlLtV = V - V ( t = O) so that - Lg~VWI 2
V = 6(x) - L t l V n
¢2¢)
V
=
6(X)
- -
L;lv
Z
nj
- -
nt-lvlul 2
j=O = 6(x) - L t l V E n j
j=0
- LtlVEAj{[ul2}. j=o
(3)
Similarly, Ltn = -V • V Ltl Ltn = -LtlV
•V
n = r(z) - L~IV.
V oo
n = T(X) - L;-1V • E S " j=o
Solving (2), (3), and (4), we identify the triplet of known terms
{
~o = ~(~) + t~(~)
Vo= 6(~)
Then ul = L-1V2Uo - L - l u o - L - 1 A o { n u } - L - 1 A o { l u ] 2 u } V1 = - L [ l n o
- L [ 1 V A o { ] u ] 2}
n l -- - L t l V
. 1Io
(4)
Klein-Gordon-Zakharov Equation
91
which are also d e t e r m i n e d b e c a u s e t h e Ao t e r m s involve o n l y uo. Next, we calculate u2 = L [ 1 V 2 U l - L - l u l
- L-l(noul
V2 : - L t l n l
- L [ 1 V A I { l U l 2}
n2 = - L ~ I V
• V1
+ nluo) - n71Al{lu12u}
W e can continue for as m a n y t e r m s as necessary; t h e c o m p u t a t i o n is s t r a i g h t f o r w a r d once t h e An are c a l c u l a t e d (which we list for convenience). However, a few t e r m s g e n e r a l l y suffice [2]. T h e n we w r i t e rn--1
m[ul :
us
5=0 rn-1
.[vl = E vj j=O m--1
~m[n] = E ?%J j=O as a p p r o x i m a n t s to u, V, n for a sufficient m for convergence t o a r e a s o n a b l e accuracy. T h e A{lul 2} = A { u . u} are given by Ao = uo • uo
A1 = 2uo - Ul A2 -- Ul • Ul + 2uo • u2 A3:2uo-u3+2Ul'U2 A4 = u 2 . u 2 + 2 u l ' u 3 + 2 u o ' u 4 A5 = 2uo - u5 + 2u2 • u3 + 2Ul • u4. T h e A{lul2u} = A { ( u . u ) u } are given by Ao = (Uo . uo)uo AI = (2uo-ul)uo
+ (uo . uo)ul
A2 = ( 2 u o ' u 2 + u l " u J u o + (2uo" U J U l + (uo " uo)u2 A3 = (2uo" u3 + 2 u l -u2)uo + (2uo • u2 + Ul " Ul)Ul + (2uo" Ul)U2 + (uo" uo)u3
92
G. ADOMIAN
A4 = (2uo" u4 + u2" u2 + 2ul • u3)uo + (2ul • u2 + 2u0 + ( u l • Ul +
• u3)ul
2u0. u2)u2 + (2u0. ul)u3 + (uo. uo)u4
A5 = (2uo • u5 + 2ua • u4 + 2u2 • u3)uo + (2ul • u3 + u2. u2 + 2uo. u4)ul + (2ul • u2 + 2uo- u3)u2 + +
( U l * U l + 2U0 "
U2)U3
(2uo- ul)u4 + (uo. uo)us.
The A j { n u } are given by Ao
=
nouo
A1 = n o u l + n l u o A2 = nou2 • n l U l -~ n2uo A3 = nou3 -~ n l u 2 • n 2 u l -~ n3uo A4 = nou4 • n l u 3 ~- n2u2 • n 3 u l • n4uo A5 = nou5 + n l u 4 -~ n2u3 -~ n3u2 + n4Ul -{- n5uo
which completes the solution with no smallness assumptions. Convergence has been previously established by a number of authors and, because of the rapidity of the convergence, few terms are necessary for sufficient accuracy. REFERENCES 1 G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method, Kluwer Academic Publishers, 1994. 2 G. Boling and Y. Guangwei, Global Smooth Solution for the Klein-GordonZakharov Equations, J. Math. Phys. 36(8) (1995).