Applied Mathematics, 2011, 2, 1005-1010
doi:10.4236/am.2011.28139 Published Online August 2011 (http://www.SciRP.org/journal/am)
Copyright © 2011 SciRes. AM
Orbital Stability of Solitary Waves for Generalized
Klein-Gordon-Schrödinger Equations
Wenhui Qi, Guoguang Lin
School of Mathematics and Statistic, Yunnan University, Kunming, China
E-mail: gglin@ynu.edu.cn
Received May 30, 201 1; revised June 13, 2011; accepted June 20, 2011
Abstract
This paper concerns the orbital stability for exact solitary waves of the Generalized Klein-Gordon-Schrö-
dinger equations. Since the abstract results of Grillakis et al. [1,2] can not be applied directly, we can extend
the abstract stability theory and use the detailed spectral analysis to obtain the stability of the solitary waves.
Keywords: Solitary Waves, Stability, Klein-Gordon-Schrödinger Equations
1. Introduction
In this paper, we consider the the stability for the exact
solitary waves of the Generalized Klein-Gordon-Schrö-
dinger equations
2
2
2
pp
txx
pp
tt xx
i
x
R
M
 
 

 
(1.1)
which describe a classical model of interaction of nu-
cleon field with a meson field [3]. Here
is a complex
scalar nucleon field,
is a real meson field, M is the
mass of a meson. By applying the abstract stability the-
ory and detailed spectral analysis in [4-6], we obtain the
orbital stability of the solitary waves.
This paper is organized as follows: in Section 2, we
state the results of the existence of the exact solitary
waves; in Section 3, we state the assumptions and the
stability results.
2. The Exact Solitary Waves
Consider the following system
2
2
2
pp
txx
pp
tt xx
i
x
R
M
 
 

 
(2.1)
Let

 
()
,ee
,
it icxct
x
tux
xtv xct


be the solitary waves of (2.1).
Put (2.2) into (2.1) and suppose ,as
,,, 0uu vv

x
,we obtain



2
22
2
22
21 0
0
10
pp
pp
cu
uc cuvuu
cvMvuvv



 

 
(2.3)
Let
1,
2ukv
(2.4)
satisfy (2.3) with constant determined later, then
we have 0k


22
2
22
22
2
20
10
p
p
pp
ucuu u
k
cvMvkvv

 

 
(2.5)
Let
1
1
1
sech p
uc cx
2
satisfy (2.4)-(2.5) and con-
stants will be determined later, then we obtain
12
,cc





2
22
222 2
22
2
222 2
22
1
1
21 ,21
1
22 1
pp
p
Mp
kcc c
c
cpcc
 

p
ct
(2.2)
Thus
1006 W. H. QI ET AL.







1
222
22
22
1
2
1
1
222
22
22
2
1
2
1
22 1
sech 21
22 1
21
sech 21
ppp
p
ppp
p
uxpc c
cp x
pcc
vx
c
cp x







(2.6)
Finally, we have
Theorem 1. For any real constants ω, c, p, M satisfy-
ing
1
01, ,1,
2
cp
 
0M
(2.7)
there exist solitary wave of (2.1) in the form of (2.2),with
satisfying (2.6).
,uv
3. Main Results
Rewrite Equation (2.1) as
2
2
2
10
2,
pp
txx
t
pp
txx
i
x
R
n
nM


 
 
(3.1)
Let
n





u, and the function space in which we
shall work is
 
11
realcomplex real
2
X
HRH RL , with
inner pro d uc t


11112 22233
,Red
xxx x
R
f
gfg fgfgfgx
fg ,
,
X
fg
(3.2)
The dual space of X is *112
realcomplex real
X
HH L

 
*
, there
is a natural isomorphism :
I
XXdefined by
,,Igffg (3.3)
where denotes the pairin g between X and ,*
X
.
112233
,Red
R
By (3.2)-(3.4), it is obvious
2
2
1
1
I
x





Because the stability in view here refers to perturba-
tions of the solitary-wave profile itself, a study of the
initial-value problem for (1.1) is necessary.
Lemma 1. Let

11
0realcomplex real
2
H
RH RL u,
there exists
**0 0TTu
and a unique solution

112
*0
[0, );,0CTHHL
 u
*
T
u
u. In addition, ei-
ther
or

*
,X
x
ttuT.
Let 12
be one-parameter groups of unitary opera-
tor on X defined by
,TT

11 1
,,TssXs R
 uu u (3.5)
 

2
22 ,e,,,
is
TsnXsR
uu

(3.6)
Obviously
 
12
0
0,0
0
x
TT
x
x







 








i
It follows from Theorem 1 and (3.1) that there exist
solitary waves


12,, ,
,,
ccc
TctTtxx nx

 
with

,,,
,,
cCc
x
xn x


defined by

 
 
,
,
,
c
icx
c
c
xvx
x
eux
nx cvx

(3.7)
Let

,,,,
,,
cccc
x
xxn


x
In this and the following sections, we shall consider
the orbital stability of solitary waves
12 ,c
TctT tx
of (3.1). Note that Equation (3.1)
is invariant under
1
T
and
2
T,we define the orbital
stability as follows:
Definition 1. The solitary wave
 
12 ,c
TctT tx
is orbitally stable if for all 0
there exists 0
with
f
gfg fgx
fg (3.4)
the following property. If 0,cX
 u
and
tu
)
is a solution of (3.1) in some interval0with [0,t
0
0
uu, then
tucan be continued to a solution in
Copyright © 2011 SciRes. AM
W. H. QI ET AL.
Copyright © 2011 SciRes. AM
1007








11
22
0, 0,
0
EtEQt Q
QtQ

uu u u
uu
0t, and
 
12 11 2 2,
0sup
t

inf infcX
sRsRtTsTsx
 u
(3.12)
Otherwise
 
12 ,c
TctT tx
is called orbitally
unstable. Note that Equation (3.1) can be written as the follow-
ing Hamiltonian system
So long as ,c
are fixed we write ,,n

for

,,
 
,
,,
ccc

d
dJE
t
uu (3.13)
x
xn x
.



Define
222
2
11 11
22 24
2
d
xx
pp
En M
x


 
where J is a skew-symmetric linear operator, E is a func-
tional (the energy).
1
R
p
u

However, by (2.4)-(2.6), we have
(3.8)

,1, 2,
0
cc c
EcQ Q
 
 

(3.14)
where ,EQ
and 2
Q
are the Frechet derivatives of E,
and , with
1
Q2
Q

1u11
dI
m d
22
xx x
RR
Qnnx x

(3.9)
 

2
2
2
1
2
pp
xx
pp
xx
M
E
n

 







u



2
21d
2R
Q
u
x (3.10)
It is easy to verify that
Eu, and

1
Qu
2
Qu
are invariant under 12
, and formally conserved under
the flow of (3.1). Namely
,TT
 


 
 
2 212
1 2 2112
1 2 2212
,,
(),,
(),,
TsEforany s sR
sT sQforanyssR
sT sQforanyssR


uu
uu
uu

1
x
x
x
n
Qi




u
,

2
0
0
Q






u
11
11
21
E Ts
QT
QT
*
Define an operator from X to
X

,,1,2ccc
HE cQQ


 
 
,c
(3.15)
(3.11)
with
123
,,
y
yy X
y, and
and for any
,tRtu is a flow of (3.1)
 
22
212
2
224
2
,1 2
2
13
12
12
22 2
pp p
x
ppppp
c
x
p
Mp yycy
x
pp
3
H
py ic
x
x
cy y
 
 




 







 







ψy
Observe that ,c
H
is self-adjoint in the sense that (3.17)
*,c,c
H
H
This means that 1,c
I
H
is a bounded self-
adjoint operator on X. The spectrum of ,c
H
consists of
the real numbers
such that ,c
H
I
is not invert-
ible. We claim that 0
belongs to the spectrum of
By (3.16), Z is contained in the kernel of ,c
H
.
Assumptio n 1. (Spectral decomposition of ,c
H
)
The space X is decomposed as a direct sum
X
NZP
 (3.18)
,c
H
. By (3.11-3.15), it is easy to prove that
where Z is defined above, N is a finite-dimensional sub-
space such that
 
 
,1 ,
,2 ,
00
0
cc
cc
HT x
HTx
0


(3.16)
,,
c
H0
uu
for (3.19) 0Nu
Let and P is a closed subspace such that
 


11,2 2,12
00,
cc
Z
kTxk TxkkR

 

2
,,
cX
Huu u
for (3.20)
Pu
1008
with some constant
W. H. QI ET AL.
0
indept of
,c
(3.21)
and define to be the Hessian of fu
ear fo
enden . u
We define

,dc RR by :R

2
,dc Q
 
 
,1,cc
E cQ 

 

,dc

bilinnction d. It
is a symmetricrm. In addition, we use
pd
to express the numbers of positive eigenv alue of d
an

,c
nH
to express the numbers of negative eigealue
d
nv
of ,c
H
.
Theorem
 
E
2. Suppose that there exist three function
 
12
,,QQuu usatisfying (3.11) and (3.12), and
 
2 ,c
tTtx
satisfying (3.14).
Moreover, suppr c
H,
given by
(3.15) satisfies Assumption 1. If
,dc
on-degen-
erative, 13p and

pd n
 then solitary
waves
solitary waves
1
Tc
ose that the operato
H
is n
c,
,
12
tT
f. According t
,
Tc t
y stable.
Proo )-(3.15), we only nee
pr
cx are orbit
o (3.8
all
d to
mption 1 d
For any
ove that Assumption 1 and


,c
pd nH

hold.
First of all, we prove that Assuhold an

n,1
c
H
X
y, let
z
(2)
then

13 ,zzzy2221 22212
,e ,,Re
icx zziz z
3.2






2
22
, 1
2
2
22131
22
12 2
4222
2
13 3
2
22
113
2
22
1
1
121
,Re 1
21
22
2
2
d
1
1
2d
pp
c
R
p
x
pp
pp
x
xx
R
pp
p
p
Mpz
x
pzz
zczz
p
pzz
x
picz z
x
cz zzx
czcz z
Mp z
pzzx











 


yy

 
 
 


H


1 21212 2222
2
22
131 21
2
11 21212 2222
,,
d, ,
p
x
R
Lz zLzz
czzp zz
LzxL zzLzz
 
 
 
where


2
22
2
121
p
p
Lc Mp
x
 

2
2
12
11
22
pp c
Lp
x



2
22
12
11
22
pp pp
c
Lp p
x

 

2
2
22
1
22
pp c
Lx
 

(3.23)
Since 2
20c
, note that


2
11
2
2
22
2
1
22
1
22
c
LM
x
c
LM
x
 
 
x
x
(3.24)
with

1
M2
0, as;0,asxxMxx 
(3.25)
Thus, by Weyl’s theorem on the essential spectrumee
[5]), we have (s


22
1
22
2
[,),
22
[,),
22
ess
ess
cc
L
cc
L
 
 
0
0


(3.26)
Following from (2.3)-( 2.5 )
12
0, 0LuL u
(3.27)
By (2.6) and (3.27), wehas a simple see that
0u zero at
x
, thiouvill theoen Sturm-L implies that 0 is the
nd eigenvalue of rem
seco 1
L,and 1
L has exactly on e strictly
ve eigenvalue negati2,
with eigenfunctionan 1
.
In virtue of (3.24)), as[3], we
lowing lemma.
Lemma 2. For any real functions
-(3.27 in have the fol-
1
21
zHR, sa-
tisfying
21 121
,,zzu

0 (38)
there exists a positive number
.2
such that
10
1
2
121 21121
,
H
Lz zz
(3.29)
1
22
zHRLemma 3. For any real functions , sa-
tisfying 22 ,0zu ,
there exists a ber positive num
Copyright © 2011 SciRes. AM
W. H. QI ET AL.
1009
20
such that
1
2
222 22222
,
H
Lz zz
For any 22
We can
se
(3.30)

123221
,e, ,
icx
zzzzzizy
223
,z sim
Choo
ply denote by
121
,,zzzy
 


1222
22
pu1
22,
ppp
p
puvpuvpuvuL
uv L
11
,0,cz


y
then
2
, 1
, 0
c
H
 yy
1
,

(3.31)
note that the kernel of s s
two vectors:
Also
following
Let
c
H,
ipanned by the


0,1 x0,2
,,, ,0,0,,0
x x
vucu nu yy



10,120,212
1234
PX

p
(3.32)
21 2 3
,
,,, ,
,,,0
NkkR
Zkk kkR
pppp
ppupu



y
yy
p
Lemma 4. For any , defined by (3.32), there
exists a constant Pp
0
such that
2
,,
cX
H
ppp
(3.33 )
with
independent of
ny p.
For a

12122 3
,,,,
X
zz z zuu
21 22
21 2112
,,
,, ,
,,
zu zu
azzbb
uu uu

 (3.34)
then u10,1 20,2
abb
 
yy y
p.
Thus under the condition of (2Assumption 1 hold
and

1nH.
.7),
the following, we shall verify that
under the condition of theorem 1.
,c
In
p
From

d


,1
c
nH
 
,1, 2
,cc
dcE cQQ 

 
,c
we have

2
2
21
1
2, 2
1dsech
22
p
c
RR
c
dQ uxx
c

 

d
x


22
1, 2d
2
cc
R
dQ uvx
 
2
22
1
11
2
22
2
2
2
11
sech d
2(1)( 1)
sech dsech
p
R
pp
RR
c
cc
cxx
ccp
xxx





Let
2
d
x


c
2
1
sech d0
p
R
xx A
th
en
22
12
sech d0
1
p
R
xx A
p
.
Thus
22
11
22
,
22
c
cc
dAdA
cc c
 


 

 

 










2
2
1
2
2
2
2
1
2
2
2
2
2
12
1
211
12
1
211
2121
11
cc
pc
c
dc
cc cp
pc
c
Accp
cp
pc







 













A
 






2
2
1
2
2
2
1
2
2
12
1
211
21
211
c
pc
c
dc
ccp
cp
c
Accp







 









A
Therefore, we obtain
For
c
ccc
dd
ddd

 



13p
,
Copyright © 2011 SciRes. AM
W. H. QI ET AL.
Copyright © 2011 SciRes. AM
1010











22
11
22
22
2
22
22
11
2
22
det
22
12 2121
111 11
21
22
11
ccc c
ddddd
cc
AA
cc
pc
cp
cp pc
cp
cc
AA
ccccp
 

 

 

 


 





 


 



 


 
Thus, theorem 2 is proved completely.
4. References
[1] M .Grillakis, J. Shatah and W. Strauss, “Stability Th
of Solitary Waves in the Presence of Symmetry, ,”
Journal of Functional Analysis, Vol. 74, No. 1, 1987, pp.
0044-9
160-197. doi:10.1016/0022-1236(87)9
] M. Grillakis, J. Shatah and W. Strauss, “Stability Theory
aves in the Presence of Symmetry, ,”
Journal of Functional Analysis, Vol. 94, No. 2, 1990, pp.
991, pp. 1-19. doi:10.1093/imamat/46.1-2.1
[2 of Solitary W
308-348.
[3] I. Fukuda and M. Tsumi, “On Coupled Klein-Gordon-
Schrodinger
 Equations,” Journal of Applied Mathe-
matics, Vol. 66, 1978, pp. 358-378.
[4] J. P. Albert and J. L. Bona, “Total Positivity and the Sta-
bility of Internal Waves in Stratified Fluids of Finite
Depth,” IMA Journal of Applied Mathematics, Vol. 46,
No. 1-2, 1

1.pd




,1
c
nH pd


eory
m
[5] M. Reed and B. Simon, “Methods of Modern Mathe-
atical Physics: Fourier Analysis, Self-Adjointness,”
Academic Press, Waltham, 1975.
[6] B. L. Guo and L. Chen, “Orbital Stability of Solitary
Waves of the Long Wave-Short Wave Resonance Equa-
tions,” Mathematical Methods in the Applied Sciences,
Vol. 21, No. 10, 1998, pp. 883-894.