THE INVARIANT MANIFOLDS FOR A PERTURBED QUINTIC-CUBIC SCHRÖDINGER EQUATION

2004,24B(4):536-548

THE INVARIANT MANIFOLDS FOR A PERTURBED QUINTIC-CUBIC SCHRODINGER EQUATION 1 )

Chen Hanlin ( '*~#. School of Science, Southwest University of Science and Technology Mianyang 621002, China Guo Boling ( :1[iJilX ) Institute of Applied Physics and Computational Mathematics, Beijing 100088, China

Abstract In this paper, the authors apply the analytical method to establish the persistence of invariant manifolds for certain perturbation of the quintic-cubic Schrodinger equation under even periodic boundary conditions. Key words

Invariant manifold, persistence, perturbation

2000 MR Subject Classification

1

35Q53

Introduction

Invariant manifolds have a long history in dynamical systems theory and have been proven to be an important tool for thinking about or formulating problems arising in a remarkable broad range of applications. In this paper, we consider a perturbed quintic-cubic Schr6dinger equation (QCS)

(1.1) which is used to describe the propagation of optical soliton in optical fibers. Where q is 27r periodic and even in x,

D is a

bounded dissipative operator and is assumed to take the form

Dq

= -aq + f3qxx = -aq + f3Bq

(1.2)

for positive constants a and f3. Here His a Fourier truncation of oxx, i.e.,

Bcos (kx,).

w E

(~J.j2 - ~,~JJ5 + 1),

1 Received

={

-k2 cos(kx), 0,

u «.« »>«

e is a dissipation parameter and 0

March 14, 2002; revised May 6, 2003.

(1.3)

~ c«

1.

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2

Preliminary Results

2.1

Existence and Regularity of Solutions Consider the following nonlinear Schr6dinger equation (2.1)

it is a equation of system (1.1) at r = 1, jj is a bounded dissipative linear operator on H~,p (The Sobolev space of even, 27r periodic functions that are square integrable with square integrable first derivative). The PDE (2.1) is well posed in H~,p as the following theorem states. Theorem 2.1 For all qo E H~,p and for t E (-00,00), there exists a unique solution q(t, qo), continuous in t with values in H~,p , of equation (2.1) such that q(t = to) = qoi moreover, q(t,',') depends smoothly on qo and e. The proof of this theorem is a standard application of the energy method, for example, see

[15]. 2.2

Analysis of Space-Independent Solutions The plane of constants II e : II e := {q(x, t) : 8x q(x , t) == O} is an invariant plane for equation (2.1), and on II e the equation takes the form (2.2)

Equivalently, in terms of polar coordinates q := .Jj exp(iB), equation (2.2) takes the form

It

= -2c:[aI +.Jj cos e],

()t _- - 2(1 -w 2) - 12 + c:sin{) .Jj .

(2.3)

When e = 0, the unperturbed equations on II e take the form

It

= 0,

()t

= -2(1 -

w2) - 12,

a circle of fixed points So is given by 12 + 2(1 - w2) = 0, i.e., I = J2w 2 + 1 - 1 == 0, the unperturbed orbits on II e are nested circles with So. For c: > 0, the fixed points ?f equations (2.3) satisfy the following equations

aVf + cos () = 0, 2Vf (I - w2 )

+ I 2Vf -

c:sin{)

= 0.

Solving the above equations, we obtain two fixed points for the system (2.3)

(2.4)

Iq

= 0 -

()q

= tan-

1

2C:

1 - a 20

(2w2

2

+ 1)0 + Ot» ),

1[~J1-;2

0

] - 7r + O(c:).

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Linearizing the equations (2.3) around (Iq,Oq), we obtain a 2 x 2 matrix about the linear system: -2E(0: + 2~ cosO) [ -2 - 2I - _ 0 _ sinO

2cV! sino] .L

2Vf3

Vi

cosO

(Iq,()q)

Thus the associated growth rate around (Iq,Oq) is given by _

2

r:

O"q-±2 vc(0+1)

[(1(2w - 0: 0) OJ t -w+O ( c 3/2) . 2+1)

(2.5a)

Similarly, we have 2

O"p

- 0: 0 ) OJ = ±2i.ji (0 + 1) [ (1(2w 2 + 1)

t

) - w + 0 ( c / 3 2

.

(2.5b)

Thus, for positive e, there are two fixed points near So : a saddle Q and a spiral sink P, their growth rates are O"q and O"p, respectively. Although the circle of fixed points So for the unperturbed (c = 0) problem does not persist as a circle of fixed points, motion near So remains slow for small positive c. By introducing the variable J, J = I - w 2 , equations (2.3) can be written as Ji = -2c[0:(J + 0) + JJ + o cosO],

Ot

= -2J(0 + 1) -

(2.6)

J2 + ~ sinO. J+O

In order to describe the flow close to the stable manifold of the point Q, we rescale the coordinates by T

= ut,

J

= vj,

where v = ...{i. This rescaling is suggested by the expressions (2.5) for the growth rates

(2.7) O"q

and

O"p.

In these scaled coordinates, the equations (2.3) on the plane fie take the form

i, = -2 [0: (vj

+ 0) + VVj + OeasO] ,

O~=-2vj(0+1)-vj2+ ~sinO, VJ +

(2.8)

which can be written in terms of the variable y := (j,O)T as Y~

= Y(y, v),

(2.9)

where Y = (Y1 , y 2 )Tis defined by equations (2.8). In terms of these variables, the point Q has coordinates Yq = (jq,Oq) with .

1/

Io = - -

2

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Linearizing the equations around Yq and writing fj = Y - Yq, we obtain fj = Y' (Yq, v)fj + O(y'l), where Y'(Yq, v) is 2 x 2 matrix whose entries are

-2v(a + [ -2 (0

b

2 y vj + fl

+ 1) -

cosO)

2vj - ~c (vj

2JV:1+l1 sin 0 ]

+ O)-~ sinO

_v-cosO VVj+fl

(0)q, (}) q

Thus we see that the eigenvalues of Y'(Yq, v) are

A = -2 (0

J.L

+

1) J2"n

= 2 (0 + 1) J2"n

1)0 t -

2

1- a 0 ] [ (2w2 + 2

1- a 0 ] [ (2w2 + 1)0

t

-

av

oa/

+0

2

(v ) , (2.10)

+0

2

(v ) ,

with eigenvectors el (v) and e2 (v), respectively. The eigenvectors depend smoothly on v and

here 1 - a 2 0 > O. From regular perturbation theory and the stable manifold theorem, we have, for c sufficiently small, an open neighborhood U of Q, independent of c, such that the stable manifold of Q in U is a smooth function of v [5]. Therefore a portion of the local stable manifold can be parametrized in terms of

y=y*(s,v),

(2.11)

s = exp (AT)

for 0::; s ::; s, = exp (AT*) , with s, small and independent of c. We note that Q is an order 0 (v) perturbation of the point Qo that has coordinates (jo, ( 0 ) jo=O,

00=tan- 1

[~J1-;20]

-7r,

and that equations (2.8) are an 0 (v) perturbation of the conservative system

jT=-2(aO+Jf!cosO),

OT=-2j(0+1).

The energy of the above system is

E(j,8) ==

~ (0 + 1) j2

- aOO -

Jf! sin 0,

(2.12)

where the curve E(j,O) = E(jo, ( 0 ) is the stable manifold of the conservative system, which we denote by

C8 : Yo (T) =

(jo (T) ,80 (T)).

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If we fix a TO < T*, we have from regular perturbation theory that for stable manifold of Q is given by

TO

<

T

< T* the

Therefore, if we denote by C; the portion of the stable manifold corresponding to T E [To, 00) , then C; can be parametrized by 8 E [0,8 ol Y = Y* (8, v) ,

8 = exp (AT) ,

where Y* is a smooth function of 8 and u, The curve Y* is of order O(v) close to Yo for An identical result follows for unstable manifold of Q (denoted by C':).

3

TO

< T.

The Equations in a Neighborhood of Sf!

In order to study the dynamics of solutions to the nonlinear problem in a neighborhood of the circle of fixed points So, we write (2.1) in terms of the coordinates that are suited for this purpose. We introduce the coordinates (J, 0, f), where J and 0 are defined in Section 2, and f E ITt, These coordinates are determined in the following manner: First, write q as (3.1) q == (p (t) + f (x, t)) expiO (t),

•

where p and 0 are polar coordinates on the plane ITc and f E ITt; that is , f has spatial mean lT fglT f (x) dx = we denote by (-) the spatial mean over one period. zero

(21

0),

2

The L -nor m is a constant of motion for unperturbed flow, therefore, it will be used as a coordinate instead of p. I :=

2~

1

2lT

q

ii dx = p2 + (f

J) .

(3.2)

Since we are working in a neighborhood of the circle of fixed points So that corresponds to I = n, it will be convenient to introduce the variable J defined by (3.3)

J=I-n

Making use of (3.1)-(3.3), we will obtain the equations in the coordinates (J,e,f). We multiply (2.1) by ii, and take the (.) operation, then we take the imaginary part and obtain (3.4)

Jt = - 2c[ 0: (J + n) + p cos 0l + 2c For small J and

(j (i5 + 0:) f) .

l, we get Jt

= - 2c [0: (J + n) + J J + n cos B] + 1 (J, B, l, c) ,

(3.5)

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here eJ.)1 (J,O,f;e) = 0 (ej2). Noticing that Substituting the above equation to (2.1), we have

+ie(D + a)fe i8

-

iw(p + f)e i 8

-

ie.

We multiply (3.6) by e- i8 , and take the (.) operation ip; - pOt

= 2 (J + 0

2

2

- w ) P + 2p (f + f J) + 2 (f J f) + p5

+p3(3f2+6f J+t)+p2(3t +6f2 J+f 3) +p (2f3 J +3

i

(3.7)

f2) + ( t f3) - iwp - iee- i8 .

Taking the real part in (3.7), we have

Ot=-2(0+I)J-J

2

esinO

+ ..;J+TI+eJ.)2(J,O,f;e), J+O

(3.8)

where eJ.)2 (J, 0, f; e) = 0 (J t? + j2) for small J and f. Finally, (3.6) x e- i8 - (3.7), we obtain

ift -

to, = fxx -

2p2 (f+ J)

+ 2 (p2 - w2) f + 4p (1 J - (f J))

+2p (12 - (12)) + p4 (2f + 3 J) + ieD f + M +p3 [(3f 2 + 6f J + t) - (3f 2 + 6f J + t)] , here M

ift

= 0 (j3).

Substituting (3'.8) to the above equation, we get

= t; +2 (0 + 0 2) (f+ J)

+2 (1 + 20) J (1+ J) +ieDf +

~ +eJ.)3 (J,O,f;e),

(3.9)

where eJ.)3 (J,O,f;e) = 0 (ej2 + j2 + Jj2 + Pf). Thus, in terms of the coordinates (J,(J, f), equation (2.1) takes the form

Jt

= - 2e [a (J + 0) + v'J + 0 cos 0]

Ot

= -2 (0 + 1) J -

e sin 0

J2 + ..;J+TI + eJ.)2 (J, 0, f; e), J+O

ift = Ld + Wd + eJ.)3 (J,(J,f;e), where

+ eJ.) 1 (J, (J, f; e) ,

(3.10)

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Wef

= 2 (1 + 2n) J ( f+

-)

f

Vol.24 Ser.B

e sinO + JT+TI'

For J and f in a small but otherwise fixed neighborhood of 0, equations (3.10) can be considered as a perturbation of (2.6), and q)k (k = 1,2,3) are 21l"-periodic function in 0 of order

=0

q)l

(J,O,f;e)

(ep) ,

q)2

(J,O,f;e) = 0 (JP

q)3

(J,O,f;e)

=0

(ep

+ P),

(3.11)

+ P + Jp + Pf).

Since we will be working with invariant real manifolds in a neighborhood of So, it will be convenient to introduce a real coordinate system: U = (Ref, 1m!) T. In terms of these variables equations (3.10) can be written as

Jt

= - 2e [a (J + n) + v'J + n cos 0] + N 1 (J, 0, U; c) , esinO

Ot=-2(n+l)J-J 2+ JT+TI+ J+n Ut

N 2(J,O,u;e),

(3.12)

= Leu + Weu + N 3 (J, 0, U; c) ,

where N 3 is interpreted as a two-vector and

L, =

Gf}; - 4 (n + n2 ) S + cD,

TV: - -4(2n e -

here G =

4

+

I)JS

esinO G

+ v'J + n '

(~1 ~ ) , s ~ (~ ~ )

Invariant Manifolds

4.1

Analysis of the linear equations In a neighborhood of So, equations (3.12) can be viewed as a perturbation of the linear system

= 0, Ot = -2 (n + 1) J, J,

(4.1)

Ut = Leu. In order to study the local behavior of solutions to the nonlinear equations (3.12), we have to analyze the spectrum of the operator L e ; therefore, we consider the eigenvalue problem

for the eigenpairs{e (x), ,\}. Using Fourier expansions, one fined a quadratic expression for the eigenvalue ,\:

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where

d(j)

= { -0: - P (3, if

j

< K,

if

j

~

-0:,

denotes the symbol of we have for j

i5.

Since

W

E

=1

K

(~Vv'2 - ~, ~ J v'5 + 1), that implies 0 a~ ,u

= =fa + ed (1) ,

their corresponded eigenvectors are e, and e u respectively at c

1

(4.2)

= 0:

T

e s ,u = 2 . r;;;ro (1, ±a) cosx, y 1r H

here a

= J4 (0 + 0 2 ) -

+ 0 2 E (t, 1) ,

(4.3)

1. For j ~ 2,

(4.4) where OJ = jJP - 4 (0 + 0 2 ) > O. In terms of this eigenbasis, the mean zero function u can be written as

u (x) = vue u (x)

+ vse s (x) + Vo (x),

(4.5)

where Vu and Vs are real scalars and Vo E [span {II c , eu, es}]L . Using these variables, linear equations (4.1) split into

(it

= -2 (0 + 1) J, (4.6)

Thus, we explicitly see that, for e = 0, the linear equations have one unstable direction (e u ) , one stable direction (e s ) , and an. infinite number of center directions (J, 8, vo). By combining these center variables as Vc = (J, 8, Vo equations (4.6) can be written as

f,

Vu,t

= a~ V'"

(4.7)

where A is defined from equations (4.6) . In a b-neighborhood of So, the nonlinear equations (3.12) can be viewed as a perturbation of the linear equations (4.1). Under the flow of this linear equations and for e = 0, So has one-dimensional stable and unstable manifolds, together with a codimension 2 center manifold. Now we focus our attention to the center manifold EC(So), together with the center-stable ECS(So) and center-unstable ECU(So) manifolds,

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An important feature of the linear equations (4.6) is that the growth rates on the invariant manifolds are separated by a wide gap. To see this, we note that for e = 0 the spectrum of the operator has real part ±a and o. Thus, for any integer n and for c: < a / 4n we have

[lexp [At] II

s nC exp [alt

l

]

,

and the invariant manifolds ECs, ECU and EC can be described by solutions whose growth rates are bounded by exp [,;-,tJ for t > 0, exp [- ,;;] for t < 0, and exp] ~] for alIt, respectively. Solutions that belong to the stable manifold span{ eu} or unstable manifold span{ es} will have growth rates of order a. 4.2 Localized equations We start with fixing an integer no sufficiently large and a localization parameter <5 = a/nfi. Here a is a constant independent of e, which will be specified later. We introduce a localization function 'l/J 6' where 'l/J is Coo and satisfies

'l/J (s)

= { 1,

lsi:::; 1, lsi ~ l.

0,

The localization function localizes the equations (3.12)

Jt

= - 2c: [a (J

8

+ 0) + 2

JJ

Ot = -2 (0 + 1) J - J 6 +

8

+ 0 cos 0] + N l

c:sinO

IT""J7)

VJ8 +0

u; 0,

U6

;

+ N 2 (Jp O, u 8 ;c:) ,

s) ,

(4.8)

Ut = Leu + [Weu 6 + N 3 (J6, O, U6iC:)], where for any variable s we denote S6 = s'l/J (s/<5) . Note that we do not cut off the variable 0; the function 'l/J6 (J, u) has the effect of cutting off the right-hand sides whenever the phase point lies outside a neighborhood Us of the circle So. Moreover, all the nonlinear terms in equations (4.8) are either multiplied by e or are at least quadratic in (J, u) and localized in a <5-neighborhood of O. This implies that equations (4.8) has a global Lipschitz constant of order 0 (s + <5). This localization has the effect of keeping the flow unchanged in a <5 neighborhood of So while changing the nonlinear equations (3.12) to become a global <5-perturbation of a linear constant-coefficient system. Using v:= (vu,vs,vcf as variables and the operator A defined in equations (4.7), we can write equations (4.8) as

+ R~ (ViC:) , Vs,t = a;v s + R~ (v;c:), Vc,t = Avc + R~ (v;c:), Vu,t = a~vu

(4.9)

where RO (v;c:) and its first derivatives are of order 0 (s + <5). We will show that the localized equations has Ci-invariant manifolds that deform from ECs, ECU and EC. In turn, for the original equations, these manifolds will be locally invariant in a o-neighborhood of So.

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Statement and proof of the persistence theorem

Theorem 4.1 There exist a 8-neighborhood U8 of So and an co (8) "Ie E [0, co], (3.12) has a locally invariant (in U8) manifold of codimension 1

>

0 such that

at

where the function hu is in all of its arguments and 21l"-periodic in O. Moreover, for e = 0, WO's intersects ECS tangentially along So. Similarly, we have a locally invariant manifold given by

(4.11)

at

where the function hs is in all of its arguments and 21l"-periodic in O. Moreover, for e = 0, WO'u intersects ECU tangentially along So. Proof of Theorem 4.1 Once the integral equations are set up, this proof is just a standard application of a fixed-point argument. We give a brief statement here. First, we write equations (4.9) in integral form Vu

(t) = exp

[(7~ (t -

Vs

(t) = exp

[(7~ (t -

Vc

(t)

+

t u )]V u (tu)

r

exp

[(7~ (t -

i. t s )] s (ts) + rt exp [(7~ (t i. t V

= exp [At] V c (0) +

1

exp [A (t - s)]

s)] R~ s)] R~

(Vj

(Vj

c) ds,

c) ds,

R~ (e; c) ds.

Because of the gap in the growth rates, we will characterize the invariant manifold W:,S and Wfu by

W;S

= {VE Hi: ~~~ (exp [~:t] IlFt(v;e)IIHl) <

-l

(4.12)

(4.13) {VE Hi : ~~~ (exp [~;] IIFt(v;e)IIHl) < oo}, respectively, where Ft (v; c) is the flow of equations (4.9). Focusing our attention upon W:,S,

us We =

for

v in a ball B of arbitrary radius p we introduce the norm IlvlI>.

=

~~~ [ex p { - :t} Ilv (t)II H l] .

From the definition of W:,S we have for v E Wfs, exp [(7~ (-t u )]V u (tu) --+ 0, as Therefore for solutions on Wfs, the integral equations can be written as

J~ exp [(7~ (t -

Vu

(t) =

Vs

(t)

= exp [(7~t] V s (U) +

Vc

(t)

= exp [At] V c (0) +

s)]

t u --+ O.

R~ (v (s) j c) ds,

1 t

1

exp

[(7~ (t -

s)]

1

exp [A (t - s)]

R~ (v (s) ;e) ds,

R~ (v (s); c) ds.

(4.14)

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To show the existence of W~s we use Newton's iterations. Letting Vo = 0 and

v:+ 1 (t) = J~ exp [O"~ (t - s)] R~ (v k (s); c) ds,

I +I

v~+l (t) = exp [O";t] vs (0) + v~+l (t) = exp [At] V c (0)

t

exp [0"; (t - s)]

t

exp [A (t - s)]

This generates a well-defined sequence of functions. If

R~ (vk (s); c) ds,

(4.15)

R~ (v k (s) ;c) ds.

Ilv k IIHl

~ C, then

where the constant C is independent of c,no, and 8. Now by fixing 8 = a/n~, here a = obtain for all e < a/n~,

4b' we (4.16)

Therefore the sequence

{vk } is well-defined and

Ilvkllno ~ 2C(p).

The sequence {v converges. Since the nonlinear term is smooth, we have a similar estimate for the difference k}

(4.17)

which implies that v k -t v, a continuous function in t and

v with values in HI, and

that

Noticing that all terms in equations (4.14) are smooth with respect to V s (0) ,vc (0), and e, we can obtain that v is smooth in initial values and c. The sequence {v k} is C I . To show this we note that the derivative Dv k satisfies

IIDvk+III Hl

1

exp

[% (t - s)] IIR' Dv k IIHl ds

L~o (t -

S)]

IIR'DvkIIHl ds.

~ C exp [~~] + C +C

I

t

exp

00

By using the bounds on R', we have

IIDvkllno

~ 2C. To estimate the difference between two terms in the sequence which implies {Dv k } , we note that by the mean value theorem

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where the constant C in the above equation depends on the norm of R". Therefore, if t5v k = v k - V k- 1, we have

IIDvk+l - Dv k IIHI

1= +c it

~C

exp

[~ (t - s)]

exp

[2~o (t - s)]

[II R' (Dt5v

k)

IIHI

[IIR' (Dt5v

k)

+ IIDvk-lII HI IIt5v k IIHI] ds

IIHI

(4.18)

+ IIDvk-lIl HI lIt5vkIIHI] ds.

Noticing that the quadratic terms in the equations lead to an increase in the growth rate

This increase in the growth rate restricts the estimate of the difference of the first derivative to

here no > 4. Therefore sequence {v k } converges in C 1 by using the 11·11 ~ -norm. This procedure 2 can be repeated to obtain bounds on {Di v k } in the II'IIT-norm provided > 2. Passing to limit we obtain v E c' for £ < [T] - 1. With the above estimates we define hu

J~ exp [-:-a~s] R~ (v (s) ; €) ds, where hu is a

f-0

(vs,V c

,€) =

Vu

(0), and

Vu

(0) satisfies

ct functional with IIDhul1 ~ ~' and

Vu

(0)

is a C t manifold. The invariant of W~s follows from the definition of h u and from the invariance of the equations under time translation. Since the integral equations are 271"-periodic in ()and have a unique solution for every () E R, we can obtain that li« is 271"-periodic in (). Finally, the tangency of Was to So follows from observing that, for e = 0, R O is at least quadratic in u and J for all () E [0,271"]. An identical argument establishes the existence of a C t function li, such that and a C t locally invariant manifold for equation (2.1) given by

IIDhsl1

~ ~

The existence of a codimension 2 "slow manifold" ME: is then given by the following: Corollary 4.2 Let AlE: denote the intersection ME: = W~s n W~u. Then ME: is a locally invariant (in Uo) manifold of codimension 2

where the functions h~ s are c' in their arguments and 271"-periodic in (). Moreover, for e = 0, !vIo intersects EC tangentially along So. Proof The intersection of W~u and W:s can by described by the solution of the following equations

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Note that IIDhsll ~ ~ and IIDhul1 ~ ~. By the implicit function theorem the above system has a unique solution given by V u = h~ (ve;c) ,V s = h; (ve;c), where h~,s are C t functions and Me = {v E H 1 : V u = h~ (ve;c) ,Vs = h; (ve;c)}. Remark For 1- 0: 2 0 > 0, the pint Q, which is stationary under the flow (2.1), is given in terms of (J, (), u) by 1

Jq =-2

v« = O.

By linearizing equations (4.8) around Q we obtain that Q is a saddle point with a twodimensional unstable manifold and a codimension 2 stable manifold.. The unstable manifold intersects the plane of constants II e along the curve C,:, and intersects W~u along a curve tangent to the vu-direction. The local stable manifold of Q intersects II e along the curve C: and therefore intersects Me in a submanifold of codimension 1. References 1 Li Y, Mclaughlin D W, Shatah J, Wiggins S. Persistent homoclinic orbits for a perturbed nonlinear Schriidinger equation. Comm Pure and Appl Math, 1996, 49: 1175-1255 2 Mclaughlin D W, Shatah J. Melnikov Analysis for Pde's. Lectures in Appl Math, 1996,31: 51-100 3 Ball J M. Saddle point analysis for an ODE in a Banach space, and an application to dynamic bucking of a beam. Nonlinear Elasticity, 1973. 93-160 4 Bates P W, Lu K, Zong C. Persistence of overflowing manifolds for semiflow. Comm Pure Appl Math, 1999, 52: 983-1046 5 Hale J. Ordinary Differential equations. New York: Robert E Krieger Publishing Co, 1980 6 Guo Boling, Jing Zhujun, Lu Bainian. Slow timeperiodic solutions of cubic quintic Ginzburg-Landau equation(I). Prog Nat Sci, 1998, 8(4): 403-415 7 Mclaughlin D W, Overman E. Whiskered tori for integrable PDEs: chaotic behavior in near integrable PDEs. Surveys in Appl Math, Vol 1. New York: Plenum, 1995. 83-203 8 Guo Boling, Jing Zhujun, Lu Bainian. Spatotemporal complexity of the cubic Ginzburg-Landau equation. Comm on Nonl Sci, and Num Simul, 1996, 2(2): 7-14 9 Yagasaki K. The method of Melnikov for perturbations of multi-degree-of-freedom Hamiltonian systems. Nonlinearity, 1999, 12: 799-822 10 Kapitula T, Rubin J. Existence and stability of standing hole solutions to Complex Ginzburg-Landau equations. Nonlinearity, 2000, 13: 77-112 11 Kovacic G. Singular perturbation theory for homoclinic orbits in a class of near integrable dissipative system. SIAM J Math Anal, 1995, 26: 1611-1643 12 Fenichel N. Geometric singular perturbation theory for ordinary differential equations. J Differential Equations, 1979, 31: 53-98 13 Wiggins S. Global Bifurcations and Chaos, Analytic Methods. New York: Springer-Verlag, 1988 14 Li Y, Wiggins S. Invariant manifolds and their fiberations for perturbed NLS equations. New York: Spring-Verlag, 1997 15 Cazenave T. An Introduction to Nonlinear Schrodinger Equations. IMUFRJ Rio De Janeiro, 1989. 22 16 Chen Hanlin, Dai Zhengde. Exponential attrctor of the Cauchy Problem of disspative Zakharov equations in R. Acta Math Sci, 2001, 20A(3): 373-383