Homoclinic Orbits and Chaos in Discretized Perturbed NLS ...faculty.missouri.edu/~liyan/lattice-horbit.pdf · Homoclinic Orbits and Chaos in Discretized Perturbed NLS Systems: Part

J. Nonlinear Sci. Vol. 7: pp. 211–269 (1997)

© 1997 Springer-Verlag New York Inc.

Homoclinic Orbits and Chaos in DiscretizedPerturbed NLS Systems:Part I. Homoclinic Orbits

Y. Li 1∗ and D. W. McLaughlin21 Department of Mathematics, University of California at Los Angeles, Los Angeles, CA 90024,

USA2 Courant Institute of Mathematical Sciences, 251 Mercer Street, New York, NY 10012, USA

Received September 1, 1995; revised manuscript accepted for publication September 11, 1996Communicated by Jerrold Marsden

Summary. The existence of homoclinic orbits, for a finite-difference discretized form ofa damped and driven perturbation of the focusing nonlinear Schroedinger equation undereven periodic boundary conditions, is established. More specifically, for external param-eters on a codimension 1 submanifold, the existence of homoclinic orbits is establishedthrough an argument which combines Melnikov analysis with a geometric singular per-turbation theory and a purely geometric argument (called the “second measurement” inthe paper). The geometric singular perturbation theory deals with persistence of invariantmanifolds and fibration of the persistent invariant manifolds. The approximate locationof the codimension 1 submanifold of parameters is calculated. (This is the material inPart I.) Then, in a neighborhood of these homoclinic orbits, the existence of “Smalehorseshoes” and the corresponding symbolic dynamics are established in Part II [21].

Key words. discrete nonlinear Schroedinger equation, spectral theory, persistent invari-ant manifolds, Fenichel fibers, Melnikov analysis, homoclinic orbits

MSC numbers. 34, 35, 39, 58

PAC numbers. 02, 03, 42, 63

∗Present address: Department of Mathematics 2-336, Massachusetts Institute of Technology, Cambridge, MA02139, USA.

212 Y. Li and D. W. McLaughlin

1. Introduction

In this paper, we prove the existence of orbits, homoclinic to a saddle fixed point, of thefollowing N particle (any 2< N 3,

3 tanπ

3< ω 0), β (> 0), 0 (> 0) are constants.The restriction onω ensures that, forN > 2, the uniform solution (|qn| = ω) to theunperturbed (² = 0) form of (1.1) has a codimension 2 center manifold, a codimension1 center-unstable manifold, and a codimension 1 center-stable manifold. (By increasingω to other intervals, one has codimensionk, for any finitek, center-unstable and center-stable manifolds for|qn| = ω. Their study is parallel to that given in this paper. Note alsothat in the caseN = 2, |qn| = ω is neutrally stable for anyω, and there is no hyperbolicstructure; hence, we will not study this case.)

This system (1.1) is a finite-difference discretization of the following perturbed NLSPDE:

iqt = qxx + 2[|q|2− ω2

]q + i ²

[− αq + βqxx + 0

], (1.2)

whereq(x + 1) = q(x), q(−x) = q(x); π < ω < 2π , ² ∈ [0, ²1), α (> 0), β (> 0),and0 (> 0) are constants. Thus, system (1.1) is of interest both as a perturbation ofa completely integrable Hamiltonian system of large, but finite, dimension; and as anapproximation to a PDE.

Homoclinic Orbits and Chaos in Discretized Perturbed NLS Systems: Part I 213

The motivation for this study comes from the numerical experiments on chaos innear-integrable systems as summarized in [22], where many references can be found.These numerical experiments show that perturbed nonlinear Schroedinger equations(1.2), actually discretizations thereof such as (1.1), possess solutions with beautifulregular spatial patterns which evolve irregularly (chaotically) in time. These numericalstudies also correlate this chaotic behavior in the perturbed system with the presence ofa hyperbolic structure in the unperturbed(² = 0) integrable NLS equation [20].

In order to begin an analytical study of the perturbed NLS system, we introduced in ref-erence [3] an extremely crude truncation of the perturbed NLS PDE to a four-dimensionaldynamical system, which we then studied both numerically and analytically [2]. In sev-eral papers [15] and [23], by ourselves and others, the existence of homoclinic orbitsand a symbol dynamics was established for the perturbed four-dimensional dynamicalsystem. In this paper, we establish the existence of homoclinic orbits in the 2(M + 1)dimensional finite-difference approximation (1.1) of the perturbed NLS PDE (1.2) forany finite M . A symbol dynamics for (1.1) is studied in a companion paper [21], andhomoclinic behavior for the PDE (1.2) is studied in [17].

There are some key differences of this study from the earlier four-dimensional sys-tem study [23]: (i) This study is for a system with an arbitrary number of N particles;therefore, when N is large enough, the behavior of such a system will be very similarto the limiting PDE. (ii) The integrable theory for the unperturbed N particles system isnovel. (iii) The “second measurement” (see below for definition) is more difficult anddelicate in this higher dimensional case. (iv) In this study we present a detailed derivationof the Melnikov distance estimate for this particular high dimensional system. (v) In acompanion paper, we establish the existence of a symbolic dynamics for this high di-mensional system which for arbitrary N is quite different from the symbol dynamics inthe four-dimensional case in that it has oscillatory motion reminiscent of a Silnikov be-havior in high dimensions. (vi) Finally, we view this work in arbitrarily large (but finite)dimensions as a necessary step toward a dynamical system analysis of the PDE (1.2).

Related studies on NLS systems can be found in the works [10] [9].Now we state our results on the system (1.1): Denote by6N (N ≥ 7) the external

parameter space,

6N ={(ω, α, β, 0)

∣∣∣∣ ω ∈ (N tan πN , N tan2πN),

0 ∈ (0, 1), α ∈ (0, α0), β ∈ (0, β0);whereα0 andβ0 are any fixed positive numbers

}.

Theorem 1.1. For any N (7 ≤ N < ∞), there exists a positive number²0, such thatfor any² ∈ (0, ²0), there exists a codimension1 submanifold E² in6N; for any externalparameters (ω, α, β, 0) on E² , there exists a homoclinic orbit asymptotic to a fixed pointq² . The submanifold E² is in an O(²ν) neighborhood of the hyperplaneβ = κ α, whereκ = κ(ω; N) is shown in Figure 1.1,ν = 1/2− δ0, 0< δ0¿ 1/2.

Remark 1.1. In the cases (3≤ N ≤ 6), κ is always negative as shown in Figure 1.1.Since we require both dissipation parametersα andβ to be positive, the relationβ = κα


shows that the existence of homoclinic orbits violates this positivity. ForN ≥ 7, κ canbe positive as shown in Figure 1.1. WhenN is even and≥ 7, there is in fact a pairof homoclinic orbits asymptotic to a fixed pointq² at the same values of the externalparameters; since for evenN, we have the symmetry: Ifqn = f (n, t) solves (1.1), thenqn = f (n+N/2, t)also solves (1.1). WhenN is odd and≥ 7, the study can not guaranteethat two homoclinic orbits exist at the same value of the external parameters.

This theorem is established in several steps: First, integrable theory in the form of Laxrepresentations and Backlund transformations is used to construct homoclinic orbits inthe integrable unperturbed (² = 0) case. In addition, through the Floquet discriminant1(z;qn) of the isospectral difference operatorLn, critical hyperbolic tori are identi-fied, together with natural representations of their stable and unstable manifolds. Thisbackground material from integrable theory is summarized in Section 2.

Next, one notes that a plane “of constants” is invariant for both the unperturbed(² = 0) and the perturbed (² > 0) flows. When restricted to this plane5, the dynamicscan be analized explicitly; and, in particular, a saddle fixed pointq² can be identifiedwhich emerges for² > 0 from a circle of fixed pointsSω when² = 0. A change ofcoordinates is then introduced which is centered upon the plane5. In these coordinates,persistent invariant manifolds are constructed in a neighborhood of the plane5. Theseconstructions are given in Section 3.

The persistent homoclinic orbits are singular deformations of the integrable (² =0) orbits. This singular nature is apparent in numerical studies [22], and it has beenestablished in the four dimensional truncated system [15] [23]. It arises because of thepresence of motion on two distinct time scalesO(t)when the phase point is far from theplane5 andO(

√²t) when the phase point is near a resonant circle on5. Because of

these two distinct time scales, we use geometric singular perturbation theory. Specifically,we represent the center-stable and center-unstable manifolds of the plane5 through thefibers of Fenichel [7]. These constructions are described in Section 4.

With these preliminaries in place, the proof of the persistent homoclinic orbits pro-ceeds with two measurements. Consider the saddle fixed pointq² on the invariant plane5. We seek an orbit, not on5, which is homoclinic toq² . The unstable manifoldWu(q²)of q² is two-dimensional. Because of the two distinct time scales, trajectories inWu(q²)leaveq² and remain near5 for a longO(1/

√²) time, before they rapidlyO(t) fly away

from 5. With the Fenichel fibers, we can define a “take-off angle”θT at which thetrajectory “flies away from5” and use this angle to label trajectories inWu(q²),

q(u,²)(t; θT ) ∈ Wu(q²).We determine ifq(u,²)(t; θT ) ∈ Wu(q²) with two measurements.

The first measurement is described in Section 5. We note that the center-stable man-ifold Wcs0 for the unperturbed (² = 0) integrable system is codimension 1, and that itpersists as a codimension 1 locally invariant manifoldWcs² for the perturbed (² > 0)system. Moreover,Ws(q²) ⊂ Wcs² . Thus, we first ask ifq(u,²)(t; θT ) ∈ Wcs² ? SinceWcs²is codimension 1, the answer requires only one measurement, which is accomplishedwith a “Melnikov integral,”

dist{q(u,²)(t; θT );Wcs²

} = ²M̂F (θT ;α, β, 0)+ o(²),


where

M̂F (θT ;α, β, 0) =∫ ∞−∞(gradF, gpert)|h(t;θT ) dt,

and wheregpert denotes the perturbation to the NLS equation which depends upon theexternal parameters (α, β, 0). The trajectoryh(t; θT ) denotes the homoclinic orbit forthe integrable system which is labeled by the take-off angleθT on the circle of fixedpoints Sω. A simple zero (inθT ) of the Melnikov function establishes, by an implicitfunction theorem argument, the existence of a trajectoryq(u,²)(t; θ∗∗T )which approachesq² as t → −∞, and resides inWcs² (provided that the behavior of this trajectory canbe controlled ast → +∞, control is required becauseWcs² is only a locally invariantmanifold). The latter control will be provided by the “ second measurement.”

The valueθ∗∗T of the “take-off” angle is a small perturbation of the simple zeroθ∗T

of the Melnikov functionM̂F . This “take-off” angleθ∗T is a function of the externalparameters (α, β, 0), and the Melnikov measurement can be interpreted as determining(approximately) that take-off angleθ∗∗T (α, β, 0) for which the trajectoryq

(u,²)(t; θ∗∗T ) ∈Wcs² .

The final step in the argument, which we call the “second measurement,” establishesthat q(u,²)(t; θ∗∗T ) ∈ Ws(q²). For θT = θ∗∗T , it is known thatq(u,²)(t; θ∗∗T ) ∈ Wcs² , andwhile Ws(q²) is codimension 2,within Wcs² , the stable manifoldW

s(q²) is codimension1. Thus, only one additional measurement is required to determine ifq(u,²)(t; θ∗∗T ) ∈Ws(q²). To set up this measurement successfully, certain terms which are quadraticallynonlinear must first be eliminated by a normal form transformation. The measurementitself is carried out using “energy-coordinates.” The technical details are described insection 6. This last measurement places one constraint on the external parameters; thus,the homoclinic orbit persists on a codimension 1 set in parameter space.

The two measurements control the behavior ofq(u,²)(t; θ∗∗T )ast →+∞and establishthe existence of an orbit homoclinic toq² . In a companion paper [21], the existence of“Smale horseshoes” and a corresponding symbolic dynamics is established, genericallyin a neighborhood of such homoclinic orbits.

2. Integrable Background

In this section, we give all the integrable preliminaries needed for studies in later chapters.Of particular importance is (i) the simple family of integrable homoclinic orbits givenby (2.12); (ii) the constant of motioñF1 given by equation (2.8); and (iii) the definitionsand representations of the integrable center-stable abd center-unstable manifoldsWcs,Wcu given in Corollary 2.

Setting² = 0 in the discretized perturbed NLS system (1.1), we get the followingHamiltonian system:

i q̇n = 1h2

[qn+1− 2qn + qn−1

]+ |qn|2(qn+1+ qn−1)− 2ω2qn, (2.1)

Fig. 1.1. The κ = κ(ω, N)curves for different values ofN.In particular, note its positivity.


which can be written in (noncanonical) Hamiltonian form,

q̇n = ρn ∂H∂q̄n

,

˙̄qn = −ρn ∂H∂qn

,

where

ρn ≡ 1+ h2|qn|2,

H ≡ − ih2

N−1∑n=0

{q̄n(qn+1+ qn−1)− 2

h2(1+ ω2h2) ln(1+ h2|qn|2)

}.

Moreover,∑N−1

n=0 {q̄n(qn+1+qn−1)} itself is also a constant of motion of the system (2.1).This invariant, together withH , implies that

∑N−1n=0 ln ρn is a constant of motion too.

Therefore,

D2 ≡N−1∏n=0

ρn (2.2)

is a constant of motion.

2.1. Lax Pair Representation

Additionally, system (2.1) is an integrable Hamiltonian system [1], with(M+1)constantsof motion. This integrability is proven with the use of the discretized Lax pair [1]:

ϕn+1 = L(z)n ϕn, (2.3)ϕ̇n = B(z)n ϕn, (2.4)

where

L (z)n ≡(

z ihqn

ihq̄n 1/z

),

B(z)n ≡i

h2

(1− z2+ 2iλh− h2qnq̄n−1+ ω2h2 −zihqn + (1/z)ihqn−1

−i zhq̄n−1+ (1/z)ihq̄n 1/z2− 1+ 2iλh+ h2q̄nqn−1− ω2h2),

and wherez≡ exp(iλh). Compatibility of the over-determined system (2.3), (2.4) givesthe “Lax representation”

L̇n = Bn+1Ln − Ln Bnof the discrete NLS (2.1).


2.2. Spectral Theory ofLn

Focusing attention upon the discrete spatial flow (2.3), we letY(1),Y(2) be the funda-mental solutions of the ODE (2.3), i.e., solutions with the initial conditions,

Y(1)0 =(

10

), Y(2)0 =

(01

).

TheFloquet discriminant,

1: C× S 7→ C, (2.5)is defined by

1(z; Eq) ≡ tr{M(N; z; Eq)}, (2.6)whereS is the phase space defined as follows:

S ≡{Eq =

(qr

) ∣∣∣∣ r = −q̄, q = (q0,q1, . . . ,qN−1)T ,qn+N = qn, qN−n = qn

},

andM(n; z; Eq) ≡ columns{Y(1)n ,Y(2)n } is the fundamental solution matrix of (2.3). InS(viewed as a vector space over the reals), we define the inner product, for any two pointsEq and Ep, as follows:

〈Eq, Ep〉 = 2 Re{ N−1∑

n=0q̄n pn

}.

And the norm ofEq is defined as‖Eq ‖2 ≡ 〈Eq, Eq〉.

Remark 2.1.1(z; Eq) is a constant of motion for the integrable system (2.1) for anyz ∈ C. Since1(z; Eq) is a meromorphic function inz of degree (+N,−N), the Floquetdiscriminant1(z; Eq) acts as a generating function for(M+1) functionally independentconstants of motion and is the key to the complete integrability of the system (2.1).

The Floquet theory here is not standard, as can be seen from the Wronskian relation,

WN(ψ+, ψ−) = D2 W0(ψ+, ψ−),

whereD is defined in (2.2),

Wn(ψ+, ψ−) ≡ ψ(+,1)n ψ(−,2)n − ψ(+,2)n ψ(−,1)n ,

ψ+ andψ− are any two solutions to the linear system (2.3). In fact,Wn+1(ψ+, ψ−) =ρnWn(ψ+, ψ−). Due to this nonstandardness, modification of the usual definitions ofspectral quantities [16] is required.

Periodic and antiperiodic pointszs are defined by

1(zs; Eq) = ±2D.


A critical point zc is defined by the condition

d1

dz|(zc;Eq) = 0.

A multiple pointzm is a critical point which is also a periodic or antiperiodic point. Thealgebraic multiplicityof zm is defined as the order of the zero of1(z)± 2D. Usually itis 2, but it can exceed 2; when it does equal 2, we call the multiple point adouble point,and denote it byzd. Thegeometric multiplicityof zm is defined as the dimension of theperiodic (or antiperiodic) eigenspace of (2.3) atzm, and is either 1 or 2.

An important sequence of constants of motionF̃j [20]

F̃j : Ä ⊂ S 7→ C (2.7)

is defined by

F̃j (Eq) = 1D1(zcj (Eq); Eq), (2.8)

whereÄ ⊂ S is the domain ofF̃j . We are going to usẽF1 to build a Melnikov integralin Section 5.

Remark 2.2(Continuum Limit). In the continuum limit (i.e.,h→ 0), the Hamiltonianhas a limit in the mannerHh → Hc, whereHc is the Hamiltonian for NLS PDE,Hc =i∫ 1

0 {qxq̄x+2ω2|q|2−|q|4} dx. The Lax pair (2.3), (2.4) also tends to the correspondingLax pair for NLS PDE with spectral parameterλ (z = eiλh) [20]. If Q ≡ maxn{|qn|}is finite, thenρn → 1 ash → 0. Therefore,D2 ≡ {

∏N−1n=0 ρn} → 1 ash → 0. The

nonstandard Floquet theory for the spatial part of the Lax pair (2.3) becomes standardFloquet theory in the continuum limit.

Some useful symmetries of the eigen-functions of the Lax pair (2.3), (2.4) (which areused in the explicit calculation of Section 2.3) may be found in Appendix A.

2.3. Hyperbolic Structure and Homoclinic Orbits

The hyperbolic structure and homoclinic orbits for (2.1) are constructed through theBacklund-Darboux transformations, which were built in [16]. First, we present theBacklund-Darboux transformations. Then, we show how to construct homoclinic or-bits.

In fact, we will present a form of Backlund-Darboux transformations which is mostrelevant to hyperbolic structure. Fix a solutionqn(t) of the system (2.1), for which thelinear operatorLn has a double pointzd of geometric multiplicity 2, which is not onthe unit circle. We denote two linearly independent solutions (Bloch functions) of thediscrete Lax pair (2.3), (2.4) atz = zd by (φ+n , φ−n ). Thus, a general solution of thediscrete Lax pair (2.3), (2.4) at(qn(t), zd) is given by

φn(t; zd, c) = φ+n + cφ−n ,


wherec is a complex parameter called a Backlund parameter. We useφn to define atransformation matrix0n by

0n =(

z+ (1/z)an bncn −1/z+ zdn

),

where

an = zd

(z̄d)21n

[|φn2|2+ |zd|2|φn1|2

],

dn = − 1zd1n

[|φn2|2+ |zd|2|φn1|2

],

bn = |zd|4− 1(z̄d)21n

φn1φ̄n2,

cn = |zd|4− 1

zdz̄d1nφ̄n1φn2,

1n = − 1z̄d

[|φn1|2+ |zd|2|φn2|2

].

From these formulae, we see that

ān = −dn, b̄n = cn.

Then we defineQn and9n by

Qn ≡ ih

bn+1− an+1qn, (2.9)

and

9n(t; z) ≡ 0n(z; zd;φn)ψn(t; z), (2.10)whereψn solves the discrete Lax pair (2.3), (2.4) at(qn(t), z). Formulas (2.9) and(2.10) are the Backlund-Darboux transformations for the potential and eigenfunctions,respectively. We have the following theorem [16].

Theorem 2.1(Backlund-Darboux Transformations).Let qn(t) denote a solution of thesystem (2.1), for which the linear operator Ln has a double point zd of geometric mul-tiplicity 2, which is not on the unit circle and which is associated with an instability.We denote two linearly independent solutions (Bloch functions) of the discrete Lax pair(2.3), (2.4) at(qn, zd) by (φ+n , φ

−n ). We define Qn(t) and9n(t; z) by (2.9) and (2.10).

Then

1. Qn(t) is also a solution of the system (2.1). (The evenness of Qn can be guaranteedby choosing the complex Backlund parameter c to lie on a certain curve, as shown inthe example below.)

2. 9n(t; z) solves the discrete Lax pair (2.3), (2.4) at(Qn(t), z).


3. 1(z; Qn) = 1(z;qn), for all z ∈ C.4. Qn(t) is homoclinic to qn(t) in the sense that Qn(t)→ ei θ± qn(t), exponentially as

exp(−σ |t |) as t→ ±∞. Hereθ± are the phase shifts,σ is a nonvanishing growthrate associated with the double point zd, and explicit formulas can be developed forthis growth rate and for the phase shiftsθ±.

This theorem is quite general, constructing homoclinic solutions from a wide class ofstarting solutionsqn(t). Its proof is by direct verification [16], [20].

Next, we study specifically the most important example—Backlund-Darboux trans-formations for the uniform solutions of (2.1). Let

qn = q, ∀n; q = a exp{−2i [(a2− ω2)t ] + i γ }, (2.11)whereN tan πN < a < N tan

2πN for N > 3, 3 tan

π3 < a 3, (2.14)

3 tanπ

3< |q0|


Fig. 2.1. Geometric illustration of the singular level sets in “figure 8⊗ A,” and theircorresponding spectral identification.

Define the “resonance circle” inA,

Sω ≡{Eq∣∣∣∣ Eq ∈ 5, |q0| = ω}. (2.15)

Sω entirely consists of fixed points under the integrable flow (2.1).

The formula in the above corollary represents the singular level set connecting to acircle in the annulusA. See Figure 2.1 for its geometric illustration and the correspondingspectral identification. For more detail, see [20] and [22].


In fact, we can spell out the following theorem from the Backlund-Darboux transfor-mation theorem 2.1.

Corollary 2. In S, under the integrable flow (2.1), there exist a codimension 1 “center-unstable manifold of Sω,” denoted by Wcu, and a codimension 1 “center-stable manifoldof Sω,” denoted by Wcs. There also exists a codimension 2 “center manifold of Sω,”denoted by Wc; Wc ∩5 = A. Moreover, Wcu = Wcs =(figure 8)⊗Wc.

Proof. This is a corollary of Theorem 2.1.

Remark 2.3. In principle, “(figure 8)⊗Wc” can be calculated through the Backlundtransformations given in theorem 2.1; the special case “(figure 8)⊗A” has been calculatedin corollary 1. We can represent “(figure 8)⊗A” by the heteroclinic orbit formulae inCorollary 1 as follows:

“(figure 8)⊗ A” =⋃

t∈(−∞,∞),γ∈[0,2π ],ω∈Iω,σ=±Qn(t; N, ω, γ, r = 0, σ ),

whereIω denotes the restriction interval onω given in (1.1), andQn is given in Corol-lary 1.

3. Persistent Invariant Manifolds

In this section, we are going to discuss persistence of the center-unstable, center-stable,and center manifolds ofSω: Wcu, Wcs, andWc, respectively. Here, “persistence” meansthat there exists a positive number²2, such that, for any² ∈ (−²2, ²2), there existcodimension 1 locally invariant manifoldsWcu² andW

cs² , and a codimension 2 locally

invariant manifoldWc² . Moreover,Wcu0 = Wcu, Wcs0 = Wcs, andWc0 = Wc. In a certain

neighborhood ofSω, Wcu² , Wcs² , andW

c² are smooth in² for ² ∈ (−²2, ²2). We will give

precise definition of “local invariancy.” The simple example in the next subsection willshow the necessity of the introduction of the concept oflocal invariancy.

3.1. Persistent Invariant Plane

It is easily seen that the plane5 remains an invariant manifold under the perturbed flow(1.1); however, motion on this plane is very different in the perturbed and integrablecases, either of which can be analyzed with phase plane methods.

On the invariant plane5, the dynamics is governed by the following two-dimensionalODE:

iqt = 2[|q|2− ω2

]q + i ²

[− αq + 0

]. (3.1)

Changing variables to an amplitude-phase representation (q = I ei γ , I = |q|) yields the


following pair of equations:

İ = ²(−α I + 0 cosγ ), (3.2)γ̇ = −2(I 2− ω2)− ²(0/I ) sinγ. (3.3)

The phase plane diagram of this system is shown in Figure 3.1. On the plane5 and inanO(

√²) neighborhood of the resonance circleI = ω, we let I ≡ ω+√²J, τ ≡ √²t ,

and obtain

J ′ = −∂H1∂γ−√²αJ, (3.4)

γ ′ = ∂H1∂ J−√²

[2J2+ 0

ω +√²J sinγ], (3.5)

where′ ≡ ∂∂τ

,H1 ≡ αγω−0 sinγ − 2ωJ2. A simple phase plane analysis shows that,to the first orderO(²0), the dynamics of (3.4) and (3.5) is thefishdynamics as shownin Figure 3.2, and to the orderO(

√²), the dynamics of (3.4) and (3.5) is the brokenfish

dynamics as shown in Figure 3.3. Under the perturbed dynamics, there are three fixedpoints on the plane5: a sinko² near the origin; a sinkp² near the resonance circle; anda saddleq² which is also near the resonance circle. In the coordinates of thefish, thesaddle fixed point (denoted bỹq²) is located at(J, γs) = (0, arccos{χαω}); thus, we seethat the condition for the existence of thefishstructure is

χα ≤ 1ω, (3.6)

while thenoseof thefish is located at(J, γn) = (0, γn). Hereγn satisfies the equationχαωγn − sinγn = χαωγs − sinγs. (3.7)

From the above phase plane analysis, we see that, although5 (2.13) is still an invariantplane,A defined in (2.14) is not an invariant subset any more. Nevertheless,A is locallyinvariant, according to the definition given in the next subsection.

3.2. Persistent Invariant Manifold Theorem

Normally hyperbolic invariant manifold theorems have been established by Fenichel [4],Kelley [13], Hirsch, Pugh, and Shub [11], Sacker [24], and others. Here, we mainly followFenichel [4], [7] for a persistence theorem of normally hyperbolic invariant manifolds,under small perturbation of the flow, which he proved through a combination of a graphtransform method first introduced by Hadamard [8] for two-dimensional maps and ageometric singular perturbation theory developed in [7].

Definition 2 (Local Invariancy). LetFt be a flow (a solution operator) defined onS, Vbe a submanifold ofS, with boundary∂V (V̄ ≡ V ∪ ∂V). We sayV is locally invariantunder the flowFt in S, if there exists a neighborhoodU of V in S, such that, for allQ ∈ V , if τ ≥ 0 and⋃t∈[0,τ ] Ft (Q) ⊂ U , then⋃t∈[0,τ ] Ft (Q) ⊂ V , and ifτ ≤ 0 and


Fig. 3.1.Phase plane diagram of the dynamics on5.

⋃t∈[τ,0] F

t (Q) ⊂ U , then⋃t∈[τ,0] Ft (Q) ⊂ V . Intuitively speaking, the orbitFt (Q),starting from the pointQ in V , can leaveV in forward or backward time; nevertheless,it can only leaveV through the boundary∂V .

Definition 3 (The Maximal Invariant Set). LetFt be a flow (a solution operator) de-fined onS, andV be a submanifold ofS. Define

A+(V) ≡{

Q ∈ V∣∣∣∣ ⋃

t∈[0,∞]Ft (Q) ⊂ V

},


A−(V) ≡{

Q ∈ V∣∣∣∣ ⋃

t∈[−∞,0]Ft (Q) ⊂ V

},

I (V) ≡{

Q ∈ V∣∣∣∣ ⋃

t∈[−∞,∞]Ft (Q) ⊂ V

}.

We callA+(V), A−(V), andI (V) the maximal positively invariant set inV , the maximalnegatively invariant set inV , and the maximal invariant set inV , respectively.

Fig. 3.2.The “fish” dynamics.

Fig. 3.3.The broken “fish” dynamics.


The persistent invariant manifold theorem stated specifically for our system (1.1) isas follows [7].

Theorem 3.1(Local Persistent Invariant Manifolds).In the phase spaceS there existsa neighborhoodU of Sω, inside of which, for every fixed k (2 ≤ k < ∞), there existsa positive number²2 = ²2(U , k), such that for any² ∈ (−²2, ²2), there exist a Ckcodimension 1 locally invariant manifold Wcu² indexed by², a C

k codimension 1 locallyinvariant manifold Wcs² indexed by², and a C

k codimension 2 locally invariant manifoldWc² indexed by², under the perturbed flow F

t² given by (1.1). These manifolds are C

k

smooth in² for ² ∈ (−²2, ²2). Moreover, Wcu0 = Wcu, Wcs0 = Wcs, and Wc0 = Wc,where Wcu, Wcs, and Wc are the “invariant manifolds of Sω” identified in Corollary 2for the integrable flow (2.1). Furthermore, A² ≡ 5 ∩ U ⊂ Wc² for any² ∈ (−²2, ²2),and

A+(U) ⊂ Wcs² ,A−(U) ⊂ Wcu² ,

I (U) ⊂ Wc² ,

where A+(U), A−(U), and I(U) denote the maximal positively invariant set inU , themaximal negatively invariant set inU , and the maximal invariant set inU , respectively.

Proof. The setup for the proof of this theorem following Fenichel [7] is given in thenext subsection.

In this paper, we use “•” to denote the action of an operator upon a set.

Theorem 3.2(Global Persistent Invariant Manifolds).The global persistent center-un-stable, center-stable, and center manifolds are defined to be

⋃t≥0 F

t² •Wcu² ,

⋃t≤0 F

t² •

Wcs² , and⋃−∞


3.3.1. An Enlarged Phase Space.In order to study smoothness in² of persistent in-variant manifolds, it is convenient to treat² as a variable, and to consider the followingenlarged system(EDPNLS):

i q̇n = 1h2

[qn+1− 2qn + qn−1

]+ |qn|2(qn+1+ qn−1)− 2ω2qn

+i ²[− αqn + β

h2(qn+1− 2qn + qn−1)+ 0

], (3.8)

²̇ = 0.

The correspondingenlarged function spacebecomesŜ:

Ŝ = S × (−²3, ²3). (3.9)

We can define a family of invariant planes parametrized by², 5̂² ,

5̂² ≡{(Eq, ²) ∈ Ŝ

∣∣∣∣ Eq ∈ 5}.It is easily seen that̂5² is invariant under the EDPNLS flow (3.8). Restricted to theinvariant planeŝ5² , the EDPNLS flow (3.8) becomes

i q̇ = 2[|q|2− ω2]q + i ² [−αq + 0],²̇ = 0.

In the enlarged function space, the resonance circleŜω is defined as follows:

Ŝω ≡ {(Eq, 0) | (Eq, 0) ∈ 50, |q| = ω}.

It is easily seen that̂Sω is a set of equilibria of (3.8).

3.3.2. A Neighborhood of the CircleŜω of Fixed Points. In order to study dynamicsnear the circlêSω of fixed points we writeqn =

[(ω + δr )+ δ fn(t)

]expi θ(t),

² = δ2²′,(3.10)

whereδ > 0 is a small parameter,r is real, and fn has spatial mean 0 (i.e.,〈 fn〉 ≡∑N−1n=0 fn = 0).


Definition 4. Define the mean zero subspace ofS as follows:

S0 ≡{Eq ∈ S

∣∣∣∣ 〈qn〉 = 0}.Definition 5. Define the phase space in terms of the new variables (r, θ, fn, ²′),

E ≡{(r, θ, fn, ²

′)∣∣∣∣ (r, ²′) ∈ R2; θ ∈ [0, 2π ], Ef ∈ S0}.

Then, through the scaling transformation (3.10), there is a one-to-one correspondencebetweenE andŜ.

Definition 6. Define the compact domainEc in the phase spaceE as follows:

Ec ≡{(r, θ, fn, ²

′) ∈ E∣∣∣∣ |r | ≤ 1, |²′| ≤ 1; θ ∈ [0, 2π ], ‖ Ef ‖ ≤ 1}.

3.3.3. The Equations in Their Final Setting. Substituting representation (3.10) intothe enlarged discrete perturbed NLS system (3.8) yields the following systemdefined onEc, with δ as thenew perturbation parameter, and with² as avariable.

i ḟn =[

1

h2( fn+1− 2 fn + fn−1)+ (ω + δr )2( fn+1+ fn−1+ 2 f̄n)

]+ δ

[2(ω + δr )( fn f̄n − 〈 fn f̄n〉)+ (ω + δr )

(( fn + f̄n)( fn+1+ fn−1)

− 〈( fn + f̄n)( fn+1+ fn−1)〉)]

+ δ2[

fn f̄n( fn+1+ fn−1)− 〈 fn f̄n( fn+1+ fn−1)〉 − fn(

2〈 fn f̄n〉

+ 12

[〈( fn + f̄n)( fn+1+ fn−1)〉 + 〈( fn + f̄n)( f̄n+1+ f̄n−1)〉])

− ²′(

0

(ω + δr ) fn sinθ + i [α fn −β

h2( fn+1− 2 fn + fn−1)]

)]− δ3

[fn

2(ω + δr )(〈 fn f̄n( fn+1+ fn−1)〉 + 〈 fn f̄n( f̄n+1+ f̄n−1)〉

)],

ṙ = δ[− ² ′

(α(ω + δr )− 0 cosθ

)− i (ω + δr )/2

(〈( fn + f̄n)( fn+1+ fn−1)〉

− 〈( fn + f̄n)( f̄n+1+ f̄n−1)〉)]

− i δ2/2[〈 fn f̄n( fn+1+ fn−1)〉 − 〈 fn f̄n( f̄n+1+ f̄n−1)〉

],


θ̇ = −2δr (2ω + δr )−δ2

[²′0

(ω + δr ) sinθ + 2〈 fn f̄n〉 + 1/2(〈( fn + f̄n)( fn+1+ fn−1)〉

+ 〈( fn + f̄n)( f̄n+1+ f̄n−1)〉)]

− δ3

2(ω + δr )[〈 fn f̄n( fn+1+ fn−1)〉 + 〈 fn f̄n( f̄n+1+ f̄n−1)〉

],

²̇′ = 0.We refer to this system as the (δ 6= 0) system. Settingδ = 0 in this system, we get the(δ = 0) system,

i ḟn =[

1

h2( fn+1− 2 fn + fn−1)+ ω2( fn+1+ fn−1+ 2 f̄n)

],

ṙ = 0,θ̇ = 0,²̇′ = 0.

For the (δ = 0) system, defined onEc, we know that the subset ofEc,

Âc ≡{(r, θ, fn, ²

′) ∈ Ec∣∣∣∣ Ef = 0},

entirely consists of fixed points. At any point in̂Ac, its stable space is one-dimensional,its unstable space is also one-dimensional, and its center space has codimension 2. Thelinear growth (= decay) rate is

Ä = 2√(1− cos2 k1)(1/h2+ ω2)

(ω2− N2 tan2 π

N

),

which is the same at any point in̂Ac, k1 = 2π /N. The center-stable, center-unstable, andcenter manifolds ofÂc are the embeddings of the corresponding center-stable, center-unstable, and center linear subbundles, respectively. We denote them byL̂cs, L̂cu, andL̂c,respectively. BotĥLcs andL̂cu have codimension 1, and̂Lc has codimension 2. Viewingδ as a perturbation parameter, and viewing the (δ 6= 0) system as a perturbed system ofthe (δ = 0) system, we can study persistence of these invariant manifolds, under thisδ-perturbation. This was done by Fenichel [7]. After completing the whole argument,we get the claim: For anyδ ∈ [0, δ0], whereδ0 is a positive number, there exist locallyinvariant manifoldsŴcsδ , Ŵ

cuδ , andŴ

cδ , which can be represented as graphs overL̂

cs,L̂cu, and L̂c, respectively; moreover,̂Wcsδ=0 = L̂cs, Ŵcuδ=0 = L̂cu, andŴcδ=0 = L̂c. Thecrucial point is thatδ0 is independent of²′. If we setδ = δ0, and transformEc into asubsetDc of Ŝ through the scaling transformation (3.10), then we have a claim for theregionDc; moreover,Dc is independent of². As ²′ → 0; thereby,² → 0, Dc is of orderO(²0).

Generalization to infinite-dimensional systems is done in the book [18]. In this book,we study persistent invariant manifolds for certain PDEs (with Eq. (1.2) as a special


example), their fibrations, smoothness of the manifolds and the fibers, using graph trans-form approaches in detail.

3.4. Stable and Unstable Manifolds of the Saddleq²

We know, from the previous section on the persistent invariant plane, that on the plane5the perturbed system has exactly three fixed points:(o²,q², p²). Of these,o² is a defor-mation of the origin, while the other two fixed pointsq² andp² deform from two pointson the circleSω. In the plane5, o² andp² are sinks whileq² is a saddle. In the full phasespaceS, linear stability analysis shows thato² is a sink, while bothp² andq² are saddles.Moreover, the unstable linear subspace ofp² is one-dimensional and the stable linearsubspace ofp² has codimension 1; the unstable linear subspace ofq² is two-dimensionaland the stable linear subspace ofq² has codimension 2. By the invariant manifold theo-rems (see, e.g., Kelley [13]),Wu(p²) exists and is one-dimensional,Ws(p²) also existsand has codimension 1,Wu(q²) exists and is two-dimensional,Ws(q²) also exists andhas codimension 2.

Remark 3.2. In this paper, we only studyq² . In fact, we are going to locate homoclinicorbits asymptotic toq² . The size of the growth and decay rates with respect to the smallperturbation parameter² is very important. Consider the saddleq² : One growth rate islargeO(²0) and is associated with the integrable instability. Its eigendirection points offthe invariant plane5 into the “cosk1n” (k1 = 2π /N) direction in the phase space. Theother growth rate isO(²

12 ), with its eigendirection in the plane5. The fractional power

of ² arises because of the resonanceγ̇ = 0 (when I = ω, ² = 0 in (3.3)). Similarly,there is one decay rateO(²0) and a secondO(²

12 ). All other decay rates areO(²). This

ordering for small² introduces a dichotomy of time scales which is central throughoutour analysis.

In Appendix B, we summarize the growth rates and eigenvectors of the linear stabilityanalysis ofq² in S.

4. Fenichel Fibers

Although the center-stable and center-unstable manifoldsWcs² andWcu² of Theorem 3.1

areCk in ², trajectories insideWcs² or Wcu² are not evenC

0 in ², due to their singularnature in time. The central object we are looking for is a homoclinic orbit connectingto q² . For this, we need better coordinates in the neighborhood ofWc² in W

cs² or W

cu² .

Fenichel [5], [6], [7] introduced such coordinates (called Fenichel fibers) by using thedichotomy of time scales and the graph transform technique of Hadamard [8]. Thesefibers, in our setting, are one-dimensional curves. The fibers have several nice properties.In particular, they are smooth in². We shall representWcs² andW

cu² as unions over these

Fenichel fibers rather than as unions over orbits.


4.1. A Simple Example Showing Fenichel Fibers

Consider the simple two-dimensional system{ẋ = −²x,ẏ = −y,

where 0≤ ² ¿ 1. For more detailed explanation of this example, see [15]. Below wegive a brief summary. Figure 4.1 is an illustration of the orbits which perturb singularly.The Fenichel fibers are defined as follows: Define a family of one-dimensional curvesindexed by pointsq0 = (x0, 0) ∈ Rx,

F (s,²)(q0) ={

q = (x0, y)∣∣∣∣ y ∈ R}.

Hereq0 is called “the base point,” and for eachq0, F (s,²)(q0) is a stable Fenichel fiberwith base pointq0. See Figure 4.2 for an illustration of the fibers. In particular, we canread off the following nice properties of the fibers:

1. The “(² 6= 0) fibers” are identical with the “(² = 0) fibers.”2. Denote byFt² the solution operator. Letq0 ∈ Rx be a base point,q0 = (x0, 0);

thenFt² • q0 = (x0e−²t , 0). ∀q1 ∈ F (s,²)(q0), thenq1 = (x0, y1), for somey1 ∈ R.Ft² • q1 = (x0e−²t , y1e−t ). Therefore, we have

‖Ft² • q1− Ft² • q0‖ = e−t‖q1− q0‖,where‖ ‖ is the Cartesian norm inR2. Thus, points on a fiber suffer the same forwardtime fate as the base point.

3. Although a fiber itself is not invariant under the flow, fibers as a family are invariantunder the flow, i.e., fibers commute with the solution operator:

Ft² • F (s,²)(q0) = F (s,²)(Ft² • q0).More importantly, similar properties hold quite generally.

4.2. Fiber Theorem

Before we state the fiber theorem, we need some definitions.

Definition 7 (Locally Positively (or Negatively) Invariant Family of Submanifolds).SupposeV is a locally invariant submanifold inS, under the flowFt² . Let {M(Q): Q ∈V} be a family of submanifolds inS parametrized byQ ∈ V . We say that{M(Q): Q ∈V} is locally positively invariant under the flowFt² if Ft² (M(Q)) ⊂M(Ft² (Q)) for allQ ∈ V and allt ≥ 0 such that⋃t ′∈[0,t ] Ft ′² (Q) ⊂ V . We say that{M(Q): Q ∈ V} islocally negatively invariant underFt² if F

t² (M(Q)) ⊂M(Ft² (Q)) for all Q ∈ V and all

t ≤ 0 such that⋃t ′∈[t,0] Ft ′² (Q) ⊂ V .Definition 8 (Cr1 Family ofCr2 Manifolds). SupposeV is a locally invariant subman-ifold in S, and let{M(Q): Q ∈ V} be a family ofCr2 submanifolds inS parametrized


Fig. 4.1.Orbits for the simple example.

Fig. 4.2.Fibers for the simple example.

by Q ∈ V . LetM = {(Q, Q′) | Q ∈ V, Q′ ∈M(Q)}.

We say that{M(Q): Q ∈ V} is a Cr1 family of Cr2 submanifolds ifM is a Cr1submanifold ofS × S.

Fibration theorems of hyperbolic invariant manifolds were proved by Fenichel [5],[6], and Hirsch, Pugh, and Shub [11]. In the setting of geometric singular perturbationsof differential systems, Fenichel [5] [6] [7] proved the fibration theorem using the graphtransform method of Hadamard [8]. Infinite-dimensional version of the fibration theoremis proved in the book [18]. The theorem stated specifically for our system (1.1) is asfollows.


Theorem 4.1. Assume the conditions in the “Local Persistent Invariant Manifold The-orem” 3.1, then inside the local persistent center-stable manifold Wcs² , there is a “C

k−1

family of Ck one-dimensional manifolds”{F (s,²)(Eq): Eq ∈ Wc² }, called stable Fenichelfibers.

• Wcs² can be represented as a union of these fibers,

Wcs² =⋃Eq∈Wc²F (s,²)(Eq).

• Each fiberF (s,²)(Eq) intersects the persistent center manifold Wc² transversally inexactly one point which is the base pointEq of the fiber.• Two fibersF (s,²)(Eq1) andF (s,²)(Eq2) are either disjoint (in which caseEq1 6= Eq2) or

identical (in which caseEq1 = Eq2), for any Eq1 and Eq2 in Wc² .Moreover, these stable fibers have the following properties:

1. {F (s,²)(Eq): Eq ∈ Wc² } is Ck−1 smooth in² for ² ∈ (−²2, ²2). The precise meaning ofthis statement is as follows: Let̂S ≡ S × (−²2, ²2), then{(F (s,²)(Eq), ²): (Eq, ²) ∈Wc² × (−²2, ²2)} is a “C k−1 family of Ck one-dimensional manifolds in̂S.”

2. {F (s,²)(Eq): Eq ∈ Wc² } is a “locally positively invariant family of submanifolds.”3. Let κs be the positive constantκs ≡

√(1− cos2 k1)(1/h2+ ω2)(ω2− N2 tan2 πN ).

There is a positive constant Cs such that ifEq ∈ Wc² and Eq1 ∈ F (s,²)(Eq), then

‖F τ² (Eq1)− F τ² (Eq)‖ ≤ Cse−κsτ‖Eq1− Eq‖,

for all τ ≥ 0 such that Ft² (Eq) ∈ Wc² , t ∈ [0, τ ]. Furthermore, ifEq belongs to themaximally positive invariant set A+(U), then

F (s,²)(Eq) ={Eq1 ∈ U : ‖Fτ² (Eq1)− F τ² (Eq)‖ ≤ Cse−κsτ‖Eq1− Eq‖, for all τ ≥ 0

}.

4. For any Eq, Ep ∈ Wc² ; Eq 6= Ep; any Eq1 ∈ F (s,²)(Eq), and anyEp1 ∈ F (s,²)( Ep); if

Ft² (Eq), Ft² ( Ep) ∈ Wc² , ∀t ∈ [0,∞);

moreover,

‖Ft² ( Ep1)− Ft² (Eq)‖ → 0, as t→∞;then, { ‖Ft² (Eq1)− Ft² (Eq)‖

‖Ft² ( Ep1)− Ft² (Eq)‖}/

e−12κst → 0, as t→∞.

Similarly, for Wcu² .

Remark 4.1(Uniqueness of Fenichel Fibers). By “uniqueness of Fenichel fibers,” wemean the uniqueness of a family of fibers having all the properties stated in Theorem 4.1.If Eq ∈ A+(U), then the Fenichel fiberF (s,²)(Eq) is unique from item 4 in Theorem 4.1.


The argument is as follows: AssumeF (s,²)1 (Eq),F (s,²)2 (Eq) are two different stable Fenichelfibers with the same base pointEq ∈ Wc² ; moreover,Eq ∈ A+(U), then there are two pointsEp1 ∈ F (s,²)1 (Eq) andEq1 ∈ F (s,²)2 (Eq), such thatEp1 does not belong toF (s,²)2 (Eq), andEq1 doesnot belong toF (s,²)1 (Eq). By item 4 in Theorem 4.1, we have{ ‖F τ² (Eq1)− F τ² (Eq)‖

‖F τ² ( Ep1)− F τ² (Eq)‖}→ 0, ast →∞,

and {‖Fτ² ( Ep1)− F τ² (Eq)‖‖F τ² (Eq1)− F τ² (Eq)‖

}→ 0, ast →∞.

This contradiction implies the “uniqueness.”

We can view these fibers as “equivalence classes” that are one-dimensional manifoldsof “initial conditions” which, under the time flow, approach each other at the “fastestrate.” Consider for example the persistent center-stable manifoldWcs² . On the persistentcenter manifoldWc² ⊂ Wcs² , all expansion and contraction rates are slow when comparedto the contraction rates off ofWc² . Two pointsP1 and P2, each on the manifoldW

cs² ,

are said tolie on the same fiberif the two solution trajectoriesq(t; P1) andq(t; P2),initialized atP1 andP2 respectively, approach each other at the fastest rate ast →+∞.Thus, a fiber for the manifoldWcs² is an equivalence class of points on the manifoldW

cs² ,

where the equivalence relation in effect “factors out” the fastest contraction.The primary use of these fibers is to “relate or connect” the fast dynamics with slow

dynamics over semi-infinite time intervals. For this connection, each fiber is labeled bythe point at which it intersects the slow manifoldWc² , i.e., the “base point” of the fiber. Alltrajectories with initial points on the fiber approach that trajectory on the slow manifoldwhich is initialized at the basepoint of the fiber.

4.3. The Unique Explicit Fenichel Fibers for “(figure 8)⊗ A”From integrable theory, we knowF (s,0)(Eq) andF (u,0)(Eq) quite well; in some cases, wecan represent them explicitly. For example, we can read off the explicit representationsof the (unique) Fenichel Fibers for “(figure 8)⊗ A” from Corollary 1.

Theorem 4.2. For the integrable discretized NLS equation (2.1), we have

1. The stable fiber in “(figure 8)⊗ A” with the base pointEq ∈ A: (qn ≡ q, ∀n;q = a exp{i γ }).

F (s,0)(Eq) = qei 2P[

G

Hn− 1

], (4.1)

where

G = 1+ cos 2P − i sin 2P tanhr,

Hn = 1± 1cosϑ

sin P sechr cos 2nϑ,


P = arctan√ρ cos2 ϑ − 1√ρ sinϑ

,

ϑ = πN, ρ = 1+ |q|

2

N2,

where r∈ (−∞,∞) parametrizes the one-dimensional fiber. As r→+∞,F (s,0)(Eq)→ q. Moreover, in this case,{F (s,0)(Eq)} are the unique stable Fenichel fibers in“(figure 8) ⊗ A” with the base points in A.

2. The unstable fiber in “(figure 8)⊗ A” with the base pointEq ∈ A: (qn ≡ q, ∀n;q = a exp{i γ }).

F (u,0)(Eq) = qe−i 2P[

G

Hn− 1

]. (4.2)

As r → −∞, F (u,0)(Eq) → q. Moreover, in this case,{F (u,0)(Eq)} are the uniqueunstable Fenichel fibers in “(figure 8)⊗ A” with the base points in A.

Remark 4.2. As for the perturbed fibersF (s,²)(Eq) andF (u,²)(Eq), we do not have explicitrepresentations. However, we know that they are smooth in² which will be enough forour later use.

5. Melnikov Measurement:Wu(q²) ∩Wcs²In this section, we present a Melnikov measurement (which we often call the “firstmeasurement”) through which we answer the question: Is there any intersection betweenWu(q²) andWcs² other thanW

u(q²)|5 (≡ Wu(q²) ∩5)?We know thatWcs² is a codimension 1 submanifold inS, and thatWu(q²) is a

two-dimensional submanifold. Generically, an intersectionWu(q²) ∩Wcs² will be one-dimensional. Since we also know that the trivial intersectionWu(q²)|5 is one-dimensional,another question arises: Is there a “nontrivial” intersection?

5.1. Main Argument

First, we rewrite equation (1.1) in the vector form:

d

dtEqn = ρn

{J gradEqn H

}+ ² Egn, (5.1)

where

Eqn ≡ (qn, rn)T , rn ≡ −q̄n, J =(

0 1−1 0

),

H ≡ − ih2

N−1∑n=0

{q̄n(qn+1+ qn−1)− 2

h2(1+ ω2h2) ln(1+ h2|qn|2)

},


Egn ≡ (gn,1, gn,2)T ,gn,1 ≡ −αqn + 0 + βN2(qn+1− 2qn + qn−1),gn,2 ≡ −αrn − 0 + βN2(rn+1− 2rn + rn−1).

Because we are working in a high-dimensional phase space, the first natural question is:How manymeasurements must be made to determine ifWu(q²) intersects withWcs² ? Theanswer is: one measurement is enough. Intuitively,Wcs² has codimension 1 (coordinatized

by grad F̃1). In order to measure the distance betweenWu(q²) andWcs² , we need only

to measure along the transverse directiongrad F̃1, which requires only one Melnikovintegral. See Figure 5.1 for the detailed geometry.

The rigorous argument begins by considering, withinS, theO(²0) neighborhoodUof the resonance circleSω, with boundary∂U , of the “local persistent invariant manifoldtheorem 3.1” and the “fiber theorem 4.1.” In this neighborhoodU , we have the followingfiber representations:

Wu(q²) =⋃{

F (u,²)(Eq)∣∣∣∣ Eq ∈ Wu(q²)|5}, (5.2)

Wu(q²)|5 ={(γ, I )

∣∣∣∣ I = ω +√²Ju(γ )}, (5.3)whereJu(γ ) denotes the curve in the plane5which represents the unstable manifold ofq² restricted to this plane; see Eq. (3.4), (3.5). Similarly,Wcs² has the fiber representation,

Wcs² =⋃{

F (s,²)(Eq)∣∣∣∣ Eq ∈ Wc² }. (5.4)

For convenience of argument, letM² be aO(²0) neighborhood of the resonance circleSω in Wc² , chosen so thatM² is properly included inU , i.e.,M² never touches theboundary ofU . Notice thatWu(q²)|5 ⊂M² ; in fact, we chooseM² large enough sothat the wholefish lies insideM² . DefineWu(M²) andWs(M²) as follows:

Wu(M²) ≡⋃{

F (u,²)(Eq)∣∣∣∣ Eq ∈M²}, (5.5)

Ws(M²) ≡⋃{

F (s,²)(Eq)∣∣∣∣ Eq ∈M²}. (5.6)

Define∂Uu² ≡ ∂U∩Wu(M²),∂Us² ≡ ∂U∩Ws(M²). The following lemma is a corollaryof the “fiber theorem 4.1”:

Lemma 5.1. In S, every point in∂Uu² is on a unique unstable fiber explicitly expressedby (5.5), with a unique base point inM² . Similarly, every point in∂Us² is on a uniquestable fiber explicitly expressed by (5.6), with a unique base point inM² .

Similarly, inS, define∂Uu(q²) ≡ ∂U ∩Wu(q²). The following lemma is also a corollaryof the “fiber theorem 4.1”:


Lemma 5.2. In S, every point in∂Uu(q²) is on a unique unstable fiber explicitly ex-pressed by (5.2), (5.3), with a unique base point in Wu(q²)|5.

We begin to study the possibility of an intersection betweenWu(q²)andWs(M²) ⊂ Wcs² ,by considering all base points inWu(q²)|5 of the unstable fibers inWu(q²). (Cf. Fig-ure 5.1.) Choose a base point(γ, I u) ∈ Wu(q²)|5; then there is a uniqueperturbedunstable fiber,F (u,²)(γ, I u) with the base point,(γ, I u). There is also a uniqueun-perturbedunstable fiber,F (u,0)(γ, I u) with the samebase point,(γ, I u). In the un-perturbed case, this fiber is explicitly represented in Theorem 4.2. By “fiber theorem4.1,” F (u,²)(γ, I u) is Ck−1 smooth in²; therefore, the fiberF (u,²)(γ, I u) is in a O(²)neighborhood ofF (u,0)(γ, I u). We know thatF (u,0)(γ, I u) intersects the codimension1 boundary∂U transversally. (If necessary, we may shrink the regionU to accomplishthis.) Thus,F (u,²)(γ, I u) also intersects the codimension 1 boundary∂U transversally.Moreover, letEq (u,²) ≡ F (u,²)(γ, I u) ∩ ∂U , Eq (u,0) ≡ F (u,0)(γ, I u) ∩ ∂U . Then we have

Eq (u,²) = Eq (u,0) + O(²). (5.7)

Starting fromEq (u,0), theunperturbeddiscrete NLS equation (2.1) produces an orbithhomoclinic to the invariant plane5, with the explicit representation given in Corollary1. After a finite time,h will intersect with∂U a second time. This unique intersectionon the “landing side” is denoted byEq (s,0). We emphasize thatEq (s,0) is on a uniqueunperturbedstable fiber with base point(γ∗, I (s,0)) ∈ 5; moreover,I (s,0) = I u. Startingfrom Eq (u,²), theperturbeddiscrete NLS equation (1.1) will also produce an orbithu² .From the smoothness property of the solution operator, in a finite time interval,hu² willstay in anO(²) neighborhood ofh. Thus,hu² will also intersect with∂U at Eq (s,²), andEq (s,²) is in O(²) neighborhood ofEq (s,0). The Melnikov measurement determines ifEq (s,²)(and thereforehu² ) lies in W

s(M²).The next step is to set up a homoclinic coordinate system at a point onh in between

Eq (u,0) andEq (s,0). For any such pointEqh, let6 be a codimension 1 hyperplane atEqh whichis transversal to the homoclinic orbith and which contains the vectorgrad F̃1. (Thismakes sense because the constant of motionF̃1 Poisson commutes with the integrableHamiltonian which defines the homoclinic orbith.) Wcs0 ∩ 6 has codimension 1 in6,andgrad F̃1 is transversal toWcs0 ∩6. Let {Er j } denote a coordinate of the tangent spaceof Wcs0 ∩6 at Eqh. Then{grad F̃1, Er j } is a coordinate system for6 with the origin atEqh.In this coordinate frame, the intersection pointEq (u,²)h ≡ hu² ∩6 has the representation

Eq (u,²)h = ²au grad F̃1+∑

j

²auj Er j , (5.8)

which follows immediately from the relation (5.7) and the regularity of the solutionoperatorFt² . (For any fixedT , 0≤ T


where{asj } parametrize the codimension 1 hypersurfaceWs(M²)∩6 in6, and f s({asj })is a smooth function. Now we choose the special values forasj : a

sj = ²auj , for all j . This

special choice determines a unique pointEq s∗ on6s, with the representation

Eq s∗ = f s({²auj }) grad F̃1+∑

j

²auj Er j . (5.10)

By the “global persistent invariant manifold theorem 3.2,” the global persistent center-stable manifoldWs(M²) ⊂ Wcs² is Ck in ²; then Ws(M²) ∩ 6 stays in anO(²)neighborhood ofWcs0 ∩6 in 6 which passes through the origin. Thus, we have

f s({²auj }) = ²as, (5.11)for some real numberas. By Eqs. (5.8), (5.10), (5.11), we define the following signeddistance betweenEq (u,²)h andEq s∗ :

d1 = Signed Distance{Eq (u,²)h , Eq s∗

}≡ ²{au − as}

∥∥∥∥grad F̃1∥∥∥∥2≡ 〈grad F̃1, Eq (u,²)h − Eq s∗ 〉. (5.12)

Now the problem is reduced to measuring the distance betweenEq (u,²)h andEq s∗ . Moreover,only one measurement, ingrad F̃1 direction, is needed. See Figure 5.1 for the detailedgeometry.

5.2. Derivation of a Melnikov Integral

In this subsection, we are going to derive a Melnikov integral as a measure of the signeddistance defined above. See Figure 5.1 for the detailed geometry.

We give the following parametrization for the orbitsh andhu² :

h ≡ h(t), −∞ < t < +∞,h(0) = Eqh;

hu² ≡ hu² (t), −∞ < t ≤ 0,hu² (0) = Eq (u,²)h .

Let hs² denote the orbitFt² (Eq s∗ ), 0≤ t < +∞, with the parametrization

hs² ≡ hs²(t), 0≤ t < +∞,hs²(0) = Eq s∗ .

Notice that


}≡ 〈grad F̃1, Eq (u,²)h − Eq s∗ 〉;


then we can decomposed1 as follows:

d1 = 〈∇ F̃1(Eqh), Eq (u,²)h − Eqh〉 − 〈∇ F̃1(Eqh), Eq s∗ − Eqh〉,

where “∇” denotes “grad”. Let

1+(t) ≡ 〈∇ F̃1(h(t)), hs²(t)− h(t)〉; 0≤ t < +∞,

1−(t) ≡ 〈∇ F̃1(h(t)), hu² (t)− h(t)〉; −∞ < t ≤ 0.

Then,

d1 = 1−(0)−1+(0). (5.13)See Figure 5.1 for an illustration. We will prove the following:

Theorem 5.1.

1−(0) = ²∫ 0−∞〈∇ F̃1(h(t)), Eg(h(t))〉 dt + O(²2),

1+(0) = −²∫ +∞

0〈∇ F̃1(h(t)), Eg(h(t))〉 dt + O(²2).

Thus,

d1 = Signed Distance{Eq (u,²)∗ , Eq s∗

}= ²MF̃1 + O(²2),

where

MF̃1 =∫ +∞−∞〈∇ F̃1(h(t)), Eg(h(t))〉 dt (5.14)

=∫ ∞−∞

{ N−1∑n=0

(δ F̃1δqn

gn,1+ δ F̃1δrn

gn,2

)∣∣∣∣(evaluated at h(t))} dt.Proof. To prove this theorem, we note thath(t), hs²(t), andh

u² (t) solve the following

equations:

ḣ(t) = J∇H(h(t)),ḣs²(t) = J∇H(hs²(t))+ ² Eg(hs²(t)),ḣu² (t) = J∇H(hu² (t))+ ² Eg(hu² (t)),

whereJ is a symplectic matrix which can be written in the block form:J = diag{J0, . . . ,JN−1}, in which,

Jn =(

0 ρn−ρn 0

),


whereρn is given in Eq. (2.1). Thus,

ḣs²(t)− ḣ(t) = J∇H(hs²(t))− J∇H(h(t))+ ² Eg(hs²(t))

= ∇(J∇H

)(h(t)) •

(hs²(t)− h(t)

)+ ² Eg(h(t))+ R1+ R2,

where

R1 ≡ J∇H(hs²(t))− J∇H(h(t))−∇(J∇H

)(h(t)) •

(hs²(t)− h(t)

),

R2 ≡ ²(Eg(hs²(t))− Eg(h(t))

).

Differentiating1+(t), we have

1̇+(t) = 〈∇ F̃1(h(t)), ḣs²(t)− ḣ(t)〉 + 〈∇2F̃1(h(t)) • ḣ(t), hs²(t)− h(t)〉

= ²〈∇ F̃1(h(t)), Eg(h(t))〉 + 〈∇ F̃1(h(t)), R1〉 + 〈∇ F̃1(h(t)), R2〉 + E,where

E =〈∇ F̃1(h(t)),∇

(J∇H

)(h(t)) •

(hs²(t)− h(t)

)〉

+〈∇2F̃1(h(t)) •

(J∇H(h(t))

), hs²(t)− h(t)

〉.

We know that the Poisson bracket

{F̃1, H} = 〈∇ F̃1,J∇H〉 = 0,then, by differentiation, we have〈

∇ F̃1(Eq),∇(J∇H

)(Eq) • δEq

〉+ 〈∇2F̃1(Eq) • δEq,J∇H(Eq)〉 = 0.

We also know that∇2F̃1(Eq) is a symmetric bilinear operator; therefore,

〈∇2F̃1(Eq) • δEq,J∇H(Eq)〉 = 〈∇2F̃1(Eq) • J∇H(Eq), δEq〉.SettingEq = h(t), δEq = hs²(t)− h(t), we see thatE = 0. Thus,

1̇+(t) = ²〈∇ F̃1(h(t)), Eg(h(t))〉+ 〈∇ F̃1(h(t)), R1〉 + 〈∇ F̃1(h(t)), R2〉. (5.15)

Similarly,

1̇−(t) = ²〈∇ F̃1(h(t)), Eg(h(t))〉+ 〈∇ F̃1(h(t)), R3〉 + 〈∇ F̃1(h(t)), R4〉, (5.16)


where

R3 ≡ J∇H(hu² (t))− J∇H(h(t))−∇(J∇H

)(h(t)) •

(hu² (t)− h(t)

),

R4 ≡ ²(Eg(hu² (t))− Eg(h(t))

).

Let Eq (s,²)∗ denote the point at whichhs²(t) intersects∂U , then Eq (s,²)∗ is on a uniqueperturbed stable fiber with base pointb² ∈ M² ⊂ Wc² . The proof of the theorem iscompleted with the following steps:

1. In Appendix C, we establish the following:

Lemma 5.3. • limt→−∞1−(t) = 0.• If F τ² (b²) ∈ Wc² for all 0 ≤ τ < +∞, thenlimt→+∞1+(t) = 0. If F τ² (b²) does not

stay in Wc² for all 0 ≤ τ < +∞, then it stays in Wc² for all 0 ≤ τ ≤ T(²), whereT(²) ∼ O( 2

κ1ln 1/²); moreover,1+(T(²)) ∼ O(²2).

2. Finally, in Appendix D, we establish the following:

Lemma 5.4. ∫ 0−∞〈∇ F̃1(h(t)), R3〉 dt ∼ O(²2),∫ 0

−∞〈∇ F̃1(h(t)), R4〉 dt ∼ O(²2),∫ +∞(or T(²))

0〈∇ F̃1(h(t)), R1〉 dt ∼ O(²2),∫ +∞(or T(²))

0〈∇ F̃1(h(t)), R2〉 dt ∼ O(²2),

∫ +∞T(²)〈∇ F̃1(h(t)), Eg(h(t))〉 dt ∼ O(²2). (5.17)

By Lemmas 5.3 and 5.4 and Eqs. (5.15), (5.16), we have the following representationsfor 1−(0) and1+(0):

1−(0) = ²∫ 0−∞〈∇ F̃1(h(t)), Eg(h(t))〉 dt + O(²2),

1+(0) = −²∫ +∞(or T(²))

0〈∇ F̃1(h(t)), Eg(h(t))〉 dt + O(²2).

Moreover, by the estimate (5.17), we can always represent1+(0) as

1+(0) = −²∫ +∞

0〈∇ F̃1(h(t)), Eg(h(t))〉 dt + O(²2).


Fig. 5.1. Geometric illustration of the Melnikov measurement. For the definition of no-tation, see the text. Note that the Melnikov function approximates the distance betweenEq (u,²)∗ andEq s∗ .

This completes the proof of the theorem.

Figure 5.1 is the geometric illustration of the argument in this section.

Remark 5.1. The integral defined in (5.14) is called the Melnikov integral, and in thenext section, we will further approximateMF̃1.

5.3. An Approximation

Next we are going to approximate the unperturbed orbith, homoclinic to a circle in theannulusA, by another unperturbed orbit which is homoclinic to the resonance circleSω,


and obtain the following approximation for the Melnikov integral:

MF̃1 = M̂F̃1 + O(√

² ln21√²

),

where

M̂F̃1 =∫ ∞−∞

{ N−1∑n=0

(δ F̃1δqn

gn,1+ δ F̃1δrn

gn,2

)∣∣∣∣(evaluated athω)} dt. (5.18)We know, by the above construction, thath intersects∂U in backward time atEq (u,0);

thus, Eq (u,0) is on a unique unperturbed unstable fiberF (u,0)(γ, I u) with base point(γ, I u) ∈ Wu(q²)|5. We know, from the representation (5.3) ofWu(q²)|5, that I u =ω +√²Ju(γ ); thus,(γ, I = ω) is in anO(√²) neighborhood of(γ, I u). (Notice thathere we take the two base points with the same phaseγ .) Moreover, the unstable fiberF (u,0)(γ, ω) is also an orbit. (Denote the orbit byhω, which is an orbit homoclinic tothe resonance circleSω; moreover, it is also an unstable fiber in backward time, and astable fiber in forward time.) LetEq (u,0)ω ≡ F (u,0)(γ, ω)∩∂U ; then, by the “fiber theorem4.1,” Eq (u,0)ω is in an O(

√²) neighborhood ofEq (u,0). By the regularity of the solution

operatorFt0, in the regionS/U (i.e., the complement ofU in S), hω stays in anO(√²)

neighborhood ofh. Recall thatEq (s,0) is the intersection point ofh to ∂U in forward time.Let Eq (s,0)ω be the intersection point ofhω to ∂U in forward time. Then,Eq (s,0)ω is in anO(√²) neighborhood ofEq (s,0). Now we are going to estimate the deviation betweenhω

andh insideU and the change in Melnikov integrals. Since the estimates are parallel forthe backward time (t → −∞) part and the forward time (t → +∞) part, we just takethe backward time (t → −∞) part as the example. By the “fiber theorem 4.1,” a pointon an unstable fiber approaches the base point exponentially under backward time flow;thus, letT ² ≡ 1/κu ln 1√² , then for allt ∈ (−∞,−T ² ],∥∥∥∥Ft0(Eq (u,0)ω )− Ft0(γ, ω)∥∥∥∥ ≤ Ĉu√²,whereĈu = Cu‖Eq (u,0)ω − (γ, ω)‖; similarly,∥∥∥∥Ft0(Eq (u,0))− Ft0(γ, I u)∥∥∥∥ ≤ Ĉ′u√²,whereĈ′u = Cu‖Eq (u,0) − (γ, I u)‖. Equivalently, for allt ∈ (−∞,−T ² ],

Ft0(Eq (u,0)ω ) = Ft0(γ, ω)+ O(√²),

Ft0(Eq (u,0)) = Ft0(γ, I u)+ O(√²).

We also know that

grad F̃1

∣∣∣∣(evaluated atFt0(γ, ω) or Ft0(γ, I u)) ≡ 0,


for anyt ∈ (−∞,∞). Moreover, from the explicit formula forgrad F̃1 below, we have∥∥∥∥grad F̃1∥∥∥∥(evaluated ath or hω) ≤ C1eκut ,for all t ∈ (−∞,−T ²), whereC1 is a constant independent of². Thus,∫ −T ²

−∞

{ N−1∑n=0

(δ F̃1δqn

gn,1+ δ F̃1δrn

gn,2

)∣∣∣∣(evaluated ath or hω)} dt ∼ O(√²).Next, we estimate the deviation betweenFt0(Eq (u,0)ω ) and Ft0(Eq (u,0)) for t ∈ [−T ², 0].From the explicit formulae in Corollary 1, and Theorem 4.2, we have the explicit repre-sentation ofFt0(Eq (u,0)ω ) andFt0(Eq (u,0)),

Ft0(Eq (u,0)ω ) = q0[

G

Hn− 1

]∣∣∣∣(given by the corollary atq = q0),Ft0(Eq (u,0)) = q1

[G

Hn− 1

]∣∣∣∣(given by the corollary atq = q1),whereq0 = ω exp{i (γ − 2P0)}, q1 = I u exp{−2i [((I u)2 − ω2)t ] + i (γ − 2P1)}, P0and P1 are given in Corollary 1 according to|q| = ω and |q| = I u, respectively. Byexplicitly checking the above representations and noticing that

2[(I u)2− ω2]T ² ∼ O(√

² ln1√²

),

we have, for allt ∈ [−T ², 0],∥∥∥∥Ft0(Eq (u,0)ω )− Ft0(Eq (u,0))∥∥∥∥ ≤ C̃√² ln 1√² ,whereC̃ is independent of². Therefore, we get∫ 0

−T ²

{ N−1∑n=0

(δ F̃1δqn

gn,1+ δ F̃1δrn

gn,2

)∣∣∣∣(evaluated ath)} dt=∫ 0−T ²

{ N−1∑n=0

(δ F̃1δqn

gn,1+ δ F̃1δrn

gn,2

)∣∣∣∣(evaluated athω)} dt+ O

(√² ln2

1√²

).

Finally, we have

MF̃1 = M̂F̃1 + O(√

² ln21√²

),

where

M̂F̃1 =∫ ∞−∞

{ N−1∑n=0

(δ F̃1δqn

gn,1+ δ F̃1δrn

gn,2

)∣∣∣∣(evaluated athω)} dt. (5.19)


5.4. Calculation ofM̂F̃1

In this subsection we calculatêMF̃1. Let−γ1 ≡ γ − 2P, where

P = arctan√ρ cos2 ϑ − 1√ρ sinϑ

,

ϑ = πN, ρ = 1+ ω

2

N2.

Then,hω has the representation, given in Corollary 1,

Qn ≡ Qn(t, r ; N, ω, γ1,±) = ωe−i γ1[

G

Hn− 1

], (5.20)

where

G = 1+ cos 2P − i sin 2P tanhτ,

Hn = 1± 1cosϑ

sin P sechτ cos 2nϑ,

τ = 4N2√ρ sinϑ√ρ cos2 ϑ − 1 t + r.

At qn ≡ ωe−i γ1, let z+ be the real double point of the Lax pair (2.3), (2.4),z+ = √ρ cosϑ +

√ρ cos2 ϑ − 1.

We have the following two linearly independent Bloch functions atz+:

ψ+n =(√

ρeiϑ)n

eÄ+t((1/z+ −√ρeiϑ)e−i (γ1/2)−i (ω/N)ei (γ1/2)

),

ψ−n =(√

ρe−iϑ)n

eÄ−t( −i (ω/N)e−i (γ1/2)(z+ −√ρe−iϑ)ei (γ1/2)

),

where

Ä+ = i N2[√ρ(1/z+ − z+)eiϑ + 2 lnz+

],

Ä− = i N2[√ρ(1/z+ − z+)e−iϑ + 2 lnz+

].

Let φn ≡ c+ψ+n + c−ψ−n ; then, following the calculation in [20], we haveδ F̃1δEqn

∣∣∣∣(evaluated athω) = Cz+ Wn1̂n An+1

((1/z2+)φ̄n1φ̄(n+1)1(1/z̄2+)φ̄n2φ̄(n+1)2

),

whereCz+ is a complex constant and

Wn = ψ+n1ψ−n2− ψ+n2ψ−n1,1̂n = |φn1|2+ |z+|2|φn2|2,An = |φn2|2+ |z+|2|φn1|2.


Let Gn ≡ δ F̃1δqn | (evaluated athω), and decomposeGn as follows:

Gn = Gen + Gon,whereGen is the even part ofGn, i.e., G

en = GeN−n, Gon is the odd part ofGn, i.e.,

Gon = −GoN−n. Gon gives no contribution to the Melnikov integral. Up to multiplicationby a real constant, we have

Gen =sechτ

3n

{cosϑ sechτ + (sin P − i cosP tanhτ) cos 2nϑ

},

where

3n = 1+ 2(1/ cosϑ) sin P sechτ cos 2nϑ cos 2ϑ+ 1/2(1/ cos2 ϑ) sin2 P sech2 τ(cos 4nϑ + cos 4ϑ).

Finally, we obtain an explicit formula for the Melnikov function̂MF̃1.

Proposition 1.

M̂F̃1 = 0{

M0 − χαMα + χβMβ}, (5.21)

whereχα ≡ α/0, χβ ≡ β/0, M0 ≡ cosγ1M̂0; moreover,

M̂0 ≡ M̂0(ω, N) =∫ ∞−∞

sechτN−1∑n=0

1

3n

{cosϑ sechτ + sin P cos 2nϑ

}dτ,

Mα ≡ Mα(ω, N) =∫ ∞−∞

N−1∑n=0

Re{GenQn} dτ,

Mβ ≡ Mβ(ω, N) = N2∫ ∞−∞

N−1∑n=0

Re{Gen(Qn+1− 2Qn + Qn−1)} dτ,

where Qn is given in (5.20).

5.5. The Intersection BetweenWu(q²) and Ws(M²) ⊂ Wcs²From the arguments in the previous subsections, we get the following characterizationof the signed distance betweenWu(q²) andWs(M²) ⊂ Wcs² :


}= ²

{M̂F̃1 + O

(√² ln2

1√²

)}, (5.22)

whereM̂F̃1 is given in Eq. (5.21). Moreover,


}= Function(²;ω, α, β, 0; γ1),


in which ² is a small positive parameter² ∈ (0, ²2), ²2 is a positive number given inTheorem 3.1,{ω, α, β, 0} are external parameters defined in the region6N defined inthe introduction, andγ1 is an internal parameter. By the relation−γ1 = γ − 2P at thebeginning of the last subsection, we can determine the “take-off”γ wheneverγ1 is given,and vice versa. From the explicit formula (5.21) forM̂F̃1, we see thatM̂F̃1 depends on

{ω, α, β, 0} andγ1 in a very simple way. SettinĝMF̃1 = 0, we have

cosγ1M̂0(ω, N)− χαMα(ω, N)+ χβMβ(ω, N) = 0. (5.23)

The following lemma is an immediate consequence of the above equation (5.23).

Lemma 5.5. For N tanπ /N < ω < N tan 2π /N, there exists a regionDω in the firstquadrant of R2, such that, for any(χα, χβ) ∈ Dω, there are two values ofγ1: γ±1 ≡γ±1 (ω, χα, χβ) = ±arccos{(χαMα − χβMβ)/M̂0}, at whichM̂F̃1 = 0. Moreover, whenM̂F̃1 is viewed as a function ofγ1, these zeros are simple.

By this lemma, Eq. (5.22), and the implicit function theorem, we have the followingtheorem:

Theorem 5.2(Intersection betweenWu(q²) andWs(M²) ⊂ Wcs² ). There exists a sub-region 6sN of 6N and a positive number²4, such that, for any fixed parameters{ω, α, β, 0; ²} ∈ 6sN × (0, ²4), there are two values ofγ1 = γ (±,²)1 which areO(√² ln2 1√

²) close toγ±1 , at which there are transversal intersections between W

u(q²)and Ws(M²) ⊂ Wcs² ; moreover, the intersections are generic, i.e., the two intersectionsets are one-dimensional curves. These two curves h(u,±)² can be determined as follows:From γ (±,²)1 , we can findγ

(±,²) = −γ (±,²)1 + 2P, as discussed above; thenγ = γ (±,²)will identify the “take-off” base points of the perturbed unstable fibers in Wu(q²) onwhich the two points h(u,±)² ∩ ∂U sit.

Remark 5.2. The distinct anglesγ = γ (±,²) can be interpreted as follows: The two-dimensional unstable manifoldWu(q²) consists in a one parameter family of orbitshu² ,indexed by the “take-off” angleγ . The two distinct members of this family of “take-off”orbits which are labeled byγ = γ (±,²) will approach, in forward time, the slow manifoldM² ; other “take-off” orbits labeled by nearby “take-off” anglesγ will not approach theslow manifoldM² in forward time. Thus, the Melnikov measurement selects thosedistinct “take-off” angles for which the orbit returns to the slow manifold.

6. Existence of Orbits Homoclinic toq² : The Second Measurement

In the last chapter, we have proved that there are generic intersections betweenWu(q²)andWs(M²) ⊂ Wcs² , so that for external parameters in a fixed open set, there are orbitswhich tend toq² in backward time, and approachM² in forward time. Denote one ofthese orbits byh² . In this chapter, we are going to show that, by restricting the externalparameters to a codimension 1 submanifold in external parameter space, these orbits are


in fact homoclinic toq² , i.e., in forward time they also tend toq² . The detailed geometryfor the argument below is shown in Figure 6.1.

Definition 9. Define the following restricted objects:

Ws(q²)|5 ≡ 5 ∩Ws(q²),Ws(q²)|M² ≡ M² ∩Ws(q²),Wu(q²)|M² ≡ M² ∩Wu(q²).

From the discussion in previous chapters, the following lemma is obvious.

Lemma 6.1. Wu(q²)|M² = Wu(q²)|5, which is one-dimensional. Ws(q²)|5 is alsoone-dimensional. Ws(q²)|M² has codimension 1 inM² .

Following the notations in the last chapter, letEq (s,²) ≡ ∂U ∩ h² (called the “landingpoint”) denote the intersection point of the orbith² with the boundary∂U . Notice, fromthe last chapter, thatEq (s,²) ∈ ∂Us² ≡ ∂U ∩Ws(M²). Then,Eq (s,²) is on a unique stablefiber in Ws(M²) with base pointEq (s,²)b ∈M² . By the “fiber theorem 4.1,”Eq (s,²) andEq (s,²)b have the same fate in forward time. Then the problem of following the orbith²in forward time is reduced to following the orbit inM² starting from the base pointEq (s,²)b . Therefore, the question: Ast → +∞, doesh² → q²? is reduced to: DoesEq (s,²)b ∈ Ws(q²)|M²? Next, we are going to answer this question affirmatively.

From the argument in the last chapter, we know thatEq (s,²) is O(²) close toEq (s,0)which is the intersection point of the unperturbed orbith with ∂U in forward time;moreover,Eq (s,0) is on an unperturbed stable fiber whose base pointEq (s,0)b certainly lieson the invariant plane5. Thus, by the “fiber theorem 4.1,”Eq (s,²)b is O(²) close to theinvariant plane5, i.e.,

Distance{Eq (s,²)b , Eq (s,0)b

}= O(²); Distance

{Eq (s,²)b ,5

}= O(²). (6.1)

Now we define an important object—the cylindrical neighborhoodVµ of 5 ∩M² inM²—as follows:

Vµ ≡{Eq∣∣∣∣ Eq ∈M², Distance{Eq,5} ≤ ²µ, 0< 1− µ¿ 1}. (6.2)

Then by Eq. (6.1), we have the lemma:

Lemma 6.2. The “landing base point”Eq (s,²)b ∈ Vµ.

We know, from Lemma 6.1, thatWs(q²)|M² has codimension 1 inM² . Here we makean assumption about the “height of the wall,”

W ≡(

Ws(q²)|M²)∩ Vµ.


Assumption 1. The height of the “wall”W shown in Figure 6.1 is large enough so thatW separates Vµ into two disconnected regions. Moreover,W intersects5 transversallyfor any² ∈ [0, ²2).

Remark 6.1. This assumption can be verified through the use of a normal form trans-formation, which can be adapted from the work of [17]. Here we omit this adaption, asthe normal form procedure is described in detail in the above reference.

Next, we consider a signed distance betweenEq (s,²)b andW, and characterize the signeddistance by a certain signed distance defined on the invariant plane5. (See Figure 6.1for the detailed geometry.) The geometric picture is as follows:W provides a sufficientlyhigh codimension 1 middlewall in Vµ. We can specify its positive and negative sides asin Figure 6.1. The pointEq (s,²)b always lies insideVµ. If, through adjustment of externalparameters, the pointEq (s,²)b can be made to sit in both regions, then by continuity, therewill exist parameters for which the pointEq (s,²)b sits on the wallW. Hence, for this specialchoice of external parameters, the orbith² must return toq² in forward time.

Let qb5 ∈M² ∩5 be a point on5 that realizes the distance betweenEq (s,²)b and5;then, from Eq. (6.1), we have

Eq (s,²)b = qb5 + ²g, (6.3)whereg is bounded. By the “height” assumption 1, we can characterize the wallW asfollows: For anyEq ∈W, there existsq5 ∈ Ws(q²)|5 that realizes the distance betweenEq andWs(q²)|5,

Eq = q5 + ²µ f, (6.4)where f is bounded. Therefore,

W ={Eq = q5 + ²µ f

∣∣∣∣ q5 ∈ Ws(q²)|5, f is bounded}.See Figure 6.1 for a geometric illustration.

Let d5 be a signed distance betweenqb5 andWs(q²)|5, inside5, andd2 be a signed

distance betweenEq (s,²)b andW in Vµ. We have

|d2| = infq5∈Ws(q² )|5, f ∈S(q5)

{‖Eq (s,²)b − (q5 + ²µ f )‖

},

= infq5∈Ws(q² )|5, f ∈S(q5)

{‖qb5 − q5 + (²g− ²µ f )‖

}, (6.5)

|d5| = infq5∈Ws(q² )|5

{‖qb5 − q5‖

},

= minq5∈Ws(q² )|5

{|qb5 − q5|

}. (6.6)

Thus, by (6.3) and (6.4), we have

d2 = d5 + O(²µ). (6.7)


Fig. 6.1. Geometry for the second measurement. For definition of notation, see thetext. The second measurement approximates the distance betweenEq (s,²)b andW.


Fig. 6.2.Geometry for the reduction.

Here,d5 is O(√²) and can be both positive and negative (which will be shown below);

hence,d2 is alsoO(√²) and can be both positive and negative. Thus, the calculation

of the signed distanced2 is reduced to the calculation of the signed distanced5 on theinvariant plane5. This calculation, on5, is identical with that described in detail in[23]. Denote byd̂5 the signed distance betweenqb5 and theunbrokenfish boundaryWs(q̃²)|5, then

d5 = d̂5 + O(²). (6.8)The detailed geometry for this reduction is shown in Figure 6.2. In Figure 6.3, we showγs andγn as functions ofχα andω. For appropriate values ofχα andω, γs− γn > 4P1.Therefore, the length of the fish parametrized byγ is big enough to capture the phaseshift 4P1. See Figure 6.4 for an illustration. Therefore, in the original coordinate (γ, I ),the signed distanced5 betweenqb5 andW

s(q²)|5, and the signed distancêd5 betweenqb5 andW

s(q̃²)|5 are bothO(√²), and can be both positive and negative. We have thefollowing:

Theorem 6.1.

d2 = d̂5 + O(²µ), (6.9)whered̂5 ≡ Signed Distance{qb5,Ws(q̃²)|5} which takes values in(−C

√²,C√²) for

some constant C. The roots of the equationd̂5 = 0 can be approximated by the roots ofthe equation,

4αωP − 20 sin 2P cosγ1 = 0.


Fig. 6.3.The functionsαs = αs(χα, ω) andαn = αn(χα, ω).


Fig. 6.4.Phase shifts of heteroclinic orbits vs. “fish” geometry.

Proof. By (6.7), (6.8), we have the relation (6.9). Notice thatd̂5 = 0 if and only if

1H1 ≡ H1(qb5)−H1(q̃²) = 0,

whereq̃² ≡ (J, γs) ≡ (0, arccos{χαω}),H1 = αγω − 0 sinγ − 2ωJ2. Moreover, wehave

H1(q̃²) = H1(γ (±,²), I u)+ O(√²),

H1(qb5) = H1(Eq (s,0)b )+ O(√²),


Fig. 6.5.Geometry for the approximation.

whereEq (s,0)b ≡ (−γ1 − 2P + O(√²), I u), γ (±,²) = −γ1 + 2P (see Corollary 1 and

Theorem 5.2), andP is given in subsection “Calculation of̂MF̃1.” Thus, we have

1H1 = 4αωP − 20 sin 2P cosγ1+ O(√²).

The detailed geometry is shown in Figure 6.5. This completes the proof of the theorem.

Now, we can combine the Melnikov measurement and the second measurement toget the approximate roots for the two distance equationsd1 = 0 andd2 = 0, given by{

cosγ1M̂0(ω, N)− χαMα(ω, N)+ χβMβ(ω, N) = 0,4αωP − 20 sin 2P cosγ1 = 0,

(6.10)

where M̂0(ω, N), Mα(ω, N), and Mβ(ω, N) are defined in Proposition 1. SolvingEq. (6.10), we get the following relation amongχα, χβ , andω:

χβ = κ(ω, N)χα,

κ(ω, N) = 1Mβ(ω, N)

{Mα(ω, N)− ωM̂0(ω, N) 2P

sin 2P

}. (6.11)

By the implicit function theorem, we have the following theorem (cf. Introduction):

Theorem 6.2. For any N (7 ≤ N < ∞), there exists a positive number²0, such thatfor any² ∈ (0, ²0), there exists a codimension1 submanifold E² in 6N, E² is an O(²ν)perturbation of the hyperplaneβ = κ α, whereκ = κ(ω; N) is shown in Figure 1.1,


ν = 1/2− δ0, 0 < δ0 ¿ 1/2. For any external parameters (ω, α, β, 0) on E² , thereexists a homoclinic orbit asymptotic to the fixed point q² .

Remark 6.2. Note that the two measurementsd1 = 0 andd2 = 0 can be satisfied asfollows: First, choose the “take-off” phaseγ = −γ1+ 2P approximately by

cosγ1 = 1M̂0(ω, N)

[χαMα(ω, N)− χβMβ(ω, N)

],

and thenχα andχβ approximately by the linear relation

χβ = κ(ω, N)χα, κ(ω, N) = 1Mβ(ω, N)

{Mα(ω, N)− ωM̂0(ω, N) 2P

sin 2P

}.

Remark 6.3. For N ≥ 7, κ can be positive. (See Figure 1.1.)

7. Conclusion

For the discretized perturbed NLS system (1.1), we have proved the existence of homo-clinic orbits. In part II, we will construct Smale horseshoes in the neighborhood of thesehomoclinic orbits.

This is a study on homoclinic and chaotic dynamics on perturbed soliton systems.The prerequisite for such study is that the unperturbed soliton system, viewed as aHamiltonian system, should have singular level sets (i.e., “Figure 8” structures). Amongsoliton systems which fall into this class are the focusing NLS equation, sine-Gordonequation, Davey-Stewartson equation [19], etc. The Backlund-Darboux representationsof the singular level set enable us to build explicit Melnikov integrals which measure thedistance between certain perturbed unstable and stable manifolds.

In the process of proving the existence of (or constructing) homoclinic orbits, severalmathematical tools from dynamical system theory are utilized. Among them are a per-sistence of invariant manifold theorem combined with a geometric singular perturbationtheory, and a Fenichel-Hadamard fibration theorem. For more on geometric singularperturbation theory, see the recent review by Jones [12].

In this paper we have built a general argument for locating homoclinic orbits in per-turbed soliton systems. This argument should be applicable to a wide class of perturbedsoliton equations.

Numerically locating homoclinic orbits, for the perturbed (lower dimensional) dis-crete NLS systems studied in this paper, is in progress [14].

8. Appendix A: Symmetries of the Eigen-Functions

In this appendix, we discuss some symmetries of the eigen-functions of the Lax pair(2.3), (2.4).


Lemma 8.1. If ψn solves the Lax pair (2.3), (2.4) at z, then Jψ̄n solves the same Laxpair at z∗ ≡ 1/z̄. Here J= ( 0 1−1 0), z∗ is the symmetry point of z with respect to the unitcircle centered at the origin. Applying this fact to the fundamental solutions, we have1̄(z) = 1(z∗).

There is a continuum counterpart (h → 0, i.e., for the Lax pair for NLS PDE) of thislemma in whichz∗ is replaced byλ∗ = λ̄, which is the symmetry point ofλwith respectto the real axis.

Lemma 8.2. If qn is even (i.e., qN−n = qn), andψn solves the Lax pair (2.3), (2.4) atz, then

φn ≡(φn,1φn,2

)= αn

(ψ̄−n+1,2ψ̄−n+1,1

),

whereαn+1αn= ρn, solves the same Lax pair atz̄. Applying this fact to the Bloch functions,

we have1̄(z) = 1(z̄).

There is a continuum counterpart of this lemma in whichz̄ is replaced by−λ̄.

Lemma 8.3. If ψn solves the Lax pair (2.3), (2.4) at (z,qn), then einπψn solves the Laxpair at (−z,−qn). Applying this fact to the fundamental solutions, we have ei Nπ1(z,qn)= 1(−z,−qn).

There is no continuum counterpart of this lemma. Due to this lemma, the spectrum inthis discrete case is a double-copy of the corresponding spectrum in the continuum case.

9. Appendix B: Linear Stability of q²

In this appendix, we study the linear stability ofq² . Consider a harmonic perturbationof q² ,

qn = q² + η q̃n, 0< η ¿ 1,

q̃n =(

Aj eÄj t + Bj eǞj t

)coskj n,

wherekj = 2π jN , Aj and Bj are complex constants. We have the followingdispersionrelation atkj :

Äj ≡ Ä±j = −²[α + 2βh−2(1− coskj )

]± 2√Ej Fj , (9.1)

where

Ej = 2|q² |2+ (|q² |2− ω2)− (1− coskj )(1/h2+ |q² |2),Fj = (1− coskj )(1/h2+ |q² |2)− (|q² |2− ω2).


From Eqs. (3.4), (3.5), we can calculate that|q² | = ω − C² + O(²2) for some positiveconstantC. We have the following asymptotic formulae for attracting or repelling rates:

1. For j = 0; hencek0 = 0, E0 = 2|q² |2+ (|q² |2−ω2) > 0, F0 = −(|q² |2−ω2) > 0;thenE0F0 > 0,

E0F0 = 4²Cω3+ O(²2).Thus,Ä±0 has the aymptotic form:

Ä±0 = ±4²1/2C1/2ω3/2+ O(²).

2. For j = 1; hence k1 = 2πN , F1 > 0. SinceN tan πN < ω < N tan2πN , E1 =(1+ cosk1)(ω2− N2 tan2 πN )+ O(²) > 0. Therefore,E1F1 > 0,

E1F1 = (1− cos2 k1)(1/h2+ ω2)(ω2− N2 tan2 π

N

)+ O(²).

Thus,Ä±1 has the aymptotic form:

Ä±1 = ±2√(1− cos2 k1)(1/h2+ ω2)

(ω2− N2 tan2 π

N

)+ O(²).

3. For j = 2, 3, . . . ,M (M = N/2, N even; M = (N − 1)/2, N odd); Fj > 0.SinceN tan πN < ω < N tan

2πN , Ej = (1+ coskj )(ω2 − N2 tan2 jπN ) + O(²) < 0.

Therefore,Ej Fj < 0. Thus,Ä±j has the form

Ä±j = −²[α + 2βh−2(1− coskj )

]± 2i√|Ej Fj |.

So we have three well-separated attracting and repelling rate scales:O(²0) (at j = 1),O(²1/2) (at j = 0), andO(²) (for all other j ). j = 2 is the slowest attracting direction.Next, we calculate the eigenvectors (and linearized flow in the linear subspaces atq²).The amplitudesAj andBj satisfy the equations[

iÄj − (2κ1(coskj − 1)+ κ2)]

Aj = 2|q² |2B̄j , (9.2)[i Ǟj − (2κ1(coskj − 1)+ κ2)

]Bj = 2|q² |2 Āj , (9.3)

whereκ1 = h−2+|q² |2+ i ²βh−2, κ2 = 4|q² |2−2ω2− i ²α. The compatibility conditionof (9.2) and (9.3) gives the dispersion relation (9.1).

1. For j = 0, 1; Äj is real, and then from the relations (9.2) and (9.3), we have∣∣∣∣i Ǟj − (2κ1(coskj − 1)+ κ2)∣∣∣∣ = 2|q² |2.Thus, we can write

i Ǟ±j − (2κ1(coskj − 1)+ κ2) = 2|q² |2eiϕ±j ,


where

ϕ±j = arctan{Ä±j − (2 Im{κ1}(coskj − 1)+ Im{κ2})

2 Re{κ1}(1− coskj )− Re{κ2}

}. (9.4)

Then,Bj = e−iϕ±j Āj . Thus, we have

q̃n = (Aj + e−iϕ±j Āj )e

Ä±j t coskj n.

Let Aj = r j ei θj , then

Aj + e−iϕ±j Āj = r j e−i

12ϕ±j

[ei (θj+

12ϕ±j ) + e−i (θj+ 12ϕ±j )

]= 2r j cos

(θj + 1

2ϕ±j

)e−i

12ϕ±j ≡ c±j e−i

12ϕ±j ,

wherec±j ≡ 2r j cos(θj + 12ϕ±j ) are real parameters. Thereforeq̃n takes the final form,

q̃n = c±j e−i12ϕ±j eÄ

±j t coskj n,

in which ϕ±j are fixed phases given in (9.4) (which identify the stable and unstabledirections) andc±j are real parameters.

2. For j = 2, . . . ,M ; Äj is complex.Bj = bj Āj , wherebj = 2|q² |2(i Ǟ+j −(2κ1(coskj−1)+ κ2))−1. Then,q̃n takes the form

q̃n =[

Aj ei Im{Ä+j }t + bj Āj e−i Im{Ä

+j }t]eRe{Ä

+j }t coskj n,

where Aj is a complex parameter. ReplacingÄ+j by Ä

−j gives the same linearized

motion.

10. Appendix C: Estimates of the Limits

In this appendix, we want to find out the limits limt→−∞1−(t) and limt→+∞1+(t), inorder to represent1−(0) by

1−(−∞)+∫ 0−∞

1̇−(t) dt,

and1+(0) by

1+(+∞)−∫ +∞

01̇+(t) dt,

respectively. Unfortunately, in some case, calculating limt→+∞1+(t) is not feasible.Nevertheless, since we are only interested in the values of1−(0) and1+(0), thecrucial


point is to find computable formulas for them. Thus, in such cases, we will represent1+(0) by

1+(T(²))−∫ T(²)

01̇+(t) dt,

for some largeT(²).First, we calculate limt→−∞1−(t), and show that limt→−∞1−(t) = 0:

limt→−∞1

−(t) = limt→−∞〈∇ F̃1(h(t)), h

u² (t)− h(t)〉.

From explicit representation of∇ F̃1(h(t)) in a later section, we see that there existT1(> 0) andC1 (> 0), such that∥∥∥∥∇ F̃1(h(t))∥∥∥∥ ≤ C1eκ1t , for all −∞ < t ≤ −T1, (10.1)whereκ1 > 0. By fiber theorem 4.1, we see that there existC2 (> 0) andC3 (> 0), suchthat ∥∥∥∥hu² (τ − Tu² )− F τ² (γ, I u)∥∥∥∥ ≤ C2eκ2τ , (10.2)∥∥∥∥h(τ − Tu0 )− F τ0 (γ, I u)∥∥∥∥ ≤ C3eκ2τ , (10.3)

for all −∞ < τ ≤ 0;where

κ2 > 0, hu² (−Tu²

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