Invariant Manifolds and Their Fibrations

for Perturbed Nonlinear Schrödinger

Equation

Yanguang (Charles) LiDepartment of Mathematics (2-336)

Massachusetts Institute of TechnologyCambridge, MA 02139

Stephen WigginsApplied Mechanics, Control and Dynamical System 116-81

California Institute of TechnologyPasadena, CA 91125

September 4, 2009

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Contents

1 Introduction 41.1 Invariant Manifolds in Infinite Dimensions . . . . . . . . . . . 41.2 Aims and Scopes of This Monograph . . . . . . . . . . . . . . 12

1.2.1 Soliton Equations . . . . . . . . . . . . . . . . . . . . . 121.2.2 Perturbed Soliton Equations . . . . . . . . . . . . . . 15

2 The Perturbed Nonlinear Schrödinger Equation 172.1 The Setting for the Perturbed Nonlinear Schrödinger Equation 172.2 Spatially Independent Solutions: An Invariant Plane . . . . . 182.3 Statement of the Persistence and Fiber Theorems . . . . . . . 212.4 Invariant Manifolds and Fibers for the Integrable NLS Equation 23

2.4.1 Integrable Preliminaries . . . . . . . . . . . . . . . . . 232.4.2 Bäcklund-Darboux Transformations . . . . . . . . . . 272.4.3 Explicit Representations for Invariant Manifolds and

Fibers . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.5 Coordinates Centered on the Resonance Circle . . . . . . . . 30

2.5.1 Definition of the H̃ Norms . . . . . . . . . . . . . . . . 302.5.2 A Neighborhood of the Circle of Fixed Points . . . . . 312.5.3 An Enlarged Phase Space . . . . . . . . . . . . . . . . 322.5.4 Scales through δ . . . . . . . . . . . . . . . . . . . . . 322.5.5 The Equations in Their Final Setting . . . . . . . . . 33

2.6 (δ = 0) Invariant Manifolds and the Introduction of a BumpFunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.6.1 (δ = 0) Invariant Manifolds . . . . . . . . . . . . . . . 372.6.2 Tangent and Transversal Bundles of M . . . . . . . . . 392.6.3 Introduction of a Bump Function . . . . . . . . . . . . 40

3 Persistent Invariant Manifolds 443.1 Statement of the Persistence Theorem and the Strategy of

Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.2 Definition of the Graph Transform . . . . . . . . . . . . . . . 473.3 The Graph Transform as a C0 Contraction . . . . . . . . . . 523.4 The Existence of the Invariant Manifolds . . . . . . . . . . . . 613.5 The Smoothness of the Invariant Manifolds . . . . . . . . . . 623.6 Completion of the Proof of the Proposition . . . . . . . . . . 72

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4 Fibrations of the Persistent Invariant Manifolds 744.1 Statement of the Fiber Theorem and the Strategy of Proof . 744.2 Rate Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . 764.3 The Existence of an Invariant Subbundle E . . . . . . . . . . 844.4 Smoothness of the Invariant Subbundle E . . . . . . . . . . . 904.5 Existence of Fibers . . . . . . . . . . . . . . . . . . . . . . . . 1024.6 Smoothness of the Fiber fE(Q) as a Submanifold . . . . . . . 1234.7 Metric Characterization of the Fibers . . . . . . . . . . . . . . 1334.8 Smoothness of the Fiber Family {fE(Q)}Q∈M With Respect

to Base Point Q . . . . . . . . . . . . . . . . . . . . . . . . . . 141

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1 Introduction

In this introductory chapter we want to give a brief survey of results oninvariant manifolds and fibers in infinite dimensions as well as describe theaims and scopes of this monograph. Recent books by Wiggins [99] andBronstein and Kopanskii [13] survey invariant manifold results in finite di-mensions. Here we will be focussing solely on surveying results in the infinitedimensional setting.

1.1 Invariant Manifolds in Infinite Dimensions

The finite dimensional invariant manifold theory is not only very general,from the point of view of mathematical assumptions, but it is broadly ap-plicable to a variety of specific systems arising in applications. Of course,this is true because the general mathematical assumptions are “reasonable”for many dynamical systems that arise in practice. For example, in the caseof vector fields on IRn, it is common for the evolution operators arising inapplications to be smooth in both time and initial data. In contrast, theevolution operators for partial differential equations arising in applicationsare often at best continuous in time. More importantly, the nature of theevolution operators for partial differential equations differs greatly from onesystem to another. This fact makes a thoery as general as in finite dimen-sional systems intractable.

From the literature one sees that there are two broad approaches toinvariant manifold theory in infinite dimensions. One is to state a list ofhypotheses satisfied by a class of infinite dimensional dynamical systemsand then prove theorems for the class of dynamical systems satisfying suchhypotheses. 1 The other approach is to take a specific infinite dimensionaldynamical system and prove the desired invariant manifold theorems specif-ically for the system (of course, one would first need some form of existence,uniqueness, and regularity theory for the specific dynamical system underinvestigation).

Many of the issues concering invariant manifold theory in infinite dimen-sions are much the same as those in finite dimensions. The first approachdescribed above is guided by the goal of “lifting” to infinite dimensions the

1Here we will interpret the phrase infinite dimensional dynamical system broadly in

that it applies to a semiflow, or flow on an appropriate infinite dimensional space. We

also refer to evolutionary partial differential equations as infinite dimensional dynamical

systems.

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finite dimensional results. It is the second approach that often results inthe discovery of phenomena that are truly infinite dimensional. Below wesurvey many of the known results related to invariant manifolds in infinitedimensions in the context of a number of specific issues that arise whenseeking to develop and apply such results. Because the literature is so largewe limit our survey to papers that prove new theorems for various classes ofinfinite dimensional dynamical systems, as well as a description of the areasof applications to which these theorems can be applied. As a result, we areleaving out a portion of the literature dealing with potential applications ofthese results.

Existence of Invariant Manifolds

As in the finite dimensional case, invariant manifold theory begins byassuming that some “basic” invariant manifold exists–the theory is thendeveloped from this point onwards by building upon this basic invariantmanifold. This may seem a bit strange, for one might think that invariantmanifold theory is concerned with determining the existence of invariantmanifolds. That is true, to a point. However, if one were to ask the generalquestion:

given a specific dynamical system (finite or infinite dimensional),what type of invariant manifolds does it possess?

it probably could not be answered in any kind of generality. Thus, invariantmanifold theory requires some type of “seed” in order to get started. Onetype of seed is knowledge of a particular orbit of the dynamical system. Suchorbits can be viewed as invariant manifolds. Examples of orbits that canoften be (relatively) easily found are:

1. Equilibrium points,

2. Periodic orbits,

3. Quasiperiodic or almost periodic orbits (invariant tori).

Starting from this point, one may then construct stable, unstable, and cen-ter manifolds associated with these basic invariant manifolds, as well as

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fibrations of these manifolds, and also consider issues such as differentia-bility of the invariant manifolds and fibrations and their peristence underperturbation.

These particular invariant manifolds all share an important property.Namely, they all admit a global coordinate description in the sense thatthey are either graphs or parametrized curves. In this case the dynamicalsystem is typically subjected to a “preparatory” coordinate transformationthat serves to localize the dynamical system about the invariant manifold.This amounts to deriving a normal form in the neighborhood of an invariantmanifold and it greatly facilitates the various estimates that are required inthe analysis. Henry [45] has derived a normal form for a class of semilinearparabolic partial differential equations in the neighborhood of a compact,finite dimensional invariant manifold, with the important condition that thenormal bundle of the invariant manifold is trivial. Bates et al. [5] considerthe situation of a compact invariant manifold in Hilbert space that is notdescribed as a graph and requires an atlas of local coordinate charts for theirdescription. We will describe the results of Henry [45] and Bates et al. [5]in more detail shortly.

The other “seed” which may serve as a starting point for invariant man-ifold theory is if the dynamical system under consideration possesses somespecial structure so that the existence of an invariant manifold is “obvious”.For example, for the nonlinear Schrödinger equation it is clear from thestructure of the equations that the solutions that are independent of thespace variable form a two dimensional invariant plane in the infinite dimen-sional function space on which the nonlinear Schrödinger equation is defined.This trivial invariant plane will play a central role in the next chapter ofthis monograph.

Behavior Near an Invariant Manifold–Stable, Unstable, andCenter Manifolds

A “stable manifold theorem” asserts that the set of points that approachan invariant manifold at an exponential rate as t → +∞ is an invariantmanifold in its own right. The exponential rate of approach is inherited fromthe linearized dynamics as the stable manifold is constructed as a graphover the linearized stable subspace or subbundle. An “unstable manifoldtheorem” asserts similar behavior in the limit as t → −∞. Obviously, onemay have problems with both of these concepts if the invariant manifold hasa boundary.

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The notion of a center manifold is more subtle. For equilibrium pointsand periodic orbits a center manifold is an invariant manifold that is tangentto the linearized subspace corresponding to eigenvalues on the imaginaryaxis, and floquet multipliers on the unit circle, respectively. In contrast tothe situation with stable and unstable manifolds, the asymptotic behaviorof orbits in the nonlinear center manifold may be very different from theasymptotic behavior of orbits in the linearized center subspaces, under thelinearized dynamics.

Once the existence of stable, unstable, and center manifolds is estab-lished then questions related to persistence and differentiability arise quitenaturally.

There are a number of papers that deal with stable, unstable, and centermanifolds of equilibrium points of partial differential equations. In particu-lar, we refer to Ball [3], Bates and Jones [4], Carr [15], Chow and Lu [18], daPrato and Lunardi [86], Henry [45], Keller [47], Mielke [76], Renardy [87],and Scarpellini [90], [91]. Below we summarize the types of equations andapplications considered by these authors.

Reference Type of Partial Differential Equation

Ball [3] semilinear hyperbolic equationsBates and Jones [4] semilinear hyperbolic and parabolic equationsCarr [15] semilinear hyperbolic equationsChow and Lu [18] semilinear hyperbolic and parabolic equationsda Prato and Lunardi [86] quasilinear parabolic equationsHenry [45] semilinear parabolic equationsKeller [47] semilinear hyperbolic equationsRenardy [87] quasilinear hyperbolic equationsScarpellini [90], [91] semilinear hyperbolic equationsMielke [76] quasilinear parabolic equationsda Prato and Lunardi [86] quasilinear parabolic equations

We remark that in dealing with center manifolds, the issue of whether ornot it is infinite dimensional can pose some technical difficulties. The workof Bates and Jones [4] deals with the infinite dimensional case.

Many of the references above treat specific applications. We list them inthe following table.

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Reference Application

Ball [3] buckling beamBates and Jones [4] nonlinear wave equations, nerve impulse equationsCarr [15] semilinear wave equationChow and Lu [18] reaction-diffusion equations, singularly perturbed wave equationHenry [45] combustion, Hodgkin-Huxley equations, plus many moreKeller [47] nonlinear wave equationsRenardy [87] Bénard problem for a viscoelastic fluidScarpellini [90], [91] Sine-Gordon Equation

Regularity of Stable, Unstable, and Center Manifolds

Existence of stable, unstable, and center manifolds is typically proven bysome form of contraction mapping argument. From this type of argumentone obtains Lipschitz manifolds, even if the nonlinearity is differentiable.To prove the existence of smooth manifolds one formally differentiates theequation describing the manifold and, in this way, derives an equation whichthe derivative of the manifold must satisfy. Using contraction mappingarguments, one then proves that this equation has a solution. Finally, usingthe definition of Frechet derivative, one proves that the solution is indeedthe derivative; thus, establishes the differentiability. One can repeat thisargument and, thus, inductively prove the existence of more derivatives,provided the structure of the partial differential equation allows this. If one isinterested in stability of the equilibrium solution, then Lipschitz is sufficient.However, for bifurcation theory one usually needs several derivatives.

We summarize the regularity of the manifolds in the references describedabove in the following table.

Reference Regularity

Ball [3] LipschitzBates and Jones [4] LipschitzCarr [15] C2

Chow and Lu [18] Ck

da Prato and Lunardi [86] C1

Henry [45] Ck

Keller [47] LipschitzRenardy [87] C1

Scarpellini [90], [91] C1

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Fibrations of Stable and Unstable Manifolds — More RefinedBehavior Near Invariant Manifolds

One may be interested in which orbits in the stable manifold approachthe invariant manifold at a specified rate. Under certain conditions theseorbits may lie on submanifolds of the stable manifold which are not invariant,but make up an invariant family of submanifolds that fiber or foliate thestable manifold. A similar situation may hold for the unstable manifold.Moreover, this fibration has the property that points in a fiber of the fibrationasymptotically approach the trajectory in the invariant manifold that passesthrough the point of intersection of the fiber with the invariant manifold (thebasepoint of the fiber). This is a generalization of the notion of asymptoticphase, that is familiar from studies of stability of periodic orbits, to arbitraryinvariant manifolds. In recent years these foliations have seen many uses inapplications in the finite dimensional setting, see Jones [46], and Wiggins[99]. Chow et al. [17] prove a fibration theorem in a very general setting.They do not consider any explicit examples in their paper, however they doremark that their results are related to inertial manifold type results. Li etal. [61] prove a fibration theorem for the nonlinear Schrödinger equation.Ruelle [88] constructs stable and unstable fibrations almost everywhere for asemi-flow in Hilbert space having a compact invariant set (he assumes thatthe linearized semi-flow is compact and injective, with dense range). Hisresults were later extended by Mañé [80].

The Persistence and Differentiability of Invariant ManifoldsUnder Perturbation

The question of whether or not an invariant manifold persists under per-turbation and, if so, if it maintains, loses, or gains differentiabilty is alsoimportant. In considering these issues it is important to characterize thestability of the unperturbed invariant manifold. This is where the notion ofnormal hyperbolicity arises. Roughly speaking, a manifold is normally hy-perbolic if, under the dynamics linearized about the invariant manifold, thegrowth rate of vectors transverse to the manifold dominates the growth rateof vectors tangent to the manifold. For equilibrium points, these growthrates can be characterized in terms of eigenvalues associated with the lin-earization at the equilibria that are not on the imaginary axis, for periodicorbits these growth rates can be characterized in terms of the Floquet mul-tipliers associated with the linearization about the periodic orbit that are

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not on the unit circle, for invariant tori or more general invariant manifoldsthese growth rates can be characterized in terms of exponential dichotomies(see Coppel [21] or Sacker and Sell [89]) or by the notion of generalizedLyapunov type numbers (Fenichel [31]).

Characterizing growth rates in the fashion described above requires knowl-edge of the linearized dynamics near orbits on the invariant manifold ast → +∞ or t → −∞. Hence, if the invariant manifold has a boundary(which an equilibrium point, periodic orbit, or invariant torus does not have),then one must understand the nature of the dynamics at the boundary. No-tions such as overflowing invariance or inflowing invariance were developedby Fenichel [31], [32], [33] to handle this. Invariant manifolds with bound-ary arise very often in applications, see Wiggins [98] for finite dimensionalexamples.

A question of obvious importance for applications is how does one com-pute whether or not an invariant manifold is normally hyperbolic? Theanswer is not satisfactory. For equilibria, the problem involves finding thespectrum of a linear operator. For invariant manifolds on which the dynam-ics is nontrivial the issues are more complicated.

However, one important class of dynamical systems which may havenontrivial invariant manifolds on which the dynamics is also nontrivial areintegrable Hamiltonian systems, see Wiggins [98] for finite dimensional ex-amples.

Bates et al. [5] have a very general result on the persistence of a compact,normally hyperbolic invariant manifold under a semiflow. They show that ifboth the unperturbed and perturbed semiflows are C1, and the unperturbedinvariant manifold is C2, then the perturbed invariant manifold is C1. Theydo not require the unperturbed invariant manifold to have a trivial normalbundle. Their work can be viewed as an infinite dimensional generalizationof Fenichel’s [31] seminal finite dimensional work. Li et al. [61] prove a per-sistence theorem for normally hyperbolic invariant manifolds, as well as theirstable and unstable manifolds and fibrations, for the nonlinear Schrödingerequation. The set-up is different than that of Bates et al. [5] in that Li etal. consider a noncompact invariant manifold invariant under a flow (ratherthan a semi-flow). Moreover, the unperturbed invariant manifold is globallyrepresented as a graph. Their invariant manifold, along with its stable andunstable manifolds, are Ck k > 3 and the fibers are Ck−2.

Inertial Manifolds

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In recent years there has been much interest in a class of global, attract-ing, finite dimensional invariant manifolds in dissipative systems–inertialmanifolds. These invariant manifolds are robust objects and contain thelong term dynamical phenomena of a system. In this way, they provide arigorous manner of reducing an infinite dimensional dynamical system to afinite dimensional dynamical system, when the questions of interest are con-cerned with asymptotic behavior. The following references give an accountof this theory: Constantin et al. [20], Chow and Lu [18], Foias and Sell [35],Foias et al. [36], Hale [41], Mallet-Paret and Sell [66], Mora [78], Mora andSolà-Morales [79], Témam [94].

Invariant Manifolds in Infinite Dimensional Hamiltonian Systems

Invariant tori (quasiperiodic solutions) are a typical type of invariantmanifold that arise in Hamiltonian systems. The KAM theorem is a well-known result describing persistence of invariant tori in perturbations of com-pletely integrable systems in finite dimensions. In recent years there havebeen a number of generalizations of this result to various infinite dimen-sional settings, see Kuksin [56], [55], [54], [53], [52], Pöschel [84], [83], andWayne [97], [95], [96], [12]. Nikolenko [81] considers the existence of asymp-totically stable tori in a dissipative perturbation of an infinite dimensionalHamiltonian system (the Korteweg-de Vries equation). Mielke [75] gives ageneral treatment of invariant manifolds in a class of Hamiltonian partialdifferential equations.

Normal Forms for Partial Differential Equations

The classical Poincaré normal form theory is not unrelated to invariantmanifold theory. Normal form theory is concerned with constructing co-ordinate changes that make the dynamical system as “simple as possible”.“Simple” could mean linear, or it might mean eliminating terms wherebythe invariant manifold structure of the resulting equation is transparent.Indeed, many invariant manifold theorems are stated in this way.

Normal forms for partial differential equations have been considered byEckmann et al. [27], Nikolenko [82], and Shatah [92], Shatah and McKean[93]. A normal form calculation for the nonlinear Schrödinger equation isgiven in Li et al. [61], and indeed is central to their persistence theoremsfor invariant manifolds and fibrations.

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Methods of Proof

As described in Wiggins [99], there are four approaches to the proof ofinvariant manifold results in finite dimensions.

• The Liapunov- Perron Method

• Hadamard’s Method-The Graph Transform

• The Lie Transform or Deformation Method

• Irwin’s Method

In infinite dimensions the Liapunov-Perron method has been the mostcommon approach. Bates and Jones [4] and Bates et al. [5] use the graphtransform approach, and the results in this monograph are based entirelyon the graph transform approach.

1.2 Aims and Scopes of This Monograph

This monograph is motivated by our work on the perturbatively dampedand driven Nonlinear Schrödinger (NLS) equation described in [8], [9], [7],[14], [28], [29], [30], [42], [44], [43], [48], [49], [50], [51], [59], [64], [65], [63],[61], [70], [71], [73], [74], and [72]. In this monograph we prove the basicexistence, persistence, and differentiability results for invariant manifoldsand fibrations of the perturbatively, damped and driven NLS equation thatwe use in our analysis. Our technique is based on an infinite dimensionalversion of the graph transform. Our setting is somewhat general, howeverthe generality is motivated by the structure of the perturbatively damped,driven NLS equation. In particular, the unperturbed NLS equation is com-pletely integrable. Consequently, a great deal is known about the invariantmanifold structure in the phase space. As a result, a number of “persis-tence of invariant manifold” questions arise naturally when one considersperturbations. Similar issues can be addressed in other completely inte-grable partial differential equations or soliton equations, and we now give abrief description on the scope of such problems.

1.2.1 Soliton Equations

Soliton equations have been a subject of much interest over the past thirtyyears since the discovery of the inverse scattering transformation (IST) for

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the Korteweg-de Vries (KdV) equation [39]. The IST is a Cauchy problemsolver. For (1+1)-dimensional (i.e. 1 temporal dimension and 1 spatialdimension) soliton equations, their IST solvers are parallel and can be viewedas nonlinear Fourier transforms [1]. For (1+2)-dimensional (i.e. 1 temporaldimension and 2 spatial dimensions) soliton equations, their IST solversare not only very different from (1+1)-dimensional IST solvers, but alsodifferent among themselves. Here there are essentially two types: One isthe “∂̄ approach”, the other is the “Riemann-Hilbert approach”. For areview, see for example [2],[37]. Aside from IST, other important tools of“soliton theory” are Backlund transformation theory [77][58][62], symplecticgeometry [85],[63], algebraic geometry theory [26], modulation theory [34][57], and long-time asymptotics theory [22],[23]. Interesting solutions ofsoliton equations include multi-solitons, breathers, kinks, anti-kinks, etc;which are of great physical importance.

Soliton equations are integrable Hamiltonian systems which can be clas-sified into two groups (under periodic boundary conditions): Type 1. Equa-tions whose phase space contains no hyperbolic structure, (i.e. all level setsare tori of elliptic stability type, referred to as “elliptic tori”), for example:

• Korteweg-de Vries equation [69],[68],

ut + 6uux + uxxx = 0,

where u is a real-valued function of (x, t).

• Defocusing nonlinear Schroedinger equation [40],

iqt = qxx − |q|2q,

where q is a complex-valued function of (x, t).

• Sinh-Gordon equation [67],

uxt = sinhu,

where u is a real-valued function of (x, t).

• The modified KdV equation [25],

ut − 6u2ux + uxxx = 0,

where u is a real-valued function of (x, t).

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Type 2. Equations whose phase space contains hyperbolic structures,for example:

• Focusing nonlinear Schroedinger equation, [63],

iqt = qxx + |q|2q,

where q is a complex-valued function of (x, t).

• Sine-Gordon equation, [29],

uxt = sinu,

where u is a real-valued function of (x, t).

• The modified KdV equation, [25],

ut + 6u2ux + uxxx = 0,

where u is a real-valued function of (x, t).

The hyperbolic structures of the type 2 soliton equations can be explic-itly represented through Backlund-Darboux transformations, see [63] whichestablishes the connection between Backlund transformations and symplec-tic geometry. Moreover, Backlund-Darboux transformations also offer rep-resentations of fibers of the normally hyperbolic invariant manifolds. Thephase block classification of the soliton equations can be studied throughMorse-Bott-Fomenko theory, [63],[10],[38], which leads to a variety of in-teresting invariant submanifolds. The Morse-Bott-Fomenko theory is builtupon invariants furnished through isospectral studies, [63]. The work [63]concentrates on the focusing nonlinear Schroedinger equation. However, itcan be easily extended to other type 2 soliton equations.

There has been a variety of applications of soliton equations. For ex-ample, the KdV equation describes shallow water waves, ion plasma waves,waves in elastic media, etc. The nonlinear Schroedinger equation describeswaves in nonlinear optics, vortex filament movement, etc. The sine-Gordonequation describes waves in Josephson junction, charge density waves in onedimensional metals, pseudo-sphere geometry, etc.

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1.2.2 Perturbed Soliton Equations

Since soliton equations can have such a rich, and well-understood phasespace structure, they are ideal “laboratories” for the development of geo-metric perturbation methods. According to the type of perturbations andthe type of the soliton equations, the questions asked may be very different.

Hamiltonian Perturbations

Under Hamiltonian perturbations, one of the most commonly asked ques-tions concerns the existence persistence of tori (or existence of periodicand/or quasiperiodic solutions) under perturbations. Under certain restric-tions on the perturbations, the infinite dimensional KAM type methodsdescribed above may apply.

For Hamiltonian perturbations of type 2 soliton equations the persistenceof normally hyperbolic invariant manifolds can be studied by the graphtransform technique developed in this monograph. Generally, persistenceof hyperbolic structures is not dependent on the Hamiltonian nature of theproblem.

Dissipative Perturbations

For soliton equations under dissipative perturbations, inertial manifoldsand attractors can be constructed, [19],[20],[6]. Inertial manifolds attracttheir neighborhoods in phase space at an exponential rate, while attractorsare the limiting attracting sets. Their Hausdorff dimensions are all finite,and can be estimated. An inertial manifold can be viewed as a graph overa subspace spanned by a finite number of Fourier modes. The constructionof an inertial manifold involves studies of spectral gaps and inequality es-timates of various types. The construction of an attractor involves energyestimates. The existence of attractors does not require strongly dissipativeperturbations, [6], while the existence of inertial manifolds usually requiresstrong dissipative perturbations in order to satisfy the spectral gap condi-tions [19].

For type 2 soliton equations under dissipative perturbations, persistenceof normally hyperbolic invariant manifolds can be studied. This monographillustrates such a study for a particular perturbed type 2 soliton equation—the perturbed nonlinear Schroedinger equation. Parallel arguments for other

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type 2 soliton equations can be easily extended. Persistent hyperbolic foli-ations and fibrations are the basic tools for the study of chaos in perturbedsoliton equations described in [61][60].

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2 The Perturbed Nonlinear Schrödinger Equation

2.1 The Setting for the Perturbed Nonlinear SchrödingerEquation

Consider the perturbatively damped and driven nonlinear Schrödinger equa-tion (PNLS):

iqt = qxx + 2

[

|q|2 − ω2]

q + iǫ

[

− αq + D̂2q + Γ]

, (2.1)

under even and periodic boundary condition

q(−x) = q(x), q(x+ 1) = q(x);where ω ∈ (π, 2π), ǫ ∈ (−ǫ0, ǫ0) is the perturbation parameter, α(> 0) and Γare real constants. The operator D̂2 is a regularized Laplacian, specificallygiven by

D̂2q ≡ −∞∑

j=1

βj k2j q̂j cos kjx,

where q̂j is the Fourier transform of q and kj ≡ 2πj. The regularizingcoefficient βj is defined by

βj ≡{

β for j ≤ Nα∗k

−2j for j > N ,

where α∗ and β are positive constants and N is a large fixed positive integer.When ǫ > 0, the terms −ǫαq and ǫD̂2q are perturbatively damping terms,the former is a linear damping, and the latter is a diffusion term; the termǫΓ is a perturbatively driving term. As mentioned in the introduction, thesystem (2.1) has been extensively studied.

Remark 2.1 The study in this book can be carried through for more generalperturbations, and even for more general systems. The arguments will beparallel. We will restrict ourselves to the particular system (2.1) and givean intensive and detailed study.

We view the pde (2.1) as defining a flow on the following function space:For any integer k (1 ≤ k

The pde (2.1) is well posed in Sk as the following two theorems state:

Theorem 2.1 (Cauchy Problem) For any ~q0 ∈ Sk, there exists a uniquesolution ~q ∈ C0[(−∞,∞);Sk] to the perturbed NLS equation (2.1), such that~q |t=t0= ~q0.

Let F tǫ denote the evolution operator of the perturbed NLS equation (2.1).Then we have

Theorem 2.2 (Dependence on Data) For any fixed t ∈ (−∞,∞), andany fixed integers n and k (1 ≤ n, k < ∞), F tǫ is a Cn diffeomorphism inSk, and is also Cn smooth in ǫ.

The above two theorems are well-known. Local well-posedness is estab-lished following standard pde methods as described in [16]. A recent workconcerning rough data is described in [11]. Global existence in Sk follows bycontrolling for the PNLS flow (2.1) the constants of motion for the integrable(ǫ = 0) system; in turn, these constants control the Hk norm of the solution.Further details about the proofs of these two theorems may be found in thethesis [59].

2.2 Spatially Independent Solutions: An Invariant Plane

The plane of constants Π (≡ {~q ∈ Sk | ddx~q ≡ 0}) is an invariant plane for thePNLS equation. Phase plane methods are sufficient to describe motion onthis plane, which is some what complicated for the perturbed PNLS system.On the other hand, the integrable NLS equation (ǫ = 0), when restricted tothis invariant plane, has very simple motion consisting of nested circles. Ofthese, the resonance circle Sω,

Sω ≡ {~q ∈ Sk | ~q ∈ Π, |q| = ω},

consists in a circle of fixed points for the NLS equation.The following simple linear stability analysis shows that, for π < ω < 2π,

at any point ~qγ ≡ ωe−iγ ∈ Sω, the stable space Es(~qγ) is 1-dimensional,the unstable space Eu(~qγ) is 1-dimensional, and the center space E

c(~qγ)has codimension 2: Explicitly, we linearize the integrable NLS equation at~qγ ∈ Sω : q = (ω + δq̃) exp (−iγ), δ

The basic solutions of (2.2) have the following form:

q̃±j ={

A±j exp(Ω±j t) +B

±j exp(Ω

±j t)

}

cos kjx, (2.3)

where

Ω±j = ±kj√

4ω2 − k2j ,kj = 2jπ, j = 0, 1, 2, · · · ,

B±j =2ω2

k2j − 2ω2 + iΩ±j

A±j .

For j = 0, 1; Ω±j is real and satisfies 2ω2 = |k2j − 2ω2 + iΩ

±j |. From this

data, one can deduce instabilities as well as construct from these linearizedsolutions a basis of Sk.

Indeed, the following is a basis for Sk:

• k0 mode:e+0 = i

(

11

)

, e−0 =

(

1−1

)

.

Remark 2.2 < e+0 , e−0 >= 0. (Here < , > denotes the L

2 innerproduct.) If ~̃q(0) = α+0 e

+0 + α

−0 e

−0 , for real numbers α

±0 , then

~̃q(t) =(α+ − 4tω2α−)e+0 + α−e−0 .

• k1 mode:

e+1 =

exp{iθ+

1

2 }− exp{−iθ

+

1

2 }

cos k1x, e−1 =

exp{iθ−

1

2 }− exp{−iθ

−

1

2 }

cos k1x,

where,

exp{iθ±1 } =2ω2

k21 − 2ω2 ± iΩ+1

,

i.e.θ±1 = ± arg{k21 − 2ω2 − iΩ+1 }.

Remark 2.3 < e+1 , e−1 > 6= 0, in general. If ~̃q(0) = α+1 e+1 + α−1 e−1 for

real numbers α±1 , then ~̃q(t) = α+1 e

Ω+1

te+1 + α−1 e

Ω−1

te−1 .

19

• kj modes, for j ≥ 2:

Ω±j = ±ikj√

k2j − 4ω2,

Denote by

Ωj ≡ kj√

k2j − 4ω2.

Let

rj ≡2ω2

k2j − 2ω2 + Ωj.

e+j =

(

1−1

)

cos kjx, e−j = i(

1 − rj1 + rj

)

(

11

)

cos kjx.

Remark 2.4 < e+j , e−j >= 0; moreover, rj → 0, as j → 0. If

~̃q(0) = α+j e+j + α

−j e

−j ,

or, in the more compact notation,

~̃q(0) =(

e+j e−j

)

(

α+jα−j

)

,

then

~̃q(t) =(

e+j e−j

)

(

cos Ωjt − sinΩjtsin Ωjt cos Ωjt

)(

α+jα−j

)

.

In this notation, we have the following

Proposition 2.1 (Tangent Spaces) A basis for the tangent space at ~qγ ∈Sω, TSk(~qγ) = Sk, is given by

{e±j , j = 0, 1, · · ·}.

20

Moreover, the linear spaces Es(~qγ), Eu(~qγ), and E

c(~qγ) are given explicitlyby

Es(~qγ) = span {e−1 }Eu(~qγ) = span {e+1 }Ec(~qγ) = span {e±j , j 6= 1}.

In addition, the center-stable and center-unstable spaces are given by

Ecs(~qγ) = span {e−1 ; e±j , j 6= 1}Ecu(~qγ) = span {e+1 ; e±j , j 6= 1}.

2.3 Statement of the Persistence and Fiber Theorems

In this subsection, we will state the two main theorems on the pde (2.1) tobe proved in this book. One is on persistent invariant manifolds, the otheris on the fibration of the persistent invariant manifolds.

First we give the definitions of invariant, overflowing invariant, inflowinginvariant, and locally invariant submanifolds.

Definition 1 Let M be a submanifold of Sk with boundary ∂M , M̄ = M ∪∂M ; F tǫ be the evolution operator of (2.1).

1. M is said to be overflowing invariant under F tǫ , if for every q ∈ M̄ ,F tǫ (q) ∈ M̄ for all t ≤ 0; and for q ∈ ∂M , F tǫ (q) /∈ M̄ for t > 0.

2. M is said to be inflowing invariant under F tǫ , if for every q ∈ M̄ ,F tǫ (q) ∈ M̄ for all t ≥ 0; and for q ∈ ∂M , F tǫ (q) /∈ M̄ for t < 0.

3. M is said to be invariant under F tǫ , if for every q ∈ M̄ , F tǫ (q) ∈ M̄ forall t ∈ R.

4. M is locally invariant under F tǫ , if there exists a neighborhood V ofM in Sk such that for all q ∈ M , if τ ≥ 0 and

⋃

t∈[0,τ ] Ftǫ (q) ⊂ V ,

then⋃

t∈[0,τ ] Ftǫ (q) ⊂ M ; and if τ ≤ 0 and

⋃

t∈[τ,0] Ftǫ (q) ⊂ V , then

⋃

t∈[τ,0] Ftǫ (q) ⊂M . (Intuitively speaking, the orbit F tǫ (q) starting from

any point q in M , can leave M in forward or backward time; never-theless, it can only leave M through the boundary ∂M . )

21

Remark 2.5 Overflowing invariant submanifolds become inflowing invari-ant under time reversal and vice versa. M can be an invariant manifoldonly if F tǫ (q) ∈ ∂M for all q ∈ ∂M , or if ∂M = ∅.Theorem 2.3 (Persistent Invariant Manifold Theorem) For any in-tegers k and n (1 ≤ k, n < ∞), there exist a positive constant ǫ0 and aneighborhood U of the resonance circle Sω in Sk, such that inside U , for anyǫ ∈ (−ǫ0, ǫ0), there exist Cn locally invariant submanifolds W cuǫ and W csǫ ofcodimension 1 and W cǫ (= W

cuǫ ∩W csǫ ) of codimension 2 under the evolution

operator F tǫ given by (2.1). When ǫ = 0, Wcu0 , W

cs0 , and W

c0 are tangent

to the center-unstable, center-stable, and center subspaces⋃

~qγ∈Sω Ecu(~qγ)

and⋃

~qγ∈Sω Ecs(~qγ) of codimension 1, and

⋃

~qγ∈Sω Ec(~qγ) of codimension 2,

along the resonance circle Sω. Wcuǫ , W

csǫ , and W

cǫ are C

n smooth in ǫ forǫ ∈ (−ǫ0, ǫ0).Remark 2.6 W cuǫ , W

csǫ , and W

cǫ are called persistent center-unstable, center-

stable, and center submanifolds near Sω.

Theorem 2.4 (Fiber Theorem) Inside the persistent center-unstable sub-manifold W cuǫ near Sω, there exists a family of 1-dimensional C

n smoothsubmanifolds (curves) {~F (u,ǫ)(~q) : ~q ∈W cǫ }, called unstable fibers:

• W cuǫ can be represented as a union of these fibers,W cuǫ =

⋃

~q∈W cǫ

~F (u,ǫ)(~q).

• ~F (u,ǫ)(~q) depends Cn−1 smoothly on both ǫ and ~q for ǫ ∈ (−ǫ0, ǫ0) and~q ∈W cǫ , in the sense that W defined by

W ={

(~q1, ~q, ǫ)

∣

∣

∣

∣

~q1 ∈ ~F (u,ǫ)(~q), ~q ∈W cǫ , ǫ ∈ (−ǫ0, ǫ0)}

is a Cn−1 smooth submanifold of Sk × Sk × (−ǫ0, ǫ0).• Each fiber ~F (u,ǫ)(~q) intersects W cǫ transversally at ~q, two fibers ~F (u,ǫ)(~q1)

and ~F (u,ǫ)(~q2) are either disjoint or identical.

• The family of unstable fibers {~F (u,ǫ)(~q) : ~q ∈W cǫ } is negatively invari-ant, in the sense that the family of fibers commutes with the evolutionoperator F tǫ in the following way:

F tǫ (~F (u,ǫ)(~q)) ⊂ ~F (u,ǫ)(F tǫ (~q))

for all ~q ∈W cǫ and all t ≤ 0 such that⋃

τ∈[t,0] Fτǫ (~q) ⊂W cǫ .

22

• There are positive constants κ (e.g. = 2π√ω2 − π2) and C which are

independent of ǫ such that if ~q ∈W cǫ and ~q1 ∈ ~F (u,ǫ)(~q), then∥

∥

∥

∥

F tǫ (~q1) − F tǫ (~q)∥

∥

∥

∥

≤ Ceκt∥

∥

∥

∥

~q1 − ~q∥

∥

∥

∥

,

for all t ≤ 0 such that ⋃τ∈[t,0] F τǫ (~q) ⊂ W cǫ , where ‖ ‖ denotes Hknorm of periodic functions of period 1.

• For any ~q, ~p ∈W cǫ , ~q 6= ~p, any ~q1 ∈ ~F (u,ǫ)(~q) and any ~p1 ∈ ~F (u,ǫ)(~p); if

F tǫ (~q), Ftǫ (~p) ∈W cǫ , ∀t ∈ (−∞, 0],

and‖F tǫ (~p1) − F tǫ (~q)‖ → 0, as t→ −∞;

then{‖F tǫ (~q1) − F tǫ (~q)‖‖F tǫ (~p1) − F tǫ (~q)‖

}/

e1

2κt → 0, as t→ −∞.

Similarly for W csǫ .

2.4 Invariant Manifolds and Fibers for the Integrable NLSEquation

When ǫ = 0, the pde (2.1) becomes the integrable nonlinear Schrödingerequation

iqt = qxx + 2

[

|q|2 − ω2]

q, (2.4)

under even and periodic boundary condition

q(−x) = q(x), q(x+ 1) = q(x).

The detailed study on the hyperbolic structure for the integrable NLS equa-tion (2.4) is given in [63] [59]. Below we will give a brief account on thatstudy.

2.4.1 Integrable Preliminaries

The NLS equation (2.4) can be integrated with the Zakharov–Shabat linearsystem [100]:

23

ϕx = U(λ) ϕ , (2.5)

ϕt = V(λ) ϕ , (2.6)

where

U (λ) = iλσ3 + i

(

0 qq̄ 0

)

; (2.7)

V (λ) = 2 i λ2σ3 + i(ω2 − |q|2)σ3 +

0 2iλq + qx

2iλq̄ − q̄x 0

, (2.8)

where λ is the spectral parameter, σ3 denotes the third Pauli matrix σ3 =diag(1,−1). Compatibility condition of (2.5,2.6) gives the NLS equation(2.4). Focusing attention upon the spatial part (2.5) of the linear system(2.5,2.6), let M = M(x;λ; ~q) be the fundamental matrix solution of the ode(2.5); The Floquet discriminant:

∆ : C × Sk 7→ C

is defined by∆(λ; ~q) ≡ trace{M(1;λ; ~q)}.

For any λ ∈ C, ∆(λ; ~q) is a constant of motion for the integrable NLSequation (2.4). ∆(λ; ~q) is an entire function in both λ and ~q. This leads to acountable number of functionally independent constants of motion. Next, wedefine some spectral points. Periodic and antiperiodic points λs are definedby

∆(λs; ~q) = ±2.A critical point λc is defined by the condition

∂∆(λc; ~q)

∂λ= 0.

A multiple point λm is a critical point which is also a periodic or antiperiodicpoint. The algebraic multiplicity of λm is defined as the order of the zero

24

of ∆(λ) ± 2. Usually it is 2, but it can exceed 2; when it does equal 2, wecall the multiple point a double point, and denote it by λd. The geometricmultiplicity of λm is defined as the dimension of the periodic (or antiperiodic)eigenspace of (2.5) at λm, and is either 1 or 2.

The following counting lemmas describe the number and locations of thespectral points.

Lemma 2.1 (Counting Lemma for Critical Points) For q ∈ H1, setN = N(‖q‖H1) ∈ Z+ by

N(‖q‖H1) = 2[

‖q‖22 cosh(

‖q‖2)

+ 3‖q‖H1 sinh(

‖q‖2)]

,

where [x] ≡ first integer greater than x. Consider

∆′(λ; q) =∂

∂λ∆ (λ; q).

Then

• ∆′(λ; q) has exactly 2N + 1 zeros (counted according to multiplicity)in the interior of the disc D ≡ {λ ∈ C: |λ| < (2N + 1)π2 }.

• ∀k ∈ Z, |k| > N,∆′(λ, q) has exactly one zero in each disc{λ ∈ C: |λ− kπ| < π4 }.

• ∆′(λ; q) has no other zeros.

• For |λ| > (2N+1)π2 , the zeros of ∆′, {λcj , |j| > N}, are all real, simple,and satisfy the asymptotics

λcj = jπ + o(1) as |j| → ∞.

Lemma 2.2 (Counting Lemma for Periodic Points) For q ∈ H1 andN as in Counting Lemma 2.1, consider

∆±(λ; q) ≡ ∆(λ; q) ± 2.

Then

25

• ∆±(λ; q) has exactly 4N + 2 zeros (counted according to multiplicity)in the interior of the disc D ≡ {λ ∈ C: |λ| < (2N + 1)π2 }.

• ∀k ∈ Z, |k| > N,∆±(λ; q) has exactly two zeros in each disc{λ ∈ C: |λ− kπ| < π2 }.

• ∆±(λ; q) has no other zeros.

• For |λ| > (2N + 1)π2 , these zeros occur in conjugate pairs. If theycoincide, they must be real. Conversely, if real, they must be double.They satisfy the asymptotics

λj = jπ + o(1) as |j| → ∞.

For the proof of the above two lemmas, see [59].The important sequence of constants of motion Fj

Fj : Ω ⊂ Sk 7→ C

is defined byFj(~q) = ∆(λ

cj(~q); ~q).

Several properties of the functional Fj must be emphasized: First, for |j| >N (given in the counting lemmas), Fj is real and Fj ∈ [−2,+2]; second, forcomplex critical points |j| ≤ N , Fj = FRj + iF Ij is not necessarily real. Thefollowing theorem characterizes the critical structures of the functionals Fj .

Theorem 2.5 1. Except for the trivial case q = 0,

δFjδ~q

∣

∣

∣

∣

~q∗

= 0 ⇐⇒ λcj(~q∗) is a multiple point

with geometric multiplicity 2.

2. If λcj(~q∗) is a real double point, then the Hessian of Fj is positive (ornegative) semi-definite.

3. If λcj(~q∗) is a complex double point, then the Hessian of Fj is complex,both its real part and imaginary part are indefinite.

26

For the proof of the theorem, see [59]. Through a Morse type study basedupon the above theorem, hyperbolic structures in the level sets of Fj ’s canbe identified. More specifically, if λcj(~q∗) is a complex double point, thenthe critical point ~q∗ of Fj resides on a level set of Fj which is a normallyhyperbolic invariant submanifold. In the subsequent subsections, we willgive explicit expressions for such hyperbolic structures through Bäcklund-Darboux transformations.

2.4.2 Bäcklund-Darboux Transformations

Starting from a level set which is a normally hyperbolic invariant submani-fold, its hyperbolic foliations can be constructed through Bäcklund-Darbouxtransformations.

Let ~q(x, t) be a solution to the NLS equation (2.4), and let ν be a com-plex double point with geometric multiplicity 2. Denote two linearly inde-pendent solutions of the Zakharov-Shabat linear system (2.5,2.6) at λ = νby (~φ+, ~φ−). Thus, a general solution of the linear system at (~q, ν) is givenby

~φ(x, t; ν; c+, c−) = c+~φ+ + c−~φ

−. (2.9)

We use ~φ to define a transformation matrix G by

G = G(λ; ν; ~φ) ≡ N(

λ− ν 00 λ− ν̄

)

N−1, (2.10)

where

N ≡[

φ1 −φ̄2φ2 φ̄1

]

(2.11)

Then we define Q and ~Ψ by

Q(x, t) ≡ q(x, t) + 2(ν − ν̄) φ1φ̄2φ1φ̄1 + φ2φ̄2

(2.12)

and~Ψ(x, t;λ) ≡ G(λ; ν; ~φ) ~ψ(x, t;λ) (2.13)

where ~ψ solves the linear system (2.5,2.6) at (~q, λ). Formulas (2.12) and(2.13) are the Bäcklund-Darboux transformations for the potential and eigen-functions, respectively. We have the following theorem.

27

Theorem 2.6 Let ~q(x, t) be a solution to the NLS equation (2.4), and let νbe a complex double point with geometric multiplicity 2. Denote two linearlyindependent solutions of the Zakharov-Shabat linear system (2.5,2.6) at λ =ν by (~φ+, ~φ−), and define Q(x, t) and ~Ψ(x, t;λ) by (2.12) and (2.13). Then

(i) Q(x, t) is also a solution to the NLS equation (2.4), and ~Ψ(x, t;λ)solves the Zakharov-Shabat linear system (2.5,2.6) at (Q(x, t), λ).

(ii) The spectra of (2.5) at Q is the same with the spectra at q.

2.4.3 Explicit Representations for Invariant Manifolds and Fibers

Consider the lowest dimensional level set: The circle Sc = {~q ∈ Π | |q| =c, c ∈ (π, 2π)}. Motion on this level set is a periodic orbit,

qγ = c exp{−i[2(c2 − ω2)t+ γ]}. (2.14)

Applying the Bäcklund-Darboux transformation (2.12,2.13) to the solution(2.14), we have the expression for the two-dimensional level set W s(Sc) =W u(Sc) (called stable and unstable manifolds of the circle Sc):

W s(Sc) = Wu(Sc) =

⋃

ρ∈(−∞,∞),γ∈[0,2π],σ=±QH(x, 0; c, γ, ρ, σ), (2.15)

where

QH = QH(x, t; c, γ, ρ,±)

= qγ

[

1 + cos 2p− i sin 2p tanh τ1 ± sin p sech τ cos(2πx) − 1

]

, (2.16)

where p = arctan

[√c2−π2

π

]

, c ∈ (π, 2π), τ = 4|ν|πt − ρ (π + ν = ceip), ρ iscalled the Bäcklund parameter. Notice that as τ → ±∞,

QH → qγe∓i2p. (2.17)

That is, the orbit QH is asymptotic to qγ up to phase shifts in both forwardand backward time. For an illustration of W s(Sc) = W

u(Sc), see Fig.2.1.QH coordinatizes W

s(Sc) = Wu(Sc) in the following way: γ coordinatizes

the circle Sc, ± selects one of the two symmetric “whiskers”, and ρ coordi-natizes the “whiskers”. From the expression (2.16,2.17), we can read off thefollowing expressions for stable and unstable fibers:

28

Figure 2.1: An illustration of the invariant submanifold W s(Sc) = Wu(Sc).

1. In W s(Sc), the stable fiber ~Fs(~q) with the base point q = ce−iγ , has

the expression

~F s(~q) = qei2p[

1 + cos 2p − i sin 2p tanh ξ1 ± sin p sech ξ cos(2πx) − 1

]

,

which is a curve parameterized by ξ, ξ ∈ (−∞,∞); as ξ → ∞, ~F s(~q) →q.

2. In W u(Sc), the unstable fiber ~Fu(~q) with the base point q = ce−iγ ,

has the expression

~F u(~q) = qe−i2p[

1 + cos 2p− i sin 2p tanh ξ1 ± sin p sech ξ cos(2πx) − 1

]

,

which is a curve parameterized by ξ, ξ ∈ (−∞,∞); as ξ → −∞,~F u(~q) → q.

For an geometric illustration of the fibers, see Fig.2.2. Fibers ~F s(~q) and~F u(~q) in contrast to orbits QH , do not spiral around the circle Sc; therefore,they provide better coordinates for W s(Sc) = W

u(Sc):

W s(Sc) =⋃

~q∈Sc

~F s(~q), W u(Sc) =⋃

~q∈Sc

~F u(~q).

29

Figure 2.2: An illustration of the fibers in W s(Sc) = Wu(Sc).

2.5 Coordinates Centered on the Resonance Circle

In this section we center the equations in a manner which is natural tostudy behavior near the invariant plane Π. This centering notation takessome time to set up.

2.5.1 Definition of the H̃ Norms

It will be useful to use the basis {e±j } to define a norm on Sk which isequivalent to the Hk norm: Set

~q =∞∑

j=0

{α+j e+j + α−j e−j },

then define

‖~q‖2H̃1

≡∞∑

j=0

{(α+j )2 + (α−j )2} +∞∑

j=0

k2j{(α+j )2 + (α−j )2},

with a similar definition for the H̃k norms, k > 1.The H̃k is equivalent to the Sobolev Hk norm; the advantage of the

H̃k norm is that its temporal evolution under the linearization (2.2) of theintegrable flow is easy to control.

From now on ‖~q‖ stands for ‖~q‖H̃k .

30

2.5.2 A Neighborhood of the Circle of Fixed Points

First, in order to study dynamics near the circle of fixed points we write

q =

[

ρ(t) + δf(x, t)

]

exp iθ(t), 0 < δ

2.5.3 An Enlarged Phase Space

In this setting we have two small parameters – an amplitude parameter δand a perturbation parameter ǫ. While both are small, we will treat thetwo very differently. First, the perturbation parameter ǫ will be treated asa variable by enlarging the system. In terms of the original variables q, thisenlargement yields the enlarged system (EPNLS):

iqt = qxx + 2[|q|2 − ω2]q + iǫ{−αq + D̂2q + Γ},ǫt = 0. (2.19)

The corresponding enlarged function space becomes Ŝk:

Ŝk = Sk × (−ǫ0, ǫ0). (2.20)

We can define a family of invariant planes parametrized by ǫ, Π̂ǫ:

Π̂ǫ ≡ {(~q, ǫ) | ~q ∈ Sk,d

dx~q ≡ 0.}

It is easily seen that Π̂ǫ is invariant under the EPNLS flow (2.19). Restrictedto the invariant planes Π̂ǫ, the (2.19) flow becomes:

iqt = 2[

|q|2 − ω2]

q + iǫ [−αq + Γ] ,ǫt = 0.

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

Ŝω ≡ {(~q, 0) | (~q, 0) ∈ Π̂0, |q| = ω.}

It is easily seen that Ŝω is a set of equilibria of (2.19).

2.5.4 Scales through δ

Next, with the amplitude parameter δ we introduce three scales: O(δ0),O(δ), and O(δ2). Along the resonance circle, the scale will be O(δ0); inthe normal direction of Sω in Sk, the H̃k scale will be O(δ); and alongthe ǫ direction, the scale will be O(δ2). Explicitly, we have the followingrepresentation:

q = (ω + δr)eiθ + δfeiθ,

ǫ = δ2ǫ1, (2.21)

< f > = 0.

32

The coordinate transformation (2.21) maps Ŝk into a new phase space S̃k:

S̃k ≡{

(r, θ, ~f , ǫ1)

∣

∣

∣

∣

(~q, ǫ) given by (2.21) belongs to Ŝk}

coordinatized by

Q ≡(

r, θ, ~f , ǫ1

)

,

where ~f ≡ (f,−f̄)T , ~q ≡ (q,−q̄)T . We use ‖Q‖ to denote the norm inheritedfrom the norm (‖~q‖, |ǫ|) through the transformation (2.21).

Definition 2 Define the bounded region D̃k in S̃k as follows:

D̃k ≡{

(r, θ, ~f , ǫ1) ∈ S̃k∣

∣

∣

∣

(

r2 + ‖~f‖2H̃k

)1/2

< 1, |ǫ1| < 1, θ ∈ [0, 2π].}

Denote by D̂k,δ the inverse image of D̃k in Ŝk under the coordinate trans-formation (2.21).

Remark 2.7 D̃k is a bounded convex subset of S̃k; moreover, it is indepen-dent of δ. On the other hand, D̂k,δ depends on δ. For a fixed value of δ,

D̂k,δ is a solid torus.

2.5.5 The Equations in Their Final Setting

Substituting representation (2.21) into the enlarged perturbed NLS system(2.19) yields

rt = δ

[

− ǫ1(

α(ω + δr) − Γ cos θ)

− i(

〈f2 − f̄2〉)

(ω + δr)

]

− iδ2[

f f̄(f − f̄)〉]

θt = −2δr(2ω + δr)

− δ2[

ǫ1Γ sin θ

(ω + δr)+ 〈f2 + f̄2 + 4f f̄〉

]

− δ3

(ω + δr)

[

〈f f̄(f + f̄)〉]

ift = fxx + 2(ω + δr)2(f + f̄)

33

+ 2δ

[

(ω + δr)

(

2(f f̄ − 〈f f̄〉) + (f2 − 〈f2〉))]

+ δ2[

2(

f f̄f − 〈f f̄f〉)−(

〈f2 + f̄2 + 4f f̄〉)

f

− ǫ1(

Γ sin θ

(ω + δr)f + i(αf − D̂2f)

)]

− δ3

(ω + δr)

[

〈f f̄(f + f̄)〉f]

ǫ1t = 0. (2.22)

The integrable linearized flow may be obtained by setting δ = 0:

rt = 0,

θt = 0,

ift = fxx + 2ω2(f + f̄), (2.23)

ǫ1t = 0.

Remark 2.8 In the equation (2.23), the f equation is the linearized equa-tion (2.2) discussed in the last subsection, but the dynamics on Π is totallydifferent from that in the last subsection, as a result of the new scaling. Infact, this scaling “slows down” motion on the annulus Π ∩ D̂k,δ to createan annulus of fixed points. Here, we want to emphasize that, from now on,we will view systems (2.22;2.23) as flows defined on D̃k. Moreover, system(2.22) is a perturbation of system (2.23) with δ as the perturbation param-eter.

Denote respectively by F̃ t and F̃ tδ the solution operators of systems (2.23)and (2.22) in S̃k.

Proposition 2.2 For any t ∈ (−∞,∞), there exist constants C = C(t)and d1 = d1(t), such that,

∥

∥

∥

∥

F̃ tδ (Q) − F̃ t(Q)∥

∥

∥

∥

≤ Cδ,∥

∥

∥

∥

∇F̃ tδ (Q) −∇F̃ t(Q)∥

∥

∥

∥

≤ Cδ,

for all Q ∈ D̃k, and all δ ∈ [0, d1];

where ∇ denotes the gradient.

34

Proof: Notice that F tǫ denotes the solution operator of the equation (2.1).By theorems (2.1;2.2),

q(t) ≡ (ω + δr(t))eiθ(t) + δf(x, t)eiθ(t)

= F tǫ=0(ωeiθ(0)) +

∂

∂qF tǫ=0(ωe

iθ(0)) • ∆q(0)

+∂

∂ǫF tǫ=0(ωe

iθ(0)) · δ2ǫ1 +R, (2.24)

where

∆q(0) = δ

[

r(0)eiθ(0) + f(x, 0)eiθ(0)]

,

R = R(t;ωeiθ(0);∆q(0); δ2ǫ1);

moreover, there exists a constant C1 = C1(t;ωeiθ(0)), such that,

‖R‖ ≤ C1‖∆q(0)‖2.

Notice that,F tǫ=0(ωe

iθ(0)) = ωeiθ(0).

Let∂

∂qF tǫ=0(ωe

iθ(0)) • ∆q(0) ≡ δeiθ(0)q1(t),

then

~q1(t) = r(0)e−0 − 4tω2r(0)e+0

+ α+1 eΩ+

1te+1 + α

−1 e

Ω−1

te−1

+∞∑

j=2

(e+j , e−j )

(

cos Ωjt − sin ΩjtsinΩjt cos Ωjt

)(

α+jα−j

)

,

where

f(x, 0) =∞∑

j=1

(α+j e+j + α

−j e

−j ).

Therefore, to order O(δ), Eq.(2.24) becomes

(ω + δr(t))eiθ(t) + δf(x, t)eiθ(t) = ωeiθ(0) + δeiθ(0)q1(t). (2.25)

Take spatial mean on both sides of (2.25), we have

(ω + δr(t))eiθ(t) = ωeiθ(0) + δeiθ(0)(r(0) − i4tω2r(0)).

35

That is,

ω + δr(t) =√

ω2 + 2ωδr(0) + δ2(r(0))2(1 + 16t2ω4),

θ(t) = θ(0) + θ1(t);

whereθ1(t) = arg{ω + δr(0) − i4δtω2r(0)}.

Thus

r(t) = r(0) + rδ(t),

θ(t) = θ(0) + θδ(t);

where there exists a constant C2 = C2(t), such that

|rδ(t)| ≤ C2δ, |θδ(t)| ≤ C2δ.

The rest (mean zero) part of (2.25) leads to

f(x, t)eiθ(t) = eiθ(0)q2(t), (2.26)

where~q2(t) = ~q1(t) − 〈~q1(t)〉.

Eq.(2.26) can be rewritten as

f(x, t) = F̃ t(f(x, 0)) + fδ(x, t),

where there exists a constant C3 = C3(t), such that,

‖fδ(x, t)‖ ≤ C3δ.

Go back to Eq.(2.24), we see that, there exists a constant C = C(t), suchthat, for all

Q ≡(

r(0), θ(0), ~f (x, 0), ǫ1

)

∈ D̃k,∥

∥

∥

∥

F̃ tδ (Q) − F̃ t(Q)∥

∥

∥

∥

≤ Cδ.

Similarly for ∇F̃ tδ and ∇F̃ t. ♣

36

Corollary 1 For any t ∈ (−∞,∞), there exist constants Λ1 = Λ1(t) andd2 = d2(t), such that,

Λ−11 ≤∥

∥

∥

∥

F̃ tδ (Q)

∥

∥

∥

∥

≤ Λ1,

Λ−11 ≤∥

∥

∥

∥

F̃ t(Q)

∥

∥

∥

∥

≤ Λ1,

Λ−11 ≤∥

∥

∥

∥

∇F̃ tδ (Q)∥

∥

∥

∥

≤ Λ1,

Λ−11 ≤∥

∥

∥

∥

∇F̃ t(Q)∥

∥

∥

∥

≤ Λ1,

for all Q ∈ D̃k, and all δ ∈ [0, d2]

Proof: The corresponding inequalities for F̃ t and ∇F̃ t are obviously truefrom the calculation in previous sections. Then by the above proposition,the corresponding inequalities for F̃ tδ and ∇F̃ tδ are also true. ♣

2.6 (δ = 0) Invariant Manifolds and the Introduction of aBump Function

In this section, we consider the unperturbed (δ = 0) flow (2.23) and certainof its invariant manifolds. Then we introduce a bump function necessary forlater studies.

2.6.1 (δ = 0) Invariant Manifolds

Consider the unperturbed (δ = 0) flow (2.23):

rt = 0,

θt = 0,

ift = fxx + 2ω2(f + f̄), (2.27)

ǫ1t = 0,

and make a decomposition of f :

f = f ′ + f ′′ (2.28)

f ′ ≡ a1 cos k1x, (2.29)

37

f ′′ ≡∞∑

j=2

aj cos kjx, (2.30)

which, in vector notation takes the form

~f ′ = α+1 e+1 + α

−1 e

−1 . (2.31)

Equation(2.27) for f becomes

if ′t = f′xx + 2ω

2(f ′ + f′), (2.32)

if ′′t = f′′xx + 2ω

2(f ′′ + f′′). (2.33)

Next in D̃k we will define the center, center-stable, and center-unstablemanifolds (subspaces) W0, W

cs0 , and W

cu0 for the equilibrium set in D̃k

defined by

S̃ω ≡{

(r, θ, ~f , ǫ1) ∈ D̃k∣

∣

∣

∣

~f = 0, r = 0, ǫ1 = 0

}

of the unperturbed system (2.27). In fact, S̃ω is the image of Ŝω under thecoordinate transformation (2.21).

Definition 3

W cs0 ≡{(

r, θ, ~f ′, ~f ′′, ǫ1

)

∈ D̃k∣

∣

∣

∣

α+1 = 0

}

,

W cu0 ≡{(

r, θ, ~f ′, ~f ′′, ǫ1

)

∈ D̃k∣

∣

∣

∣

α−1 = 0}

,

W0 ≡ W cs0 ∩W cu0 ={(

r, θ, ~f ′, ~f ′′, ǫ1

)

∈ D̃k∣

∣

∣

∣

~f ′ = 0}

.

From now on, we will concentrate our discussion on W cu0 . The discussionon W cs0 goes similarly. For convenience of notation, we write M for W

cu0 :

M ≡W cu0 .

38

2.6.2 Tangent and Transversal Bundles of M

Next, we consider the tangent and transversal bundles of M(≡ W cu0 ), de-noted respectively by TM and N . Since the linear flow ∇F̃ t(m) is indepen-dent of m ∈ M , these bundles are trivial. Moreover, since ∇F̃ t(m) = F̃ t,we have the simple representations

TM(m) = span{ê+1 ; ê±j , j = 0, 2, 3, ...; êǫ}N(m) = span{ê−1 },

where

ê+0 ≡ (1, 0,~0, 0) ∈ S̃k,ê−0 ≡ (0, 1,~0, 0) ∈ S̃k,êǫ ≡ (0, 0,~0, 1) ∈ S̃k,ê±j ≡ (0, 0, e±j , 0) ∈ S̃k,

j = 1, 2, ....

In addition, the temporal flows can be explicitly computed: Let w0 ∈N(m), w0 = α

−1 ê

−1 ; then

w−t ≡ ∇F̃−t(m)w0 = F̃−tw0 = e−Ω−

1tw0 = e

Ω+1

tw0. (2.34)

Let v0 ∈ TM(m),v0 = α

+1 ê

+1 +

∑

j 6=1{α+j ê+j + α−j ê−j } + ǫ1êǫ,

then,

v−t ≡ F̃−tv0 = α+1 e−Ω+

1tê+1 + α

+0 ê

+0 + α

−0 ê

−0 +

+∞∑

j=2

(

ê+j ê−j

)

(

cos Ωjt − sin Ωjtsin Ωjt cos Ωjt

)(

α+jα−j

)

+

+ǫ1êǫ. (2.35)

From these explicit formulas and definition (3), one sees that the manifoldM is overflowing invariant for the flow (2.27), but that it is not strictly over-flowing invariant. At some points on the boundary ∂M , the unperturbedvector field is tangent to the boundary, while at other points it is pointingstrictly outward. The points “of tangency” are too sensitive to perturba-tions. As is standard, we “bump the vector field” near the boundary of Mto make M strictly overflowing (or inflowing) invariant.

39

Figure 2.3: The bump function

2.6.3 Introduction of a Bump Function

To make the manifold M strictly overflowing (or inflowing) invariant, wemodify the flows defined by equations (2.23) and (2.22) in some neighbor-hood of the boundary of M . We emphasize that the modified flows will beidentical with the original flows in the unmodified region of M .

Choose real numbers a1, a2, a3, a4, and a5, satisfying

0 < a5 < a4 < a3 < a2 < a1 = 1,

and choose a C∞ “bump” function ψ : [0, 1] 7→ R, such that

ψ(a) = 0, for a ∈ [0, a4],ψ(a3) > 0,

ψ(a2) < 0,

ψ(a1) > 0.

See Fig.2.3. Let Rc ≡ (r2 + ‖~f ′′‖2H̃k)1/2. Then we modify the systems (2.23)

and (2.22) into the following systems:

rt = δ1ψ(Rc)r,

40

θt = 0,

if ′t = f′xx + 2ω

2(f ′ + f′), (2.36)

if ′′t = f′′xx + 2ω

2(f ′′ + f′′) + iδ1ψ(Rc)f

′′

ǫ1t = 0.

rt = δ1ψ(Rc)r + δGr,

θt = δGθ,

if ′t = f′xx + 2ω

2(f ′ + f′) + δGf ′ , (2.37)

if ′′t = f′′xx + 2ω

2(f ′′ + f′′) + iδ1ψ(Rc)f

′′ + δGf ′′ ,

ǫ1t = 0.

If we drop the two terms

δ1ψ(Rc)r and iδ1ψ(Rc)f′′

in the above systems (2.36;2.37), then they are identical to the systems(2.23;2.22). Thus,

Gr, Gθ, Gf ′ , and Gf ′′

represent the corresponding parts of the right hand side of (2.22). Here δ1is a small positive number. In fact, these are the final unperturbed andperturbed systems that we are going to study.

Remark 2.9 We are not forced to modify the ǫ1 equation with a “bumpfunction” because the ǫ1 equation is trivial; the later argument can go throughwithout bumping the ǫ1 equation.

Denote the solution operators of (2.36) and (2.37) by

F tδ1 and Ftδ,δ1,

respectively. Following similar argument as for theorems (2.1; 2.2), propo-sition (2.2), and corollary (1), we have the following claims on systems(2.36;2.37):

Theorem 2.7 (Cauchy Problem) ∀Q0 ∈ S̃k and ∀t ∈ (−∞,∞), ∃ aunique solution, Q(t, ·) ∈ S̃k, to (2.36) and (2.37), respectively; such thatQ |t=t0= Q0; moreover, Q(t, ·) depends continuously upon t.

41

Theorem 2.8 (Dependence on Data) For any fixed t ∈ (−∞,∞), anyfixed n (n < ∞), and any fixed k (k < ∞), both F tδ1 and F tδ,δ1 are Cndiffeomorphisms in S̃k.

Theorem 2.9 For any t ∈ (−∞,∞), there exist constants C1 = C1(t) andd3 = d3(t), such that,

∥

∥

∥

∥

F tδ1(Q) − F̃ t(Q)∥

∥

∥

∥

≤ C1δ1,∥

∥

∥

∥

F tδ,δ1(Q) − F̃ tδ (Q)∥

∥

∥

∥

≤ C1δ1,∥

∥

∥

∥

∇F tδ1(Q) −∇F̃ t(Q)∥

∥

∥

∥

≤ C1δ1,∥

∥

∥

∥

∇F tδ,δ1(Q) −∇F̃ tδ (Q)∥

∥

∥

∥

≤ C1δ1,

for all Q ∈ D̃k, all δ1 ∈ [0, d3],and all δ ∈ [0, d3].

Theorem 2.10 For any t ∈ (−∞,∞), there exist constants C2 = C2(t)and d4 = d4(t), such that,

∥

∥

∥

∥

F tδ,δ1(Q) − F tδ1(Q)∥

∥

∥

∥

≤ C2δ,∥

∥

∥

∥

∇F tδ,δ1(Q) −∇F tδ1(Q)∥

∥

∥

∥

≤ C2δ,

for all Q ∈ D̃k, all δ ∈ [0, d4],and all δ1 ∈ [0, d4].

Corollary 2 For any t ∈ (−∞,∞), there exist constants Λ = Λ(t) andd5 = d5(t), such that,

Λ−1 ≤∥

∥

∥

∥

F tδ1(Q)

∥

∥

∥

∥

≤ Λ,

Λ−1 ≤∥

∥

∥

∥

∇F tδ1(Q)∥

∥

∥

∥

≤ Λ,

42

Λ−1 ≤∥

∥

∥

∥

F tδ,δ1(Q)

∥

∥

∥

∥

≤ Λ,

Λ−1 ≤∥

∥

∥

∥

∇F tδ,δ1(Q)∥

∥

∥

∥

≤ Λ,

for all Q ∈ D̃k, all δ1 ∈ [0, d5],and all δ ∈ [0, d5].

Remark 2.10 For any fixed δ1 ∈ (0, d4], when δ is sufficiently small, sys-tem (2.37) is a C1 perturbation of system (2.36), in the sense described inTheorem (2.10), in D̃k; moreover, M is strictly overflowing invariant under(2.36).

Theorem 2.11 For any t ∈ (−∞,∞), there exist constants Λ∗ = Λ∗(t) andd∗ = d∗(t), such that

∥

∥

∥

∥

∇2F tδ,δ1(Q)∥

∥

∥

∥

≤ Λ∗,∥

∥

∥

∥

∇3F tδ,δ1(Q)∥

∥

∥

∥

≤ Λ∗;

for all Q ∈ D̃k, all δ1 ∈ [0, d∗], and all δ ∈ [0, d∗].

Proof: By theorem (2.8), F tδ,δ1 is a Cn diffeomorphism. In the original

coordinate (q, ǫ) (2.21), for any fixed t ∈ (−∞,∞),∥

∥

∥

∥

∇2F tδ,δ1(q = ωeiθ, ǫ = 0)∥

∥

∥

∥

≤ 12Λ∗.

Since ‖∇2F tδ,δ1(q, ǫ)‖ is a continuous function of (q, ǫ), then when δ is suffi-ciently small,

∥

∥

∥

∥

∇2F tδ,δ1(q = (ω + δr)eiθ + δfeiθ, ǫ = δ2ǫ1)∥

∥

∥

∥

≤ Λ∗,

which is equivalent to the claim of the theorem. ♣From now on, for simplicity of notations, we abbreviate

F tδ1 and Ftδ,δ1

asF t0 and F

tδ

respectively.

43

3 Persistent Invariant Manifolds

In this section, we establish the existence of certain infinite dimensional(codimension 1 and 2) invariant manifolds for the enlarged, perturbed sys-tem (EPNLS) (2.19), using a dichotomy of time scales; i.e., using normalhyperbolicity.

3.1 Statement of the Persistence Theorem and the Strategyof Proof

The following definitions describe the nature of the invariant manifoldswhich we will show to persist.

Definition 4 (Locally Invariant Manifold) Let V be a submanifold inS̃k with boundary ∂V , F t be a flow defined in S̃k. If V has the followingproperty: Whenever there exist a point Q ∈ V , and time t, such that

⋃

τ∈[0,t)F τ (Q) ⊂ V, F t(Q) 6∈ V ; (t > 0)

or⋃

τ∈(t,0]F τ (Q) ⊂ V, F t(Q) 6∈ V ; (t < 0)

thenF t(Q) ∈ ∂V.

Then, we call V a locally invariant submanifold under F t.

Notice that S̃ω is an equilibrium manifold for systems (2.36; 2.37;2.23;2.22),we define its center-unstable, center-stable, and center manifolds as follows:

Definition 5 (Invariant Manifolds of S̃ω) We call Wcuδ a center-unstable

manifold of S̃ω under the flow F̃tδ (2.22), if:

1. S̃ω ⊂W cuδ ;

2. W cuδ is locally invariant under the flow F̃tδ (2.22);

3. As δ → 0, W cuδ coincides with W cu0 .Its center-stable manifold W csδ is defined similarly. Its center manifold Wδis defined as the intersection

Wδ ≡W cuδ⋂

W csδ .

44

Denote the center-unstable, center-stable, and center manifolds of S̃ω underthe bumped flow (2.37) by

W cuδ1,δ, Wcsδ1,δ, and Wδ1,δ

respectively. Under the coordinate transformation (2.21), for any fixed δ,W cuδ , W

csδ , and Wδ are transformed into center-unstable, center-stable, and

center manifolds of Ŝω in D̂k,δ under the flow (2.19). Denote them respec-tively by

W cu, W cs, and W.

For any fixed ǫ, the center-unstable, center-stable, and center manifolds ofSω in Sk under (2.1) are defined by restriction of W cu, W cs, and W to thefixed value of ǫ,

W cuǫ ≡W cu |ǫ, W csǫ ≡W cs |ǫ, Wǫ ≡W |ǫ .

Definition 6 Define the following subregions of D̃k:

D̃(j)k ≡

{

(r, θ, ~f , ǫ1) ∈ D̃k∣

∣

∣

∣

(r2 + ‖~f ′′‖2H̃k

)1/2 < aj

}

(j = 1, ..., 5)

where aj ’s are given in the definition of the bump function ψ.

Proposition 3.1 There exists a positive constant d̃1, for any δ1 ∈ (0, d̃1],there exists a positive constant d̃ = d̃(δ1), such that, for any δ ∈ [0, d̃], thereexist a Cn codimension 1 center-unstable, a Cn codimension 1 center-stable,and a Cn codimension 2 center manifolds

W cuδ1,δ, Wcsδ1,δ, and Wδ1,δ

of S̃ω in D̃(2)k under the bumped perturbed flow (2.37).

Theorem 3.1 There exist a Cn codimension 1 center-unstable, a Cn codi-mension 1 center-stable, and a Cn codimension 2 center manifolds,

W cuδ ≡ D̃(5)k⋂

W cuδ1,δ,

W csδ ≡ D̃(5)k

⋂

W csδ1,δ,

Wδ ≡ D̃(5)k⋂

Wδ1,δ,

45

of S̃ω in D̃(5)k under the perturbed flow (2.22), where

δ ∈ [0, d̃], d̃ = d̃(δ1), δ1 ∈ (0, d̃1]

are given in proposition (3.1).

Denote by D̂(j)k,δ (j = 1, ..., 5) the image of D̃

(j)k (j = 1, ..., 5) in Ŝk under the

coordinate transformation (2.21). Let

δ∗ ≡ d̃(δ1 = d̃1),

then D̂(5)k,δ∗

is a fixed solid torus neighborhood of Ŝω in Ŝk.

Theorem 3.2 In the solid torus neighborhood D̂(5)k,δ∗

of Ŝω in Ŝk, there exista Cn codimension 1 center-unstable, a Cn codimension 1 center-stable, anda Cn codimension 2 center manifolds,

W cu, W cs, and W

of Ŝω under the perturbed flow (2.19).

Theorem 3.3 For any ǫ ∈ (−δ2∗ , δ2∗), there exist a Cn codimension 1 center-unstable, a Cn codimension 1 center-stable, and a Cn codimension 2 centermanifolds,

W cuǫ , Wcsǫ , and Wǫ

of Sω under the perturbed flow (2.1), inside the solid torus neighborhood ofSω,

D̂(5)k,δ∗

|ǫ⊂ Sk.

Moreover, W cuǫ , Wcsǫ , and Wǫ are C

n in ǫ for ǫ ∈ (−δ2∗ , δ2∗).

Remark 3.1 Theorems (3.1;3.2;3.3) are corollaries of proposition (3.1).

Inside the interior region D̃(5)k of D̃k, the bump function ψ is identically equal

to zero; therefore, the bumped flow (2.37) coincides with the unbumped flow

(2.22) in D̃(5)k . Thus theorem (3.1) follows immediately from proposition

(3.1). If we fix the value of δ (= δ∗) in theorem (3.1), and restate thistheorem in Ŝk, we get theorem (3.2). Theorem (3.3) follows immediatelyfrom theorem (3.2) through restriction.

The strategy of proof in next subsection can be summarized as follows:

46

1. Define graph transform on the section space of a transversal bundleover the unperturbed overflowing invariant manifold. Based upon cer-tain rate inequalities, a contraction map argument is carried on thegraph transform. The fixed point of the graph transform gives thepersistent overflowing invariant manifold.

2. Prove the Cn smoothness of the persistent invariant manifold: Firstformally differentiate the graph transform equation that the persistentinvariant manifold satisfies; then, a contraction map argument leadsto a solution; finally, by the definition of Frechet derivative and aweighted inequality argument, the C1 smoothness is proved. Similarlyfor Cn smoothness. The approach for proving smoothness here isused everywhere in this book, e.g. in proving smoothness of fibersand smoothness of fibers with respect to their base points in latersubsections.

Next subsection is devoted to the proof of proposition (3.1).As remarked above, we only need to prove proposition (3.1). Moreover,

we will concentrate on proving the existence of W cuδ1,δ in D̃(1)k (≡ D̃k), the

existence of W csδ1,δ in D̃(2)k follows similarly through reversing the time (t →

−t).

3.2 Definition of the Graph Transform

Fix the manifold M(≡W cu0 ). It is overflowing invariant under the “bumpedlinear flow” (2.36). We have defined the transversal bundle N in section4.2. Let Nκ be the subset of N such that Nκ consists of pairs (Q,w) with‖w‖ ≤ κ; i.e.,

Nκ ≡ {(Q,w) | Q ∈M, (Q,w) ∈ N, ‖w‖ ≤ κ.}

(Here, and throughout this proof, ‖·‖ will always mean ‖·‖H̃k .) Alternatively,we can view the transversal bundle as it is embedded in S̃k:

em : Nκ 7→ S̃kem(Q,w) = Q+ w.

In either case, the “transversal fiber” w can be realized explicitly throughthe basis element ê−1 :

w = {αê−1 , α ∈ R},

47

where ê−1 is defined in section 4.2. This representation emphasizes that thefibers of the transversal bundle N are one dimensional.

There exists a κ0, such that for 0 < κ < κ0, emmapsNκ C∞-diffeomorphically

onto a neighborhood of M in S̃k. Let Σ be the space of sections of Nκ |M :

Σ = {σ | σ : M 7→ Nκ, σ(Q) = (Q,wσ(Q))},

where, for each σ, the function wσ(Q) can be explicitly realized in terms ofthe basis element ê−1 :

σ(Q) =

(

Q,α{σ}(Q)ê−1

)

, α{σ} : M → R.

Next, we define a Lipschitz semi-norm on Σ:

Lip{σ} ≡ supQ,Q′∈M̄

‖wσ(Q) − wσ(Q′)‖‖Q−Q′‖ .

Let Σζ be a subset of Σ defined by

Σζ ≡ {σ ∈ Σ | Lip{σ} ≤ ζ}. (3.1)

Next, let M1 ≡ F 10 (M). By changing time scale, we can assume that M1 isstill overflowing invariant. For details see the following remark.

Remark 3.2 F t0(M) will be overflowing invariant for sufficiently small t,t < t0 for example. In this case, the time t0 = t0(δ1). Recall that withrespect to δ, δ1 is O(1). Rescale t so that t0 is scaled into 1, then F

10 (M) is

overflowing invariant, and δ is still a small parameter. So without loss ofgenerality, we can assume M1 is overflowing invariant.

We will need the following lemma in the proof of the next two propositions[24].

Lemma 3.1 (Mean Value Theorem) Let E1, E2 be two Banach spaces,F be a continuous mapping from a neighborhood of a segment l joining twopoints q0, q0 + q1 of E1, into E2. If F is differentiable at every point of l,then

∥

∥

∥

∥

F (q0 + q1) − F (q0)∥

∥

∥

∥

≤ ‖q1‖ · sup0≤α≤1

∥

∥

∥

∥

∇F (q0 + αq1)∥

∥

∥

∥

.

Denote F tδ • em • σ simply by F tδσ. The following lemma is an importantprerequisite for the definition of the graph transform.

48

Lemma 3.2 Let b : Nκ 7→ M be the base projection; i.e., b(Q,w) = Q.For any finite T > 0, define Φ(Q) ≡ bF Tδ σF−T0 (Q). There exists a positiveconstant d6 = d6(T ), such that, when δ ∈ [0, d6], for all σ ∈ Σζ (defined inNκ):

1. Φ(Q) is defined for all Q ∈M1,

2. M̄ ⊂ Φ(M1),

3. Each point in M is the Φ image of only one point in M1.

Proof: When δ = 0, by the discussion in sections 4.2 and 4.3,

Φ |δ=0≡ bF T0 σF−T0 = identity map.

Then,

Φ(Q) = bF T0 σF−T0 (Q) + (bF

Tδ − bF T0 )σF−T0 (Q)

≡ Q+ Φ1(Q).

By lemma (3.1),

∥

∥

∥

∥

Φ1(Q1) − Φ1(Q2)∥

∥

∥

∥

≤∥

∥

∥

∥

σF−T0 (Q1) − σF−T0 (Q2)∥

∥

∥

∥

·

· sup0≤α≤1

∥

∥

∥

∥

(

∇bF Tδ −∇bF T0)

(ασF−T0 (Q1) + (1 − α)σF−T0 (Q2))∥

∥

∥

∥

.

Then, by theorem (2.10), when δ ∈ [0, d4],∥

∥

∥

∥

Φ1(Q1) − Φ1(Q2)∥

∥

∥

∥

≤ C2(T )δ∥

∥

∥

∥

σF−T0 (Q1) − σF−T0 (Q2)∥

∥

∥

∥

.

By lemma (3.1) again,

∥

∥

∥

∥

σF−T0 (Q1) − σF−T0 (Q2)∥

∥

∥

∥

≤∥

∥

∥

∥

bσF−T0 (Q1) − bσF−T0 (Q2)∥

∥

∥

∥

+

∥

∥

∥

∥

hσF−T0 (Q1) − hσF−T0 (Q2)∥

∥

∥

∥

≤ (1 + ζ)∥

∥

∥

∥

F−T0 (Q1) − F−T0 (Q2)∥

∥

∥

∥

≤ (1 + ζ)‖Q1 −Q2‖ sup0≤α≤1

∥

∥

∥

∥

∇F−T0 (αQ1 + (1 − α)Q2)∥

∥

∥

∥

,

49

where h is the projection on fibers; i.e., h(Q,w) = w. By corollary (2), whenδ ∈ [0, d5],

sup0≤α≤1

∥

∥

∥

∥

∇F−T0 (αQ1 + (1 − α)Q2)∥

∥

∥

∥

≤ Λ.

Finally, we have∥

∥

∥

∥

Φ1(Q1) − Φ1(Q2)∥

∥

∥

∥

≤ δC2(1 + ζ)Λ‖Q1 −Q2‖.

Thus, when δ is small enough, Φ is a perturbation of the identity map inthe Lipschitz topology for map. Then the lemma follows immediately [31][24]. ♣

Next, we define the Graph Transform,

G : Σζ 7→ ΣGσ(bF Tδ (σ(Q))) ≡ (bF Tδ (σ(Q)), hF Tδ (σ(Q))), (3.2)

where h is the projection on fibers; i.e., h(Q,w) = w. See Figure *** forintuition.

Lemma 3.3 There exists a constant d7 = d7(T ), such that, for any δ ∈[0, d7], the graph transform G defined in (3.2) is well-defined as a map G :Σζ 7→ Σ, in the sense that, for σ ∈ Σζ, Gσ ∈ Σ; i.e., Gσ : M 7→ Nκ.

Proof: By lemma (3.2), for any Q ∈ M , there exists a unique Q1 ∈M1,such that,

Q = bF Tδ σF−T0 (Q1).

SetQ0 ≡ F−T0 (Q1),

if T > 1, thenQ0 ∈M.

Moreover,Q = bF Tδ σ(Q0).

In order to prove Gσ : M 7→ Nκ, we need to show that∥

∥

∥

∥

hF Tδ σ(Q0)

∥

∥

∥

∥

≤ κ.

We can rewrite hF Tδ σ(Q0) as:

hF Tδ σ(Q0) = hFT0 σ(Q0) + hF

Tδ σ(Q0) − hF T0 σ(Q0).

50

Then,

∥

∥

∥

∥

hF Tδ σ(Q0)

∥

∥

∥

∥

≤∥

∥

∥

∥

hF T0 σ(Q0)

∥

∥

∥

∥

+

∥

∥

∥

∥

hF Tδ σ(Q0) − hF T0 σ(Q0)∥

∥

∥

∥

.

By the calculation in sections 4.2 and 4.3, we see that when T is large enough(≥ 1

Ω+1

ln 2),∥

∥

∥

∥

hF T0 σ(Q0)

∥

∥

∥

∥

≤ 12κ.

By theorem (2.10),

∥

∥

∥

∥

hF Tδ σ(Q0) − hF T0 σ(Q0)∥

∥

∥

∥

≤ C2δ,

where C2 = C2(T ). Let

d7 =κ

2C2,

then∥

∥

∥

∥

hF Tδ σ(Q0) − hF T0 σ(Q0)∥

∥

∥

∥

≤ 12κ,

when δ ∈ [0, d7]. Therefore,∥

∥

∥

∥

hF Tδ σ(Q0)

∥

∥

∥

∥

≤ κ.

The proof is completed. ♣

Remark 3.3 Fixed points of the graph transform define invariant manifoldsfor the flow F Tδ . To see this, consider a section σ, which consists of theordered pairs

σ ≡{(

Q, wσ(Q)

)

∀Q ∈ M̄}

;

and its graph transform Gσ

Gσ ≡{(

Q′, hF Tδ [Q+ wσ(Q)])

∀Q′ ∈ M̄}

,

where Q and Q′ are related by

Q′ = bF Tδ [Q+ wσ(Q)].

51

If σ is a fixed point of the graph transform G, these two sections are thesame; that is,

Q′ = bF Tδ [Q+wσ(Q)]

wσ(Q′) = hF Tδ [Q+ wσ(Q)],

which, when embedded in the function space Ŝk yields

Q′ + wσ(Q′) = F Tδ (Q+ wσ(Q)).

The latter certainly shows that the fixed point (section) σ defines an invariantmanifold for the flow F Tδ .

3.3 The Graph Transform as a C0 Contraction

Here we are going to define a new norm for elements of Σ, the C0 norm, andwe will prove that the graph transform is a contraction map in this norm.

Definition 7 The sections have the representation σ(Q) = (Q,wσ(Q)) interms of which the C0 norm is defined as follows:

‖σ‖C0 = supQ∈M̄

‖wσ(Q)‖.

Lemma 3.4 Σζ is closed under the C0 norm.

Proof: Suppose σj is a Cauchy sequence (under the C0 norm) in Σζ . Then

for any Q ∈ M̄ , wσj (Q) is a Cauchy sequence (under H̃k norm). We knowthat wσj (Q) have the representation

wσj (Q) = α−1{σj}ê

−1 ;

thus, α−1{σj} is a real Cauchy sequence, which has a limit α−1 . Define a new

section σ byσ(Q) ≡ (Q,wσ(Q)),

wherewσ(Q) = α

−1 ê

−1 .

First, we want to show that σ ∈ Σ. Since,

(Q,wσj (Q)) ∈ Nκ, for all j;

52

i.e.,‖α−1{σj}ê

−1 ‖ ≤ κ, for all j;

in another word,|α−1{σj}| ≤ κ‖ê

−1 ‖−1,

we have|α−1 | ≤ κ‖ê−1 ‖−1;

i.e.,‖α−1 ê−1 ‖ ≤ κ.

Then,(Q,wσ(Q)) ∈ Nκ.

Thus, σ ∈ Σ. Then, we want to show that σ ∈ Σζ . We calculate Lip{σ}:

Lip{σ} = supQ,Q′∈M̄

‖wσ(Q) − wσ(Q′)‖‖Q−Q′‖

= supQ,Q′∈M̄

|α−1 (Q) − α−1 (Q′)|‖ê−1 ‖‖Q−Q′‖

= supQ,Q′∈M̄

limj→∞

|α−1{σj}(Q) − α−1{σj}(Q

′)|‖ê−1 ‖‖Q−Q′‖ .

We know that|α−1{σj}(Q) − α

−1{σj}(Q

′)|‖ê−1 ‖‖Q−Q′‖ ≤ ζ,

so

limj→∞

|α−1{σj}(Q) − α−1{σj}(Q

′)|‖ê−1 ‖‖Q−Q′‖ ≤ ζ,

thus

supQ,Q′∈M̄

limj→∞

|α−1{σj}(Q) − α−1{σj}(Q

′)|‖ê−1 ‖‖Q−Q′‖ ≤ ζ;

i.e. σ ∈ Σζ . Σζ is closed under the C0 norm. ♣

Proposition 3.2 The graph transform G defined above in (3.2) maps Σζinto Σζ .

53

Proof: Denote F tz • em • (Q,w) by F tz(Q,w), where z = 0, δ; Q ∈ M ,w ∈ N(Q). From the explicit calculation in sections 4.2 and 4.3, we chooseT large enough; moreover, fixed from now on, such that, for some κ0 > 0,

sup(Q,w)∈Nκ0

∥

∥

∥

∥

∇whF T0 (Q,w)∥

∥

∥

∥

< 1/4, (3.3)

sup(Q,w)∈Nκ0

∥

∥

∥

∥

{

∇QbF T0 (Q,w)}−1∥

∥

∥

∥

n

·

· sup(Q,w)∈Nκ0

∥

∥

∥

∥

∇whF T0 (Q,w)∥

∥

∥

∥

< 1/4, (3.4)

sup(Q,w)∈Nκ0

∥

∥

∥

∥

{

∇QbF T0 (Q,w)}−1∥

∥

∥

∥

<1

2Λ2, (3.5)

where Λ2 = Λ2(T ). Without loss of generality, we can take Λ2 = Λ, whereΛ is given in corollary (2). By theorem (2.10), when δ is sufficiently small,we have

sup(Q,w)∈Nκ0

∥

∥

∥

∥

∇whF Tδ (Q,w)∥

∥

∥

∥

< 1/2, (3.6)

sup(Q,w)∈Nκ0

∥

∥

∥

∥

{

∇QbF Tδ (Q,w)}−1∥

∥

∥

∥

n

·

· sup(Q,w)∈Nκ0

∥

∥

∥

∥

∇whF Tδ (Q,w)∥

∥

∥

∥

< 1/2, (3.7)

sup(Q,w)∈Nκ0

∥

∥

∥

∥

{

∇QbF Tδ (Q,w)}−1∥

∥

∥

∥

< Λ. (3.8)

Also from the explicit calculation in sections 4.2 and 4.3, we know that

∇QhF T0 (Q,w) = 0, ∀(Q,w) ∈ Nκ0. (3.9)

Thus, for any η > 0, there exists δ̂ > 0, such that, when δ ∈ [0, δ̂];

sup(Q,w)∈Nκ0

∥

∥

∥

∥

∇QhF Tδ (Q,w)∥

∥

∥

∥

< η. (3.10)

(Note that η is independent of ζ. Later we will first choose ζ small enough,then choose η < ζ8Λ .) Denote by

A ≡ sup(Q,w)∈Nκ0

∥

∥

∥

∥

∇whF Tδ (Q,w)∥

∥

∥

∥

, (3.11)

54

B ≡ sup(Q,w)∈Nκ0

∥

∥

∥

∥

∇QhF Tδ (Q,w)∥

∥

∥

∥

, (3.12)

C ≡ sup(Q,w)∈Nκ0

∥

∥

∥

∥

{

∇QbF Tδ (Q,w)}−1∥

∥

∥

∥

. (3.13)

Then, inequalities (3.6;3.7;3.8;3.10) can be rewritten as

A < 1/2, (3.14)CnA < 1/2, (3.15)

C < Λ, (3.16)B < η. (3.17)

SetQ̂ ≡ bF Tδ (σ(Q)), Q̂′ ≡ bF Tδ (σ(Q′)).

Then the claim of the proposition amounts to

∥

∥

∥

∥

hF Tδ (σ(Q)) − hF Tδ (σ(Q′))∥

∥

∥

∥

≤ ζ‖Q̂− Q̂′‖,

for any Q̂, Q̂′ ∈M .∥

∥

∥

∥

hF Tδ (σ(Q)) − hF Tδ (σ(Q′))∥

∥

∥

∥

=

∥

∥

∥

∥

hF Tδ (Q,hσ(Q)) − hF Tδ (Q′, hσ(Q))

+ hF Tδ (Q′, hσ(Q)) − hF Tδ (Q′, hσ(Q′))

∥

∥

∥

∥

≤∥

∥

∥

∥

hF Tδ (Q,hσ(Q)) − hF Tδ (Q′, hσ(Q))∥

∥

∥

∥

+

∥

∥

∥

∥

hF Tδ (Q′, hσ(Q)) − hF Tδ (Q′, hσ(Q′))

∥

∥

∥

∥

.

By lemma (3.1),

∥

∥

∥

∥

hF Tδ (Q,hσ(Q)) − hF Tδ (Q′, hσ(Q))∥

∥

∥

∥

≤ ‖Q−Q′‖ ·

· sup0≤α≤1

∥

∥

∥

∥

∇QhF Tδ (αQ+ (1 − α)Q′, hσ(Q))∥

∥

∥

∥

≤ ‖Q−Q′‖B. (3.18)

55

By lemma (3.1),

∥

∥

∥

∥

hF Tδ (Q′, h