CONSTRUCTION OF NONLINEAR QUASIMODES NEAR ELLIPTIC PERIODIC ORBITS. 1. Introduction

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1 CONSTRUCTION OF NONLINEAR QUASIMODES NEAR ELLIPTIC PERIODIC ORBITS PIERRE ALBIN, HANS CHRISTIANSON, JEREMY L. MARZUOLA, AND LAURENT THOMANN Abstract. We generalize to arbitrary manifolds wit elliptic periodic geodesics a quasimode construction for te nonlinear Scrödinger equation first discovered by te last autor. 1. Introduction In tis result, we study te Nonlinear Scrödinger Equation (NLS) on a compact, Riemannian manifold wit positive curvature and in particular wit periodic, elliptic geodesics. From enceforward, we will assume tat tese periodic geodesics satisfy a standard, generic nonresonance condition, referred to wen necessary as (NRC), related to te eigenvalues of te so-called Poincaré map, wic we state fully in Appendix A. Toug te Poincaré map is a muc studied object wit a ric istory in dynamical systems, our presentation will stem from previous work of te second autor [Cr11], wic contains a toroug collection of results and references. Specifically, we study solutions to { i t ψ + g ψ = σ ψ p ψ, (1.1) ψ(x, 0) = ψ 0, were g is te standard (negative semidefinite) Laplace-Beltrami operator and te solution is of te form ψ(x, t) = e iλt u(x). To solve te resulting nonlinear elliptic equation, wic can be analyzed using constrained variations, we will use Fermi coordinates to construct a nonlinear quasimode for an arbitrary manifold wit a periodic, elliptic orbit similar to te one constructed for p = 2 on 2-dimensional surfaces presented in [To08b]. To do so, we must analyze te metric geometry in a neigbourood of a periodic orbit, for wic we use te presentation in [Gra90]. For furter references to construction of quasimodes along elliptic geodesics, see te seminal works Ralston [Ral82], Babič [Bab68], Guillemin-Weinstein [GW76], Cardoso-Popov [CP02], as well as te toroug survey on spectral teory by Zelditc [Zel08]. For experimental evidence of te existence of suc Gaussian beam type solutions in nonlinear optics as solutions to nonlinear Maxwell s equations, see te recent paper by Sculteiss et al [SBS + 10]. Henceforward, we assume tat (M, g) is a compact Riemannian manifold of dimension d 2 witout boundary, and tat it admits an elliptic periodic geodesic Γ M. For β > 0, we introduce te notation U β = {x M : dist g (x, Γ) < β}. Let L > 0 be te period of Γ. Trougout te paper, eac time we refer to te small parameter > 0, tis means tat takes te form (1.2) = L 2πN, for some large integer N N. 1

2 2 P. ALBIN, H. CHRISTIANSON, J.L. MARZUOLA, AND L. THOMANN Our first result is te existence, in te case of a smoot nonlinearity, of quasimodes wic are igly concentrated near te elliptic periodic geodesic Γ satisfying te (N RC). More precisely : Teorem 1.1. Let (M, g) be a compact Riemannian manifold of dimension d 2, witout boundary and wic as an elliptic periodic geodesic. Let p be an even integer, let s 0 and assume tat p ( d 1 s ) < 1. 4 Ten for 1 sufficiently small, and any δ > 0, tere exists ϕ (x) H (M) satisfying (i) Frequency localization : For all r 0, tere exist C 1, C 2 independent of suc tat (ii) Spatial localization near Γ : C 1 s r ϕ Hr (M) C 2 s r. ϕ H s (M\U 1/2 δ ) = O( ) ϕ H s (M). (iii) Nonlinear quasimode : Tere exists λ() R so tat ϕ satisfies te equation g ϕ = λ()ϕ σ ϕ p ϕ + O( ). Note, in te above, by O( ) to mean a constant C() suc tat for any N N. C() C N N Te quasimode ϕ and te nonlinear eigenvalue λ() will be constructed tanks to a WKB metod, tis will give a precise description of tese objects (see Section 7.3). For instance, λ() reads λ() = 2 E o( 1 ), for some E 0 R. As a consequence of Teorem 1.1, and under te same assumptions, we can state : Teorem 1.2. Consider te function ϕ given by Teorem 1.1. Tere exists c 0 > 0, suc tat if we denote d 1 p( by T = c 0 4 s) ln( 1 ), te solution u of te Caucy problem (1.1) wit initial condition ϕ satisfies u L ([0,T ];H s (M\U 1/2 δ )) C (d+1)/4. To construct a quasimode wit O( ) error, we need tat te nonlinearity is of polynomial type, tat is p 2N. However, in te special case were d = 2 and s = 0, we are able to obtain a result for any 0 < p < 4, wic is a weaker version of Teorem 1.2. Teorem 1.3. Let (M, g) be a compact Riemannian surface witout boundary wic admits an elliptic periodic geodesic and let 0 < p < 4. Ten for 1 sufficiently small, and any δ, ɛ > 0, tere exist ϕ (x) H (M) and ν > 0 satisfying (i) Frequency localization : For all r 0, tere exist C 1, C 2 independent of suc tat (ii) Spatial localization near Γ : C 1 r ϕ H r (M) C 2 r. ϕ L2 (M\U 1/2 δ ) = O( ) ϕ L2 (M), suc tat te corresponding solution u to (1.1) satisfies u L ([0,T ];L 2 (M\U 1/2 δ )) C ν, for T = p in te case p (0, 4)\{1}, and T = 1+ɛ in te case p = 1.

3 NONLINEAR QUASIMODES 3 Remark 1.1. In our current notations, te Ḣ1 critical exponent for (1.1) is p = 4 d 2 = 4 n 1, were n is te dimension of te geodesic normal ypersurface to Γ. Te L 2 critical exponent is p = 4/d wic is always in our range. Te Ḣ1/2 critical exponent is p = 4/(d 1), wic is our endpoint. In general we construct igly concentrated solutions along an elliptic orbit, wic is effectively a d 1 dimensional soliton. Since stable soliton solutions exist on R d for monomial Scrödinger of te type in (1.1) precisely for p < 4 d, tis numerology matces well tat required for te expected local existence of stable nonlinear states on tese lower dimensional manifolds. In addition, solving (1.1) on compact manifolds wit te power p = 4/(d 2) plays an important role in te celebrated Yamabe problem, see [SY94]. Hence, te autors ope tat te tecniques ere can be relaxed to include tis critical case and peraps give insigt into te energy minimizers of suc a problem related to te geometry. Remark 1.2. In [CM10], te existence of nonlinear bound states on yperbolic space, H d, was explored. In tat case, it was found tat te geometry made compactness arguments at rater simple, ence te only driving force for te existence of a nonlinear bound state was te local beavior of te nonlinearity. Work in progress wit te second autor, tird autor, Micael Taylor and Jason Metcalfe is attempting to generalize tis observation to any rank one symmetric space. In tis note, we find nonlinear quasimodes based purely on te local geometry, were ere te nonlinearity becomes a lower order correction, wic is a nicely symmetric result. Remark 1.3. In a private conversation wit Nicolas Burq, e as pointed out tat in settings were one assumes radial symmetry of te manifold, it is possible to construct exact nonlinear bound states. Notations. In tis paper c, C denote constants te value of wic may cange from line to line. We use te notations a b, a b if 1 C b a Cb, a Cb respectively. Acknowledgments. P. A. was supported by an NSF joint institutes postdoctoral fellowsip (NSF grant DMS ) and a postdoctoral fellowsip of te Foundation Sciences Matématiques de Paris. J.L.M. was supported in part by an NSF postdoctoral fellowsip (NSF grant DMS ) at Columbia University and a Hausdorff Center Postdoctoral Fellowsip at te University of Bonn. He would also like to tank bot te Institut Henri Poincaré and te Courant Institute for being generous osts during part of te completion of tis work. L.T. was supported in part by te grants ANR-07-BLAN-0250 and ANR-10-JCJC 0109 of te Agence Nationale de la Recerce. In addition, te autors wit to tank Rafe Mazzeo and especially Colin Guillarmou for elpful conversations trougout te preparation of tis work and te referee for extremely elpful suggestions on te exposition. 2. Comparison to Previous Results Te proof of Teorem 1.2 is to solve approximately te associated stationary equation. Tat is, by separating variables in te t direction, we write from wic we get te stationary equation ψ(x, t) = e iλt u(x), (λ + g )u = σ u p u. For 1 of te form (1.2), te construction in te proof finds a family of functions u (x) = (d 1)/4 g( 1/2 x)

4 4 P. ALBIN, H. CHRISTIANSON, J.L. MARZUOLA, AND L. THOMANN suc tat g is rapidly decaying away from Γ, C, g is normalized in L 2, and (λ + g )u = σ u p u + Q(u ), were λ 2 and were te error Q(u ) is expressed by a truncation of an asymptotic series similar to tat in [To08b] and is of lower order in. To do so, te autors expand te metric using normal coordinates about te orbit and carefully bound te error terms using a presentation from [Gra90] and reviewed in Section 5. Wile te normal coordinate expansion is valid in a neigborood of any point on te manifold, te fact tat we work near a periodic, elliptic geodesic allows us to expand in normal coordinates uniformly about te orbit as well as apply te macinery of te Poincaré map in order to justify te existence of eac term in our asymptotic series, see Section 7.1, Appendix A and [Cr11] for furter details. Te point is tat Teorem 1.2 (as well as Teorem 1.3) is an improvement over te standard Gaussian beam approximate solution. For instance, it is well known tat tere exist quasimodes for te linear equation localized near Γ of te form v (x) = (d 1)/4 e is/ f(s, 1/2 x), (0 < 1), wit f a function rapidly decaying away from Γ, and s a parametrization around Γ, so tat v (x) satisfies in any seminorm, see [Ral82]. Ten were te error Q 2 (v λ ) = v p v satisfies (λ + g )v = O( ) v (λ + g )v = σ v p v + Q 2 (v ), Q 2 (v ) Ḣs = O( s p(d 1)/4 ). 3. A toy model In tis section we consider a toy model in two dimensions, and we give an idea of te proof of Teorem 1.3. For simplicity, we moreover assume tat 2 p < 4. As it is a toy model, we will not dwell on error analysis, and instead make Taylor approximations at will witout remarking on te error terms. Consider te manifold M = R x /2πZ R θ /2πZ, equipped wit a metric of te form ds 2 = dx 2 + A 2 (x)dθ 2, were A C is a smoot function, A ɛ > 0 for some ɛ. From tis metric, we get te volume form dvol = A(x)dxdθ, and te Laplace-Beltrami operator acting on 0-forms g f = ( 2 x + A 2 2 θ + A 1 A x )f. We observe tat we can conjugate g by an isometry of metric spaces and separate variables so tat spectral analysis of g is equivalent to a one-variable semiclassical problem wit potential. Tat is, let S : L 2 (X, dvol) L 2 (X, dxdθ) be te isometry given by Su(x, θ) = A 1/2 (x)u(x, θ). Ten = S S 1 is essentially self-adjoint on L 2 (X, dxdθ) wit mild assumptions on A. A simple calculation gives f = ( x 2 A 2 (x) θ 2 + V 1 (x))f,

5 NONLINEAR QUASIMODES 5 were te potential V 1 (x) = 1 2 A A (A ) 2 A 2. We are interested in te nonlinear Scrödinger equation (1.1), so we make a separated Ansatz: u λ (t, x, θ) = e itλ e ikθ ψ(x), were k Z and ψ is to be determined (depending on bot λ and k). Applying te Scrödinger operator (wit replacing ) to u λ yields te equation ( D t + )e itλ e ikθ ψ(x) = (λ + 2 x k 2 A 2 (x) + V 1 (x))e itλ e ikθ ψ(x) = σ ψ p e itλ e ikθ ψ(x), were we ave used te standard notation D = i. We are interested in te beaviour of a solution or approximate solution near an elliptic periodic geodesic, wic occurs at a maximum of te function A. For simplicity, let A(x) = (1 + cos 2 (x))/2, so tat in a neigbourood of x = 0, A 2 1 x 2 and A x 2. Te function V 1 (x) const. in a neigbourood of x = 0, so we will neglect V 1. If we assume ψ(x) is localized near x = 0, we get te stationary reduced equation (λ + 2 x k 2 (1 + x 2 ))ψ = σ ψ p ψ. Let = k 1 and use te rescaling operator T ψ(x) = T,0 ψ(x) = 1/4 ψ( 1/2 x) (see Lemma 4.5 below wit n = 1) to conjugate: or were ϕ = T 1 ψ. Let us now multiply by : were and T 1 (λ + 2 x k 2 (1 + x 2 ))T T 1 ψ = T 1 (σ ψ p ψ) (λ x k 2 (1 + x 2 )ϕ = σ p/4 ϕ p ϕ, ( 2 x x 2 E)ϕ = σ q ϕ p ϕ, E := 1 λ2 q := 1 p d 1 = 1 p 4 4. Observe te range restriction on p is precisely so tat 0 < q 1/2. We make a WKB type Ansatz, altoug in practice we will only take two terms (more is possible if te nonlinearity is smoot as in te context of Teorems 1.1 and 1.2): Te first two equations are ϕ = ϕ 0 + q ϕ 1, E = E 0 + q E 1. 0 : ( 2 x + x 2 E 0 )ϕ 0 = 0, q : ( 2 x + x 2 E 0 q E 1 )ϕ 1 = E 1 ϕ 0 + σ ϕ 0 p ϕ 0. Observe we ave included te q E 1 ϕ 1 term on te left and side. Te first equation is easy: ϕ 0 (x) = e x2 /2, E 0 = 1.

6 6 P. ALBIN, H. CHRISTIANSON, J.L. MARZUOLA, AND L. THOMANN For te second equation, we want to project away from ϕ 0 wic is in te kernel of te operator on te left and side. Tat is, coose E 1 satisfying E 1 ϕ 0 + σ ϕ 0 p ϕ 0, ϕ 0 = 0, so tat te rigt and side is in ϕ 0 L 2. Ten since te spectrum of te one-dimensional armonic oscillator is simple (and of te form (2m + 1), m Z), te operator ( x 2 + x 2 E 0 q E 1 ) is invertible on ϕ 0 L 2 wit inverse bounded by (2 q ) 1. Hence for > 0 sufficiently small, we can find ϕ 1 L 2 satisfying te second equation above (ere we ave used tat ϕ 0 is Scwartz wit bounded H s norms). Furter, since ϕ 0 is Scwartz and strictly positive, so is ϕ 0 p ϕ 0, so by propagation of singularities, ϕ 1 is also Scwartz. In particular, bot ϕ 0 and ϕ 1 are rapidly decaying away from x = 0. Let now ψ(x) = T (ϕ 0 (x) + q ϕ 1 (x)), and observe tat by te above considerations, ψ(x) is O( ) in any seminorm, outside an 1/2 δ neigbourood of x = 0. Let u = e itλ e ikθ ψ(x) so tat u L2 (dxdθ) 1, and u is O( ) outside an 1/2 δ neigbourood of x = 0. Furtermore, u satisfies te equation (again neglecting smaller terms) ( D t + )u = 1 T T 1 (λ + 2 x k 2 A 2 (x))t T 1 u = 1 T (σ q T 1 u p T 1 u + O( T 1 u p+1 ) = σ u p u + Q, were Q satisfies te pointwise bound Q = O( q u p+1 ). We now let ũ be te actual solution to (1.1) wit te same initial profile: { ( D t + )ũ = σ ũ p ũ ũ t=0 = e ikθ ψ(x). Set T = p, ten wit te Stricartz estimates of Burq-Gérard-Tzvetkov [BGT04] we prove (see Proposition 8.2) tat tere exists ν > 0 so tat and terefore we can compute: u ũ L ([0,T ];L 2 (M)) C ν, ũ L ([0,T ];L 2 (M\U 1/2 δ )) u L ([0,T ];L 2 (M\U 1/2 δ )) + ũ u L ([0,T ];L 2 (M\U 1/2 δ )) wic gives te result. = O( ) + O( ν ) = O( ν ), Remark 3.1. It is very important to point out tat te sources of additional error in tis euristic exposition ave been ignored, and indeed, to apply a similar idea in te general case, a microlocal reduction to a tubular neigbourood of Γ in cotangent space is employed. Te function ϕ 0 is no longer so simple, and te nonlinearity ϕ 0 p ϕ 0 is no longer necessarily smoot. Because of tis, te semiclassical wavefront sets are no longer necessarily compact, so a cutoff in frequency results in a fixed loss. Remark 3.2. We remark tat te Stricartz estimates from [BGT04] are sarp on te spere S d for a particular Stricartz pair, but tis is not necessarily true on a generic Riemannian manifold. See [BSS08] for a toroug discussion of tis fact.

7 NONLINEAR QUASIMODES 7 4. Preliminaries 4.1. Symbol calculus on manifolds. Tis section contains some basic definitions and results from semiclassical and microlocal analysis wic we will be using trougout te paper. Tis is essentially standard, but we include it for completeness. Te tecniques presented ave been establised in multiple references, including but not limited to te previous works of te second autor [Cr07, Cr11], Evans-Zworski [EZ07], Guillemin-Sternberg [GS77, GS10], Hörmander [Hör03, Hör05], Sjöstrand-Zworski [SZ02], Taylor [Tay81], and many more. To begin we present results from [EZ07], Capter 8 and Appendix E, to wic we refer readers for toroug introduction to te notation of wave front sets, a tecnique to be applied eavily in te sequel. Let X be a smoot, compact manifold. We will be operating ( on alf-densities, ) u(x) dx 1 2 C X, Ω 1 2 X, wit te informal cange of variables formula u(x) dx 1 2 = v(y) dy 1 2, for y = κ(x) v(κ(x)) κ (x) 1 2 = u(x), were κ (x) as te canonical interpretation as te Jacobian det( κ). By symbols on X we mean te set ( ) S k,m T X, Ω 1 2 (4.1) T X := { = a C (T X (0, 1], Ω 1 2 T X ) : α x β ξ a(x, ξ; ) Cαβ m ξ k β }. Essentially tis is interpreted as saying tat on eac coordinate cart, U α, of X, a a α were is a α S k,m in a standard symbol on R d. Tere is a corresponding class of pseudodifferential operators Ψ k,m (X, Ω 1 2 X ) acting on alf-densities defined by te local formula (Weyl calculus) in R n : 1 Op w (a)u(x) = 1 (2π) n We will occasionally use te sortand notations a w ambiguity in doing so. ( ) x + y a, ξ; e i x y,ξ / u(y)dydξ. 2 We ave te principal symbol map ) σ : Ψ k,m (X, / ( ) Ω 1 2X S k,m S k 1,m 1 T X, Ω 1 2 T X, wic gives te left inverse of Op w in te sense tat σ Op w : S k,m S k,m /S k 1,m 1 := Op w (a) and A := Op w (a) wen tere is no is te natural projection. Acting on alf-densities in te Weyl calculus, te principal symbol is actually well-defined in S k,m /S k 2,m 2, tat is, up to O( 2 ) in (see, for example [EZ07], Appendix E). We will use te notion of semiclassical wave front sets for pseudodifferential operators on manifolds, see Hörmander [Hör05], [GS77]. If a S k,m (T X, Ω 1 2 T X ), we define te singular support or essential support for a: ess-supp a T X S X, 1 We use te semiclassical, or rescaled unitary Fourier transform trougout: F u(ξ) = (2π) d/2 Z e i x,ξ / u(x)dx.

8 8 P. ALBIN, H. CHRISTIANSON, J.L. MARZUOLA, AND L. THOMANN were S X = (T X \ {0})/R + is te cospere bundle (quotient taken wit respect to te usual multiplication in te fibers), and te union is disjoint. Te ess-supp a is defined using complements: ess-supp a := { } (x, ξ) T X : ɛ > 0, ( x α β ξ a)(x, ξ ) = O( ), d(x, x ) + ξ ξ < ɛ {(x, ξ) T X \ 0 : ɛ > 0, ( x α β ξ a)(x, ξ ) = O( ξ ), d(x, x ) + 1/ ξ + ξ/ ξ ξ / ξ < ɛ}/r +. We ten define te wave front set of a pseudodifferential operator A Ψ k,m (X, Ω 1 2 X ): (4.2) WF (A) := ess-supp (a), for A = Op w (a). If u() is a family of distributional alf-densities, u C ((0, 1], D (X, Ω 1 2 X )), we say u() is -tempered if tere is an N 0 so tat N0 u is bounded in D (X, Ω 1 2 X ). If u = u() is an -tempered family of distributions, we can define te semiclassical wave front set of u, again by complement: (4.3) WF (u) := {(x, ξ) : A Ψ 0,0, wit σ (A)(x, ξ) 0, and Au C ((0, 1], C (X, Ω 1 2 X ))}. For A = Op w (a) and B = Op w (b), a Sk,m, b S k,m we ave te composition formula (see, for example, [DS99]) (4.4) were (4.5) S k+k,m+m wit ω te standard symplectic form. A B = Op w (a#b), a#b(x, ξ) := e i 2 ω(dx,d ξ;d y,d η) (a(x, ξ)b(y, η)) We record some useful results, wic are stated and proved as Teorems 8.10 and 8.12 in [EZ07], to wic we refer te reader for a more detailed discussion of wavefront set analysis. Lemma 4.1. Suppose (x 0, ξ 0 ) / WF (u). Ten b C c (T R n ) wit support sufficiently close to (x 0, ξ 0 ) we ave b(x, D)u = O S ( ). Here O S ( ) means O( ) in any Scwartz semi-norm. Teorem Suppose a S(m) and u() is -tempered. Ten 2 If a S(m) is real-valued, ten also WF (a w u) WF (u) ess-supp (a). WF (u) WF (a w u) {ess-supp a}. Proof. Assertion 1 is straigtforward. Te proof of assertion 2 is standard, owever we present it ere so we can use it for te analogous result for te blown-up wavefront set. Using Lemma 4.1, we will sow if (x 0, ξ 0 ) / WF (a w u) and a(x 0, ξ 0 ) 0, ten for all b, b(x 0, ξ 0 ) 0 so tat b w u = O L 2( ). Tere exists a neigbourood U (x 0, ξ 0 ), a real-valued function χ, and a positive number γ > 0 suc tat supp χ U = and a + iχ γ everywere. x=y ξ=η,

9 NONLINEAR QUASIMODES 9 Ten P = a w + iχ w as an approximate left inverse c w so tat c w P = id + R w, were R w = O L2 L 2( ). Coose b S so tat supp (b) U and b(x 0, ξ 0 ) 0. Ten b w χ w = O( ) as an operator on L 2. Hence b w u =b w c w P u b w R w u =b w c w a w u + ib w c w χ w u b w R w u =O( ), were we ave used te Weyl composition formula to conclude ess-supp (c#χ) supp b = Exotic symbol calculi. Following ideas from [SZ02], since rescaling often means dealing wit symbols wit bad decay properties, we introduce weigted wave front sets as well. Let us first recall te non-classical symbol classes: ( ) S k,m δ,γ T X, Ω 1 2 (4.6) T X := { = a C (T X (0, 1], Ω 1 2 T X ) : α x β ξ a(x, ξ; ) Cαβ δ α γ β m ξ k β }. Tat is, symbols wic lose δ powers of upon differentiation in x and γ powers of upon differentiation in ξ. Note te simplest way to acieve tis is to take a symbol a(x, ξ) S and rescale (x, ξ) ( δ x, γ ξ), wic ten localizes on a scale δ+γ in pase space. We tus make te restriction tat 0 δ, γ 1, 0 δ + γ 1, and to gain powers of by integrations by parts, we usually also require δ + γ < 1. We can define wavefront sets using S δ,γ = S 0,0 δ,γ symbols, but te localization of te wavefront sets is stronger. Definition 4.3. If u() is -tempered and 0 δ, γ 1, 0 δ + γ 1, (4.7) WF,δ,γ (u) = {(x 0 ξ 0 ) : a S δ,γ C c, a(x 0, ξ 0 ) 0, a(x, ξ) = ã( δ x, γ ξ) We ave te following immediate corollary. for some ã S and a w u = O L 2( )}. Corollary 4.4. If a S δ,γ (m), 0 δ + γ < 1, and a is real-valued, ten WF,δ,γ (u) WF,δ,γ (a w u) {ess-supp a}. Te proof is exactly te same as in Teorem 4.2 only all symbols must scale te same, so tey must be in S δ,γ. In order to conclude te existence of approximate inverses, we need te restriction δ + γ < 1, and te rescaling operators from 4.3 wic can be used to reduce to te familiar δ calculus, were δ = (δ + γ)/2, by replacing wit δ γ Rescaling operators. We would like to introduce -dependent rescaling operators. Te rescaling operators sould be unitary wit respect to natural Scrödinger energy norms, namely te omogeneous Ḣ s spaces. Let us recall in R n, te Ḣs space is defined as te completion of S wit respect to te topology induced by te inner product u, v Ḣs = ξ 2s û(ξ)ˆv(ξ)dξ, R n were as usual û denotes te Fourier transform. Te Ḣ0 norm is just te L 2 norm, and te Ḣ1 norm is u Ḣ1 u L 2. Te purpose in taking te omogeneous norms instead of te usual Sobolev norms is to make te rescaling operators in te next Lemma unitary.

10 10 P. ALBIN, H. CHRISTIANSON, J.L. MARZUOLA, AND L. THOMANN Lemma 4.5. For any s R, > 0, te linear operator T,s defined by (4.8) is unitary on Ḣs (R n ), and for any r R, T,s w(x) = s/2 n/4 w( 1/2 x) T,s w Ḣr = (s r)/2 w Ḣr, T,0 w Ḣs = s/2 w Ḣs. Moreover, for any pseudodifferential operator P (x, D) in te Weyl calculus, T 1,s P (x, D)T,s = P ( 1/2 x, 1/2 D). Remark 4.1. Observe tat for tis lemma, te usual assumption tat be small is not necessary. Proof. Te proof is simple rescaling, but we include it ere for te convenience of te reader. To ceck unitarity, we just cange variables: u, T,s w Ḣs = ξ s û, ξ s T,s w = s/2+n/4 ξ 2s û(ξ)ŵ( 1/2 ξ)dξ = n/4 s/2 ξ 2s û( 1/2 ξ)ŵ(ξ)dξ = n/4 s/2 ξ 2s u( 1/2 )(ξ)ŵ(ξ)dξ = T 1,s Ḣ u, w. s To ceck te conjugation property, we again compute T 1,s P (x, D)T,sw(x) = T 1,s (2π) n s/2 n/4 = T 1,s (2π) n s/2+n/4 = T 1,s (2π) n s/2 n/4 = (2π) n e i x y,ξ P ( x + y, ξ)w( 1/2 y)dydξ 2 e y,ξ x + 1/2 y i x 1/2 P (, ξ)w(y)dydξ 2 e i x 1/2 y, 1/2 ξ x + 1/2 y P (, 1/2 ξ)w(y)dydξ 2 e i x y,ξ P ( 1/2 (x + y), 1/2 ξ)w(y)dydξ 2 = P ( 1/2 x, 1/2 D)w(x). Te purpose of using te rescaling operators T,s is tat if u Ḣs as -wavefront set as in (4.7) WF,δ,γ (u) { x α() and ξ β()}, were, according to te uncertainty principle, α()β(), ten T,s u as -wavefront set WF,δ 1/2,γ+1/2 (T,s u) { x α() 1/2 and ξ β() 1/2 }, provided, of course, tat δ 1/2. To see tis, we just observe tat for any ψ Cc (R 2n ), ψ 0 on { x 1, ξ 1}, we ave ψ(x/α(), D/β())u = O( ) u L 2,

11 NONLINEAR QUASIMODES 11 or any oter semi-norm, and ence ψ( 1/2 x/α(), 1/2 D/β())T,s u = T,s ψ(x/α(), D/β())T 1,s T,su = T,s ψ(x/α(), D/β())u = s/2 O( ) u L 2 = O( ) u L 2. Finally, we note tat te symbol of te operator on te left-and side is zero on te set { x 1/2 α(), and ξ 1/2 β()}, and any suc symbol in S δ 1/2,γ+1/2 as defined in (4.6), elliptic at a point outside tis set, can be locally obtained in tis fasion. 5. Geometry near an elliptic geodesic Suppose Γ is a geodesic in a (n + 1)-dimensional Riemannian manifold. Following [MP05, 2.1] (cf. [Gra90]) we fix an arclengt parametrization γ(t) of Γ, and a parallel ortonormal frame E 1,..., E n for te normal bundle NΓ to Γ in M. Tis determines a coordinate system x = (x 0, x 1,..., x n ) exp γ(x0)(x 1 E x n E n ) = F (x). We write x = (x 1,..., x n ) and use indices j, k, l {1,..., n}, α, β, δ {0,..., n}. We also use X α = F ( xα ). Note tat r(x) = x x2 n is te geodesic distance from x to Γ and r is te unit normal to te geodesic tubes {x : d(x, Γ) = cst}. Let p = F (x 0, 0), q = F (x 0, x ), and r = r(q) = d(p, q) ten we ave [MP05, Proposition 2.1], wic states (5.1) g jk (q) = δ ij g(r(x s, X j )X l, X k ) p x s x l + O(r 3 ), g 0k (q) = O(r 2 ), g 00 (q) = 1 g(r(x k, X 0 )X 0, X l ) p x k x l + O(r 3 ), Γ δ αβ = O(r), n Γ k 00 = g(r(x k, X 0 )X j, X 0 ) p x j + O(r 2 ), j=1 were Γ δ αβ = 1 2 gδη (X α g ηβ + X β g αη X η g αβ ). (5.2) In tese coordinates, te Laplacian as te form g := 1 det g div( det gg 1 ) = g jk X j X k g jk Γ l jkx l = g 00 X 0 X 0 + 2g k0 X k X 0 g jk Γ 0 jkx 0 + 2g 0j Γ k 0jX k + Γ, were Γ is te Laplacian in te directions transverse to Γ. Denote te geodesic flow on SM, te unit tangent bundle, by ϕ t. Let γ(0) = p Γ and ζ = γ (0) SM. Associated to Γ is a periodic orbit ϕ t ζ of te geodesic flow on SM. Tis orbit ϕ t ζ is called stable if, wenever V is a tubular neigbourood of ϕ t ζ, tere is a neigbourood U of ζ suc tat ζ U implies ϕ t ζ V. Given a ypersurface Σ in SM containing ζ and transverse to ϕ t ζ, we can define a Poincaré map P near

12 12 P. ALBIN, H. CHRISTIANSON, J.L. MARZUOLA, AND L. THOMANN ζ, by assigning to eac ζ te next point on ϕ t (ζ ) tat lies in Σ. Two Poincaré maps of ϕ t ζ are locally conjugate and ence te eigenvalues of dp at ζ are invariants of te periodic orbit ϕ t ζ. Te Levi-Civita connection allows us to identify T T M wit te sum of a orizontal space and a vertical space, eac of wic can be identified wit T M. Tus we can coose Σ so tat T ζ Σ is equal to E E, were E is te ortogonal complement of ζ in T p M. Te linearized Poincaré map is ten given by E E (V, W ) dp E E, (J(lengt(γ)), X0 J(lengt(γ))) were J is te unique Jacobi field along γ wit J(0) = V and X0 J(0) = W, i.e., J solves { X0 X0 J + R(J, X 0 )X 0 = 0 J(0) = V, X0 J(0) = W. Te linearized map dp = dp ζ preserves te symplectic form on E E, ω((v 1, W 1 ), (V 2, W 2 )) = V 1, W 2 W 1, V 2, and so its eigenvalues come in complex conjugate pairs. We say tat Γ is a non-degenerate elliptic closed geodesic if te eigenvalues of dp ave te form {e ±iαj : j = 1,..., n} were eac α j is real and te set {α 1,..., α n, π} is linearly independent over Q. From (5.1), te Hessian of te function g 00 (q) as a function of x is (minus) te transformation appearing in te Jacobi operator E V R(V, X 0 )X 0 E. Notice tat if V is a normalized eigenvector for tis operator, wit eigenvalue λ, ten Te very useful property sec p (X 0, V ) = g p (R(V, X 0 )X 0, V ) = λg p (V, V ) = λ. (5.3) p Γ, V E = sec p (X 0, V ) > 0, olds for any elliptic closed geodesic. Remark 5.1. One can verify (5.3) by means of te Birkoff normal form (see [Zel08, 10.3]). Indeed, if one of te sectional curvatures were negative, ten te Birkoff normal form of te linearized Poincare map (in T M) must ave an eigenvalue off te unit circle. If so, ten tere is at least one nearby orbit wic does not stay nearby, ence te periodic geodesic is not elliptic. Similarly, if one of te curvatures vanises, ten te linearized Poincare map as a zero eigenvalue, and ence te logs of te eigenvalues are not independent from π over te rationals (in oter words, te Poincare map is degenerate, and not even symplectic). 6. Compact Solitons: Te nonlinear Ansatz We are interested in finding quasimodes for te non-linear Scrödinger equation g u = λu σ u p u, were σ = ±1 determines if we are in te focussing or defocussing case. We will construct u approximately solving tis equation wit u concentrated near Γ in a sense to be made precise below. We take as Ansatz F (x, ) = e ix0/ f(x, ), 1 = 2πN L,,

13 NONLINEAR QUASIMODES 13 were L > 0 is te period of Γ, and assume for te time being tat te function f is concentrated in an -dependent neigbourood of Γ. We are going to employ a semiclassical reduction, and we are interested in fast oscillations ( 0), so we assume WF,1/2 δ,0 f(x, ) { x ɛ 1/2 δ, ξ ɛ} for some ɛ, δ > 0 2. Te localization property of f as constructed later will be verified in Section 7. We compute from (5.2), wit te non-positive Laplacian, (6.1) F = e ix0/ [ g 00 ( 1 2 f + 2i X 0f + X 0 X 0 f ) ( ) i + 2g k0 X k f + X 0f g kj Γ 0 kj ( ) ] i f + X 0f + 2g 0j Γ k 0jX k f + Γ f. Remark 6.1. One may initially be inclined to use te Ansatz of te original Gaussian beam from Ralston [Ral82], wic is e iψ(x)/ (a 0 + a a N N ), te standard geometric optics quasimode construction. After all, Ralston is able to make very nice use of te geometry to construct a pase function of te form ψ(x) = i/(x x B(x 0 )x ) wit Im B(x 0 ) > 0 (for x 0 0, vanising oterwise). In suc a regime, owever, te non-linear term in te Scrödinger equation (1.1) vanises to infinite order in. Tus wile suc a solution always exists, it fails to capture te effects of te nonlinearity tat we are interested in. We analyze (6.1) by applying te operator T 1,s from (4.8) in te variables transversal to Γ. We normalize everyting in te L 2 sense, so we take ere s = 0. Let z = 1/2 x and set v(x 0, z, ) = T 1,0 f(x 0, z, ) = n/4 f(x 0, 1/2 z, ). Notice tat te distance to te geodesic r = x is scaled to ρ = z = 1/2 r, as described above. In particular, now (6.2) WF,0,1/2 v { z δ ɛ, ζ 1/2 ɛ}, as defined in (4.7) if ζ is te (semiclassical) Fourier dual to z. We conjugate (6.1) to get [ ( T 1,0 T,0T 1,0 F = eix0/ g v + 2i ) ( ) i X 0v + X 0 X 0 v + 2g k0 1/2 zk v + X 0v ( ) ] i g kj Γ 0 kj v + X 0v + 2g 0j Γ k 0j 1/2 zk v + 1 Γ v, were te metric components and Cristoffel symbols are evaluated at (x 0, 1/2 z). On te oter and, from (5.1), expanding in Taylor polynomials, we know tat g 00 (x) = 1 + R 2 (x) + R 3 (x) + R 4 (x) + O(r 5 ) = 1 + R 2 (z) + 3/2 R 3 (z) + 2 R 4 (z) + O( 5/2 ρ 5 ) g 0k (x) = g 0k 2 (z) + O( 3/2 ρ 3 ) g jk (x) = δ jk + O(ρ 2 ) Γ 0 jk(x) = 1/2 Γ0 jk1 (z) + Γ 0 jk2(z) + O( 3/2 ρ 3 ) Γ k 0j(x) = 1/2 Γk 0j1 (z) + O(ρ 2 ), 2 Te reason for te weaker concentration in frequency ξ is tat te nonlinearity forces working wit non-smoot functions, so some decay at infinity in frequency is lost.

14 14 P. ALBIN, H. CHRISTIANSON, J.L. MARZUOLA, AND L. THOMANN for some smoot functions R l, g l 0k, Γ α jkl omogeneous of degree l respectively. Hence [ T 1,0 T,0T 1,0 F = eix0/ 1 2 v 1 R 2(z)v 1 R 3(z)v R 1/2 4 (z)v + 2i X 0v+ (6.3) + 2i +P v, 1/2 gk0 2 (z) zk v i 1/2 gjk Γ0 jk1 (z)v ig jk Γ0 jk2 (z)v + 1 ] Γ v were P contains te remaining terms from te Taylor expansion. Let us record te following Lemma. Lemma 6.1. Te operator P as te following expansion: P = O( 1/2 z 5 ) + X 0 X 0 + O( z 2 ) 1/2 zk X 0 + O( z 3 ) zk + O( 1/2 z 3 ) In particular, if v satisfies (6.2), ten +O( 1/2 z + z 2 )X 0 + O( 3/2 z 3 ) 1/2 zk. P v L 2 C 1 2 5δ v L 2 + C X 0 X 0 v L 2 + C 2δ X 0 v L 2. Remark 6.2. We will sow later tat for te particular coice of v we construct, te operators X 0 and X 2 0 are bounded operators for δ sufficiently small, so tat te error 1 2 P v is bounded in L 2 by v (see Remark 7.3) as required in our error analysis below. For te purposes of exposition, let us ten assume for now tat te term 1 2 P v is bounded for some σ > 0 and proceed (justification of te bound will be made clear below). Applying T 1,0 to te equation F = λf σ F p F yields T 1,0 T,0T 1,0 so tat multiplying (6.3) by e ix0/, we get 1 1 F = λt,0f σt,0 ( F p F ) = λt 1,0 F σ pn/4 T 1,0 F p T 1,0 F, (6.4) (2iX 0 + Γ R 2 (z))v = 1 2 λ ( ) v + 1/2 ig jk Γ0 jk1 2i g k0 (z) zk + R 3 (z) v + (R 4 (z) + ig jk Γ0 jk2 )v + 1 pn/4 σ v p v + e ix0/ P v, were P is te same as above. Remark 6.3. In order to ensure tat te nonlinearity appears ere as a lower order term, we require (6.5) q := 1 pn/4 > 0, or p < 4 n = 4 d 1 as stated in te teorems. We want to tink of te left and side as similar to a time-dependent armonic oscillator were x 0 plays te role of te time variable. Let q = 1 pn/4, 0 < q < 1. We would like to assume tat v and λ ave expansions in q, owever te spreading of wavefront sets due to te nonlinearity allows us to only take te first two terms wen 0 < q 1/2 and te first tree terms oterwise. Case 1: 0 < q 1/2. Assume tat v as a two-term expansion v = v 0 + q v 1

15 NONLINEAR QUASIMODES 15 and moreover tat tere exist E k, k = 0, 1, satisfying 1 2 λ = E 0 + q E 1 + O( 2q ). Since q 1/2, ten te O( 1/2 ) term in (6.4) is of equal or lesser order tan te nonlinear term, and substituting into (6.4) we get te following equations according to powers of : (6.6) were Lv 0 = 0 : (2iX 0 + Γ R 2 (z) E 0 )v 0 = 0, q : (2iX 0 + Γ R 2 (z) E 0 q E 1 )v 1 = E 1 v 0 + σ v 0 p v 0 + 1/2 q Lv 0, ( ) ig jk Γ0 jk1 2i g k0 (z) zk + R 3 (z) + 1/2 R 4 (z) + i 1/2 g jk Γ0 jk2 + 1/2 e ix0/ P v 0. We will sow te error terms are O( 2q ) in te appropriate H s space. See Case 2: 1/2 < q < 1. In te case q > 1/2, te O( 1/2 ) term becomes potentially larger tan te nonlinearity, so we take tree terms in te expansions of v and E = 1 2 λ : We ten want to solve (6.7) v = v 0 + 1/2 v 1 + q v 2, E = E 0 + 1/2 E 1 + q E 2 + O(). 0 : (2iX 0 + Γ R 2 (z) E 0 )v 0 = 0, 1/2 : (2iX 0 + Γ R 2 (z) E 0 )v 1 = E 1 v 0 + Lv 0 q : (2iX 0 + Γ R 2 (z) E 0 1/2 E 1 q E 2 )v 2 = E 2 v q E 1 v 1 + 1/2 E 2 v 1 + σ v 0 p v q Lv 1. In tis case we will sow te error is O( 1/2+q ) in te appropriate H s space. See In tis case, we coose = 1 q 7. Quasimodes We begin by approximately solving te 0 equation by undoing our previous rescaling. Tat is, let w 0 (x 0, x, ) = T.0 v 0 (x 0, x, ) = n/4 v 0 (x 0, 1/2 x, ), and conjugate te 0 equation by T,0 to get: 0 = T,0 (2iX 0 + Γ R 2 (z) E 0 )T 1,0 T,0v 0 = (2iX 0 + Γ 1 R 2 (x) E 0 )w 0, were now te coefficients in Γ are independent of, and we ave used te omogeneity of R 2 in te x variables. Multiplying by, we ave te following equation: (7.1) (2iX Γ R 2 (z) E 0 )w 0 = 0, Hence, (7.1) is a semiclassical equation in a fixed neigbourood of Γ wit symbols in te 0 calculus (i.e. no loss upon taking derivatives). Te principal symbol of te operator in (7.1) is p = τ ζ 2 g R 2 (z), were g is te metric in te transversal directions to Γ, and τ is te dual variable to te x 0 direction. If we let Γ be te (unit speed) lift of Γ to T M, and if exp(sh p ) is te Hamiltonian flow of p, ten Γ = {ζ = z = 0, x 0 = s R/Z}. Since, in te transversal directions, p is a negative definite quadratic form, te linearization of te Poincaré map S is easy to compute: S = exp(h q ),

16 16 P. ALBIN, H. CHRISTIANSON, J.L. MARZUOLA, AND L. THOMANN were so tat q = ζ 2 g R 2 (z), H q = 2a(x 0, z) j,k ζ j zk + 2b j,k (x 0 )z j ζk, were a and b are symmetric, positive definite matrices. Linearizing S about, say, x 0 and z = ζ = 0 we get tat ds(0, 0) as all eigenvalues on te unit circle, in complex conjugate pairs. Tat is, Γ is still a periodic elliptic orbit of te classical flow of p. Since p is defined on a fixed scale, we can glue p togeter wit an operator wic is elliptic at infinity so tat p is of real principal type so tat we can apply Teorem A.1 in te appendix to construct linear quasimodes. Note tat since we ave quasi-eigenvalue of order O(), Teorem A.1 implies te quasimodes are concentrated on a scale z 1/2, ζ 1/2. Tis is made precise in te following proposition. Proposition 7.1. Tere exists w 0 L 2, w 0 L 2 = 1, and E 0 = O(1) suc tat (2iX Γ R 2 (z) E 0 )w 0 = e 0 (). Here te error e 0 () = O( ) L 2 (or in any oter seminorm), and w 0 as -wavefront set sarply localized on Γ in te sense tat if ϕ Ψ 0 1/2 δ,1/2 δ is 1 near Γ, ten for any δ > 0, ϕw 0 = w 0 + O( ), and if δ = 0, ϕw 0 c 0 w 0 for some positive c 0 depending on te support of ϕ. Moreover, w 0 H (M) and satisfies te estimate w 0 H s (M) = O( s/2 ). Proof. Te construction follows from Teorem A.1, and is well-known in oter settings, see for instance [Ral82], [Bab68], and [CP02]. To get te sarp localization, apply Lemmas B.1 and B.2 from te appendix to get te localization on w 0. Once we know tat w 0 is so localized, we can replace w 0 wit ϕw 0, were ϕ Ψ 0 1/2 δ is as in te proposition. Ten ϕw 0 satisfies were (2iX Γ R 2 (z) E 0 )ϕw 0 = ẽ 0 (), ẽ 0 () = ϕe 0 () + [(2iX Γ R 2 (z)), ϕ]w 0. But now ϕe 0 () = O( ), wile te commutator is supported outside of an 1/2 δ neigbourood of Γ, so by te localization of w 0 is O( ) and localized in a sligtly larger set on te scale 1/2 δ. Now recalling v 0 = T 1,0 w 0, ten v 0 satisfies (2iX 0 + Γ R 2 (z) E 0 )v 0 = 1 T 1,0 (2iX Γ R 2 (z) E 0 )T,0 v 0 = 1 T 1,0 (2iX Γ R 2 (z) E 0 )w 0 = 1 T 1,0 e 0(). Te error 1 T 1,s e 0() = O( ) in any seminorm still, but te function v 0 is now localized on a scale δ in space. Tat is, WF,0,1 δ v 0 { x ɛ δ, ξ ɛ 1 δ }, again as defined in (4.7).

17 NONLINEAR QUASIMODES Te inomogeneous equation. We are now in a position to solve te lower order inomogeneous equations in (6.6). Te quasimode v 0 as been constructed as a Gaussian beam (see [Ral82]); it is a armonic oscillator eigenfunction extended in te x 0 direction by te quantum monodromy operator from [SZ02], wic is defined in (7.4) below. From tis construction, te boundedness of te error term 1 2 P v 0 follows as stated in Lemma 6.1. In wat follows we construct v 1 in te case 0 q 1 2 and v 1, v 2 in te case 1 2 < q < 1 and sow tat a similar bound olds for q P v 1 and 1 2 P v 1 + q P v 2 respectively. We want to solve (7.2) (2iX 0 + Γ R 2 (z) Ẽ0)v 1 = E 1 v 0 + G, were Ẽ0 = E 0 + η E 1 for some η > 0, E 0 and v 0 ave been fixed by te previous construction, and G = G(x 0, x) is a given function (periodic in x 0 ) wit WF,0,1/2 G { x ɛ δ, ξ ɛ 1/2 }. Note, te localization of G follows from te nonlinearity as well as geometric multipliers in te operator L, see Appendix E and (6.2). Conjugating by T,0 as before, we get te equation (7.3) (2iX Γ R 2 (x) Ẽ0)w 1 = G 2, were We observe ten tat G 2 = T,0 (E 1 v 0 + G). WF,1/2 δ,0 G 2 { x ɛ 1/2 δ, ξ ɛ}. Specifically, let us parametrize motion along Γ by te parameter 0 < t < L. Let M(t) be te deformation family to te quantum monodromy operator defined as te solution to: { (2iX0 + 2 Γ R 2 (x) Ẽ0)M(t) = 0, (7.4) M(0) = id, wic exists microlocally in a neigbourood of Γ T X (for furter discussion see also Appendix A and references terein). From ere onward, by te term microlocally, we mean up to terms tat are O( ). By te Duamel formula, we write w 1 = M(t)w 1,0 + i t 0 M(t)M( t) G 2 ( t, )d t. We ave to coose w 1,0 and E 1 (implicit in G 2 ) in suc a fasion to make w 1 periodic in t; in oter words, to solve te equation (approximately) around Γ. Let L be te primitive period of Γ, so tat t = 0 corresponds to t = L. Ten we require w 1 (L, ) = w 1 (0, ), or L w 1,0 = M(L)w 1,0 + i M(L)M( t) G 2 ( t, )d t. 0 In oter words, we want to be able to invert te operator (1 M(L)). Te problem is tat w 0,0 := w 0 (0, ) = T,0 v 0 (0, ) is in te kernel of (1 M(L)), so we need to coose E 1 in suc a fasion to kill te contribution of G 2 in te direction of w 0,0. Recall tat G 2 = T,0 (E 1 v 0 + G) = (E 1 w 0 + G),

18 18 P. ALBIN, H. CHRISTIANSON, J.L. MARZUOLA, AND L. THOMANN were We want to solve microlocally (7.5) Let (1 M(L))w 1,0 = i so tat (7.5) is ortogonal to w 0,0. If we denote = i L 0 L 0 G = T,0 G. M(L)M( t) ((E 1 w 0 + G))d t M(L)M( t) (E 1 M( t)w 0,0 + G)d t = ilm(l)e 1 w 0,0 + i L E 1 = 1 L M( t) Gd t, w 0,0, L 0 L 2 w 0,0 = {u L 2 : u, w 0,0 = 0}, 0 M(L)M( t) Gd t. ten since E 0 + η E 1 is a small perturbation of E 0 by te (NRC) and M(L) is unitary on L 2, (I M(L)) 1 is a bounded operator (see [Cr11]) Hence satisfies and (1 M(L)) 1 : L 2 w L 2 0,0 w0,0. w 1,0 = (1 M(L)) 1 i L L w 1,0 L 2 C 1 M( t) G 2 ( t, )d t 0 0 w 1,0 L 2 w 0,0. Furtermore, we ave te estimate L w 1,0 Ḣs C 1 s/2 M( t) G 2 ( t, )d t 0 M(L)M( t) G 2 ( t, )d t L 2 L 2 CL 1/2 1 G 2 L2 (t)l 2 CL 1/2 1 s/2 G 2 L 2 (t)l 2, tat is, te Ḣs norm is controlled, but not by te omogeneous Sobolev norm. We ave proved te following Proposition, wic follows simply from tracing back te definitions. Proposition 7.2. Let v 0 be as constructed in te previous section, and let G H s for s 0 sufficiently large satisfy WF,0,1/2 G { x ɛ δ, ξ ɛ 1/2 }. Ten for any η > 0, tere exists v 1 L 2 and E 1 = O( G L 2) suc tat and moreover (2iX 0 + Γ R 2 (z) E 0 η E 1 )v 1 = E 1 v 0 + G, v 1 Ḣs C( G H s + v 0 H s).

19 NONLINEAR QUASIMODES 19 Remark 7.3. We note ere tat by construction of v 1 we ave implicitly microlocalized into a periodic tube in te x 0 variable. Using te fact tat te Quantum Monodromy Operator is a microlocally unitary operator (see [SZ02, Cr11]), te bound X j 0 v 1 L 2 v 1 L 2 follows easily for any j, wic is required for proof of Lemma Construction of quasimodes in te context of Teorem 1.3. We now ave all te tools to construct te quasimodes wic will be used to prove Teorem 1.3. Let d 2 and 0 < p < 4/(d 1). As previously, denote by q = 1 p(d 1)/4. Te main results of tis part will be stated in Propositions 7.4 and Te case 0 < q 1/2, i.e. d 1 p < 4 d 1. In tis subsection, we see ow to apply Proposition 7.2 in te case 0 < q 1/2. As described previously, in tis case, te nonlinearity is te next largest term, and we ave only one inomogeneous equation so solve (see (6.6)). According to Propostion E.2 and Corollary E.1 from Appendix E, if G = σ v 0 p v 0 1/2 q Lv 0 is te nonlinear term on te rigt-and side, ten G is sarply localized in space but weakly localized in frequency. Tat is, if χ Cc (T X) is equal to 1 in a neigbourood of Γ, ten for any 0 δ < 1/2 and any 0 γ 1, χ( δ x, 1 γ D x )G(x 0, x) = G(x 0, x) + E, were for any 0 r 3/2, E Ḣr C (1 γ)(3/2+p r). We are interested in te case were γ = 1/2, since in tat case G is weakly concentrated in frequencies comparable to 1/2, so by cutting off, satisfies te assumptions of Proposition 7.2. Tat is, take γ = 1/2, and replace G wit G = χ( δ x, 1 γ D x )G, and apply Proposition 7.2 to get v 1 and E 1 satisfying or in oter words were (2iX 0 + Γ R 2 (z) E 0 q E 1 )v 1 = E 1 v 0 + G, (2iX 0 + Γ R 2 (z) E 0 q E 1 )v 1 = E 1 v 0 + σ v 0 p v 0 + 1/2 q Lv 0 + Q 1, Q 1 Ḣr C (1/2)(3/2+p r). Now letting v = v 0 + q v 1 and E = E 0 + q E 1, we ave solved were Te remainder Q 2 satisfies (2iX 0 + Γ R 2 (z) E)v = q σ v 0 p v 0 + 1/2 Lv 0 + q Q1 = q σ v p v + 1/2 Lv + Q 2, Q 2 = q Q1 1/2+q Lv 1 + O( 2q v p+1 ). Q 2 Ḣs C s/2+min{2q,1/2+q,q+3/4+p/2} = C s/2+2q, since q 1/2. Recalling te definitions, ϕ = e ix0/ T,0 v satisfies (i) WF,1/2 δ,0 ϕ { x ɛ 1/2 δ, ξ ɛ};

20 20 P. ALBIN, H. CHRISTIANSON, J.L. MARZUOLA, AND L. THOMANN (ii) (iii) and or ϕ L 2 1, D l x 0 ϕ L 2 l, D l x ϕ L 2 C l/2 ; ( + λ)ϕ = 1 e ix0/ T,0 e ix0/ T 1,0 T,0T 1,0 ϕ = 1 e ix0/ T,0 (2iX 0 + Γ R 2 E 1/2 L)v = 1 e ix0/ T,0 (σ q v p v + Q 2 ), ( + λ)ϕ = σ ϕ p ϕ + α(p) Q, were Q = 1 e ix0/ T,0 Q2 satisfies Q Ḣs C s and were ( ) d 1 (7.6) α(p) := 1 + 2q = 1 p. 2 We now sum up wat we ave proven in a proposition. Consider te objects we ave just defined : v = v 0 + q v 1, ϕ = e ix0/ T,0 v and λ() = 2 E 0 1 E 1 1+q. Ten we can state Proposition 7.4. Let 2/(d 1) p < 4/(d 1) and α(p) be given by (7.6). Ten te function ϕ satisfies te equation ( + λ() ) ϕ = σ ϕ p ϕ + α(p) Q(), were Q() is an error term wic satisfies Q() Ḣs s, for all s Te case 1/2 < q 1, i.e. 0 p < 2 d 1. We again construct v 0 as a Gaussian beam using te quantum monodromy operator. We ten set G 1 = E 1 v 0 + Lv 0, wic is smoot wit compact wavefront set contained in te wavefront set of v 0, so no pase space cutoff is necessary to apply te inomogeneous argument to get v 1 wit wavefront set contained in te wavefront set of v 0. Now let G 2 = E 2 v q E 1 v 1 + 1/2 E 2 v 1 + σ v 0 p v q Lv 1, and solve for v 2 as in te previous subsection. Tis is possible since v 1 is ortogonal to v 0 by construction. We ten ave were (2iX 0 + Γ R 2 (z) E 0 q E 1 )v 2 = E 2 v q E 1 v 1 + 1/2 E 2 v 1 + σ v 0 p v q Lv 1 + Q 1, Q 1 Ḣr C (1/2)(3/2+p r). Letting v = v 0 + 1/2 v 1 + q v 2 and E = E 0 + 1/2 E 1 + q E 2, we ave solved were We now ave te remainder estimate (2iX 0 + Γ R 2 (z) E)v = q σ v 0 p v 0 + 1/2 Lv 0 + Lv 1 + Q 1 = q σ v p v + 1/2 Lv + Q 2, Q 2 = q Q1 1/2+q Lv 2 + O( 1/2+q v p+1 ). Q 2 Ḣs C s/2+min{1/2+q,q+3/4+p/2} = C s/2+1/2+q.

21 NONLINEAR QUASIMODES 21 Recalling te definitions, ϕ := e ix0/ T,0 v satisfies (i) (ii) (iii) and WF,1/2 δ,0 ϕ { x ɛ 1/2 δ, ξ ɛ}; ϕ L 2 1, D l x 0 ϕ L 2 l, D l x ϕ L 2 C l/2 ; ( + λ)ϕ = 1 e ix0/ T 0 e ix0/ T 1 0 T 0 T 1 0 ϕ = 1 e ix0/ T 0 (2iX 0 + Γ R 2 E 1/2 L)v = 1 e ix0/ T 0 (σ q v p v + Q 2 ), or ( + λ)ϕ = σ ϕ p ϕ + α(p) Q, were Q = 1 e ix0/ T 0 Q2 satisfies Q Ḣs C s and were (7.7) α(p) := 1/2 + q = 1 ( ) d 1 2 p. 4 Once again, by construction we ave for j = 0, 1, 2. X j 0 v j L 2 v j L 2 Consider v = v v 1 + q v 2, ϕ = e ix0/ T,0 v and λ() = 2 E 0 1 E E 2 1+q defined previously, ten we ave proven Proposition 7.5. Let 0 < p 2/(d 1) and α(p) be given by (7.7). Ten te function ϕ satisfies te equation ( + λ() ) ϕ = σ ϕ p ϕ + α(p) Q(), were Q() is an error term wic satisfies Q() Ḣs s, for all s Higer order expansion and proof of Tereorem Te two dimensional cubic equation. We first deal wit te simpler case d = 2, p = 2 and s = 0. As in te previous section, we define q = 1 p(d 1)/4, tus for tis coice we ave q = 1 2 allowing us to matc powers of te asymptotic parameters in a canonical way. Te general algoritm for any smoot nonlinearity arising wen rescaling in te appropriate H s space will follow similarly. Using (6.4) and te Taylor expansions of te geometric components g ij and Γ i jk for i, j, k = 0,..., d in (6.3), we look for an asymptotic series solution of te form for N sufficiently large. v = v v1 + 1 v m 1 2 vm + + N 1 2 vn + ṽ,

22 22 P. ALBIN, H. CHRISTIANSON, J.L. MARZUOLA, AND L. THOMANN Ten, we ave te following equations: were 0 : (2iX 0 + Γ R 2 (z) E 0 )v 0 = 0, 1 2 : (2iX0 + Γ R 2 (z) E 0 )v 1 = E 1 v 0 + (iδ jk Γ0 jk1 2i g k0 zk + R 3 (z))v 0 + σ v 0 2 v 0, 1 : (2iX 0 + Γ R 2 (z) E 0 )v 2 = E 2 v 0 + E 1 v 1 + (iδ jk Γ0 jk1 2i g k0 zk + R 3 (z))v 1. +(iδ jk Γ0 jk2 + R 4 )v 0 + σ(2 v 0 2 v 1 + v 2 0 v 1 ), m 2 : (2iX0 + Γ R 2 (z) E 0 )v m = m 1 + j=0 m 1 j=0 (f z k j,m (z) z k + f X0 j,m X 0 + f 1 j,m(z))v j, j=1 E m j v j + σ( m 1 j,k,l=0., N N N 2 : (2iX0 + Γ R 2 (z) E 0 j 2 Ej )v N = E N j v j + σ( N 1 + j=0 for j, m = 0,..., N and j=0 (f z k j,n (z) z k + f X0 j,n X 0 + f 1 j,n (z))v j + P N v, f z k j,m = O N ( z m j ), f X0 j,m = O N( z m j ), f 1 j,m = O N ( z m j ) P N = O( N/2 2 z N ) + X 0 X 0 + O( N/2 z N ) 1/2 zk X 0 (c m jklv j v k v l )) N 1 j,k,l=0 (c N jklv j v k v l )) +O( N/2 3/2 z N ) zk + O( N/2 1 z N ) + O( N/2 z N )X 0 + O( N/2 z N ) 1/2 zk. Note tat all constants ave implicit dependence upon N relating te number of terms in te expansion at eac order. Te expansion is valid provided first of all tat N j 2 Ej < E 0 j=1 in order to justify te solvability of te O( N/2 ) equation. Remark 7.6. We note ere tat in tis expansion, te sign of σ can effect te sign and value of E 1, wic will impact te remaining asymptotic expansion and in particular te order of quasimode expansion possible. It is possible tat te focussing/defocussing problem enters in to te stability analysis of tese quasimodes troug tis point. Applying Proposition 7.2 at eac asymptotic order and bounds similar to tose in Lemma 6.1 at order N/2 as in Section 7.2.1, we ave by a simple calculation tat v is a quasimode for te nonlinear elliptic equation wit remainder Q N suc tat Q N Ḣs C N s 1+N/2.

23 NONLINEAR QUASIMODES 23 As a result, for sufficiently smoot nonlinearities, one is capable of constructing iger order asymptotic expansions and ence a quasimode of iger order accuracy Te general case. Let d 2, p 2N and s 0. We define ere q s = 1 + p(s d 1 4 ). Assume tat p( d 1 4 s) < 1, or equivalently tat q s > 0. Firstly, write w = s v. Ten v as to satisfy (6.4) but were te power in front of te nonlinearity is qs. Hence, we can look for v and E of te form N N v = j/2+lqs v j,l + ṽ and E = j/2+lqs E j,l + Ẽ. j,l=0 j,l=0 Since q s > 0, te nonlinearity does not affect te equation giving v 0,0, and we ave (2iX 0 + Γ R 2 (z) E 0,0 )v 0,0 = 0, wic is te same equation as before. Ten, using Taylor expansions, we write all te equations, similarly to te previous case, in powers of j/2+lqs. Again, we can solve eac equation and obtain bounds of te solutions and of te error terms. Moreover, it is clear tat we can go as far as we want in te asymptotics, so tat we can construct a O( ) quasimode, and tis proves Teorem Error estimates 8.1. Te regular case. In tis section, we assume tat p is an even integer. Fix an integer k > d/2 (te fact tat k is an integer is not necessary). We ten define te semiclassical norm f H k = ( 1 2 ) k/2 f L 2 (M). In te previous section we ave sown te following : Given α R, tere exist two functions ϕ H k (M) and Q() H k (M) and λ() R so tat ( ) + λ() ϕ = σ ϕ p ϕ + α Q(), were (8.1) α > d s + p( d 1 + s). 4 4 Moreover, microlocally te function ϕ takes te form (8.2) ϕ (σ, x, ) = d 1 4 +s e iσ/ f(σ, 1/2 x, ), and we ave Q() H k C. We set u app (t, ) = e itλ() ϕ. Ten if we denote by Q() := e itλ Q(), te following equation is satisfied (8.3) i t u app + u app = σ u app p u app + α Q(). Proposition 8.1. Let s 0. Consider te function ϕ given by (8.2). Let u be solution of { i t u + u = σ u p u, (8.4) u(0, ) = ϕ. Assume tat α is as in (8.1). Ten tere exists C > 0 and c 0 > 0 independent of so tat d 1 p( for 0 T c 0 4 s) ln( 1 ). u u app L ([0,T ];H s (M)) C (d+1)/4,

24 24 P. ALBIN, H. CHRISTIANSON, J.L. MARZUOLA, AND L. THOMANN Tis result sows tat u app is a good approximation of u, provided tat te quasimode ϕ as been computed at a sufficient order α. Proof. Here we follow te main lines of [To08a, Corollary 3.3]. Wit te Leibniz rule and interpolation we ceck tat for all f H k (M) and g W k, (M) (8.5) f g H k f H k g L (M) + f L 2 (M) ( 1 2 ) k/2 g L (M). Moreover, as k > d/2, for all f 1, f 2 H k (M) (8.6) f 1 f 2 H k d/2 f 1 H k f 2 H k. Let u be te solution of (8.4) and define w = u u app. Ten, by (8.3), w satisfies { i t w + w = σ ( w + u app p (w + u app ) u app p ) u app α Q() (8.7) w(0, x) = 0. We expand te r..s. of (8.7), apply te operator ( 1 2 ) k/2 to te equation, and take te L 2 - scalar product wit ( 1 2 ) k/2 w. Ten we obtain (8.8) d dt w H p+1 w j u p+1 j k app H k + α. We now ave to estimate te terms w j u p+1 j app H k, for 1 j p + 1. From (8.5) we deduce j=1 (8.9) w j u p+1 j app H k w j H k u p+1 j app L (M) + w j L 2 (M) ( 1 2 ) k/2 u p+1 j app L (M). By (8.6), and as we ave (8.10) u p+1 j d 1 (p+1 j)( app L (M) tus inequality (8.9) yields Terefore, from (8.8) we ave 4 +s), ( 1 2 ) k/2 u p+1 j app w j u p+1 j app H k d(j 1)/2 d 1 (p+1 j)( 4 +s) w j. H k d dt w d 1 H k p( 4 +s) w H k + dp/2 w p+1 + α. H k Observe tat w(0) H k = 0. Now, for times t so tat (8.11) dp/2 w p+1 H k d 1 p( 4 +s) w H k, d 1 (p+1 j)( L (M) 4 +s), i.e. w H k C (d+1)/4+s, we can remove te nonlinear term in (8.9), and by te Gronwall Lemma, d 1 α p( (8.12) w H k C 4 +s) d 1 Cp( 4 e +s)t. d 1 p( If c 0 > 0 is small enoug, and t c 0 4 s) ln 1, ten d 1 α p( C 4 +s) d 1 Cp( 4 e +s)t C (d+1)/4+s, so tat inequality (8.11) is satisfied. By te usual bootstrap argument, we infer tat for all d 1 p( t c 0 4 s) ln 1