aa r X i v : . [ m a t h . A P ] A p r Long time behavior of the NLS-Szeg˝o equation
Ruoci Sun ∗ April 25, 2019
Abstract
We are interested in the influence of filtering the positive Fourier modes to the integrablenon linear Schr¨odinger equation. Equivalently, we want to study the effect of dispersion added to thecubic Szeg˝o equation, leading to the NLS-Szeg˝o equation on the circle S i∂ t u + ǫ α ∂ x u = Π( | u | u ) , < ǫ < , α ≥ . There are two sets of results in this paper. The first result concerns the long time Sobolev estimatesfor small data. The second set of results concerns the orbital stability of plane wave solutions. Someinstability results are also obtained, leading to the wave turbulence phenomenon.
Keywords
Cubic Schr¨odinger equation, Szeg˝o projector, small dispersion, stability, wave turbulence,Birkhoff normal form
Contents α ≥ α > ≤ α < ≤ α < ∗ Laboratoire de Math´ematiques d’Orsay, Univ. Paris-Sud XI, CNRS, Universit´e Paris-Saclay, F-91405 Orsay, France([email protected]). Orbital stability of the traveling plane wave e m Theorem . H s -stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.2.1 Proof of Proposition . H s -stability in the case α = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.3.1 The Birkhoff normal form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.3.2 End of the proof of Theorem . We consider the NLS-Szeg˝o equation defined on the circle S i∂ t u + ∂ x u = Π( | u | u ) , u (0 , · ) = u . (1.1)Here Π : L ( S ) → L ( S ) denotes the orthogonal projector from L ( S ) onto the space of L boundaryvalues of holomorphic functions on the unit disc,Π : X k ∈ Z u k e ikx X k ≥ u k e ikx . We denote by L := Π( L ( S )) ⊂ L ( S ), H s + := H s ( S ) T L , for all s ≥
0, and C ∞ + := C ∞ ( S ) T L . The NLS-Szeg˝o equation can be seen as the combination of two completely integrable systems: thedefocusing cubic Schr¨odinger equation i∂ t u + ∂ x u = | u | u, ( t, x ) ∈ R × S , (1.2)and the cubic Szeg˝o equation i∂ t V = Π( | V | V ) , ( t, x ) ∈ R × S . (1.3)They have both a Lax Pair structure and the action-angle coordinates, which can be used to obtaintheir explicit formulas with the inversed spectral method(see Zakharov–Shabat [28], Faddeev–Takhtajan[7], Gr´ebert–Kappeler [19], G´erard [11], for the NLS equation and G´erard–Grellier [12 , , ,
17] for thecubic Szeg˝o equation). However, these two Lax pairs cannot be combined in order to give a Lax pair for(1 . .
2) has a sequence of conservation laws controlling every Sobolev norms(see Faddeev–Takhtajan [7], Gr´ebert–Kappeler [19], G´erard [11]), so all the solutions are uniformly bounded in every H s space. Moreover, Gr´ebert and Kappeler [19] have proved the existence of the global Birkhoff coordi-nates for NLS equation. So the solutions of (1 .
2) are actually almost periodic on R valued into H s ( S ).2ompared to (1 . , , G δ dense subset of initial data in C ∞ + , the solutions may blow up in H s , for every s > with super–polynomialgrowth on some sequence of times, while they go back to their initial data on another sequence of timestending to infinity. However, all the H -solutions are almost periodic. (see also Theorem . Remark 1.1.
Consider the following equation without the Szeg˝o projector Π on S : ( i∂ t V = | V | V,V (0 , · ) = V . (1.4) Then V ( t, x ) = e it | V | V ( x ) and we have k V ( t ) k H s ≃ | t | s , for all s ≥ , if | V | is not a constant function.Hence, the Szeg˝o projector both accelerates the energy transfer to high frequencies, and facilitates thetransition to low frequencies for (1 . . One wonders about whether filtering the positive Fourier modes can change the long time Sobolev esti-mates of the cubic defocusing Schr¨odinger equation. So we introduce equation (1 . ∂ x to its linear part.In order to see the gradual change of the dispersion, we add the parameter ǫ α in front of the Laplacian ∂ x to get a more general model, the NLS-Szeg˝o equation (with small dispersion): i∂ t u + ǫ α ∂ x u = Π( | u | u ) , u (0 , · ) = u , < ǫ < , α ≥ . (1.5)Equation (1 .
1) is the special case α = 0 for (1 . L with the canonical symplectic form ω ( u, v ) = Im R S u ¯ v π . Equation (1 .
5) has the Hamiltonianformalism with the energy functional E α,ǫ ( u ) = ǫ α k ∂ x u k L + 14 k u k L , u ∈ H . (1.6)Besides E α,ǫ , equation (1 .
5) has two other conservation laws, ( Q ( u ) = k u k L ,I ( u ) = Im R S u∂ x u = k u k H , which give the estimate of the solution for low frequencies:sup t ∈ R k u ( t ) k H s ≤ k u k − sL k u k sH , ∀ s ∈ [0 ,
12 ] . Proceeding as in the case of equation (1 . Theorem
II.5.16 in Boyer–Fabrie [3]) and theTrudinger type inequality (see Yudovich [27], Vladimirov [26], Ogawa [24] and G´erard–Grellier [12]).Its well-posedness problem in low frequency Sobolev spaces can be dealt with Strichartz’s inequalityintroduced in Bourgain [2]. Only the high frequency Sobolev estimates are considered in this paper.
Proposition 1.2.
For every s ≥ , given u ∈ H s + , there exists a unique solution u ∈ C ( R , H s + ) of (1 . such that u (0) = u . For every T > , the mapping u ∈ H s + u ∈ C ([ − T, T ] , H s + ) is continuous. .2 Main results The first result concerns the long time stability around the null solution of the NLS-Szeg˝o equation (1 . u is bounded by ǫ , we look for a time interval I αǫ , in which the solution u ( t ) is stillbounded by O ( ǫ ). Now we state the first result of this paper. Theorem 1.3.
For every s > , there exist two constants a s ∈ (0 , and K s > such that for all < ǫ ≪ and u ∈ H s + , if k u k H s = ǫ and u denotes the solution of (1 . , then ( sup | t |≤ asǫ − α k u ( t ) k H s ≤ K s ǫ, if α ∈ [0 , | t |≤ asǫ k u ( t ) k H s ≤ K s ǫ, if α > . (1.7) Moreover, the time interval I αǫ = [ − a s ǫ , a s ǫ ] is maximal for the case α > and s ≥ in the followingsense: for every < ǫ ≪ , there exists u ǫ ∈ C ∞ + such that k u ǫ k H s ≃ ǫ and for every β > , we have sup | t |≤ ǫ β k u ( t ) k H s & ǫ | ln ǫ | ≫ ǫ, u (0) = u ǫ . Remark 1.4.
In the case α ∈ [0 , , the proof is based on the Birkhoff normal form method, similarlyto Bambusi [1] , Gr´ebert [18] , G´erard–Grellier [13] and Faou–Gauckler–Lubich [8] for instance. However,the time interval [ − a s ǫ − α , a s ǫ − α ] may not be optimal. The resonant term of indices in the homologicalequation can not be cancelled by the Birkhoff normal form transform.(see subsubsection . . ) The second set of results concerns the long time H s -estimates for the solutions of (1 . e m : x e imx , for some m ∈ N and s ≥
1. Let u = u ( t, x ) be thesolution of equation (1 .
5) such that k u (0) − e m k H s = ǫ . Its energy functional (1 .
6) gives the followingestimate: sup t ∈ R k u ( t ) k H . k u k H ǫ − α , ∀ < ǫ < , α ≥ . (1.8)However, no information on the stability of the plane waves e m is obtained from (1 .
8) during the process ǫ → + . Consider the super-polynomial growth of Sobolev norms in the cubic Szeg˝o equation case(see G´erard–Grellier [15 ,
17] and
Proposition . .
5) depends on the level of its dispersion. We begin with three long time stabilityresults for the polynomial dispersion ǫ α ∂ x case with 0 ≤ α ≤
2. The following theorem indicates H -orbital stability of the traveling waves e m for equation (1 . Theorem 1.5.
For all ǫ ∈ (0 , , α ∈ [0 , and m ∈ N , there exists C m > such that if k u (0) − e m k H = ǫ ,then we have sup t ∈ R inf θ ∈ R k u ( t ) − e iθ e m k H ≤ C m ǫ − α . For each t ∈ R , the infimum can be attained when θ = arg u m ( t ). A similar result is established byZhidkov [29, Sect. 3 .
3] and Gallay–Haragus [9 ,
10] for the 1D cubic Schr¨odinger equation. In smalldispersion case,
Theorem . . S α,ǫ thenon linear evolution group defined by (1 .
5) on H + . In other words, for every φ ∈ H + , t S α,ǫ ( t ) φ isthe solution u ∈ C ( R , H + ) of equation (1 .
5) such that u (0) = φ . Corollary 1.6.
For every m ∈ N , we have sup <ǫ< ≤ α ≤ sup k φ − e m k H ≤ ǫ sup t ∈ R k S α,ǫ ( t ) φ k H < ∞ . Proposition . , , ǫ α ∂ x counteracts thewave turbulence phenomenon in H norm for equation (1 . ≤ α ≤
2. After the change of variable u ( t ) = e i arg u m ( t ) ( e m + ǫ − α v ( t )), we use a bootstrap argument to get long time orbital stability of thetraveling waves e m with respect to higher Sobolev norms. Proposition 1.7.
For all s ≥ and m ∈ N , there exist two constants b m,s ∈ (0 , and L m,s > suchthat if ≤ α < and k u (0) − e m k H s = ǫ ∈ (0 , , then we have sup | t |≤ bm,sǫ − α inf θ ∈ R k u ( t ) − e iθ e m k H s ≤ L m,s ǫ − α . (1.9)We also look for a larger time interval in which the estimate (1 .
9) holds, by using the Birkhoff normalform transformation. But the coefficients in front of the high frequency Fourier modes in the homologicalequation may be arbitrarily large, if α ∈ (0 , α = 0 and considerequation (1 . i∂ t u + ∂ x u = Π( | u | u ) , u (0 , · ) = u . Then the time interval can be enlarged as [ − d m,s ǫ , d m,s ǫ ] in this case. Theorem 1.8.
In the case α = 0 , for all s ≥ and m ∈ N , there exist three constants d m,s , ǫ m,s ∈ (0 , and K m,s > such that if k u (0) − e m k H s = ǫ ∈ (0 , ǫ m,s ) , then we have sup | t |≤ dm,sǫ inf θ ∈ R k u ( t ) − e iθ e m k H s ≤ K m,s ǫ. A similar result is obtained in Faou–Gauckler–Lubich [8] for the focusing or defocusing cubic Schr¨odingerequation on the arbitrarily dimensional torus. (see
Section .
1) and (1 . .
5) with respect to the initialdata, if the level of dispersion is exponentially small with respect to the level of perturbation of the planewave e : x e ix . We state the last result of this paper. Theorem 1.9.
There exists a constant
K > such that for all < δ ≪ , we denote by U the solutionof the following NLS-Szeg˝o equation with small dispersion i∂ t U + ν ∂ x U = Π( | U | U ) , U (0 , x ) = e ix + δ, (1.10) where ν = e − πK δ , then we have k U ( t δ ) k H ≃ δ with t δ := πδ √ δ . This H -instability result indicates that the support of the energy functional of equation (1 .
10) is trans-ferred to higher Fourier modes. This phenomenon is similar to the cubic Szeg˝o equation case (seeG´erard–Grellier [15 ,
16, 17]) and the 2D cubic NLS case (see Colliander–Keel–Staffilani–Takaoka–Tao[5]). Compared to
Theorem .
5, adding the low-level dispersion e − πKδ ∂ x fails to change the quality ofwave turbulence phenomenon ( Proposition .
4) for the cubic Szeg˝o equation.The second part of
Theorem . Theorem .
9. Indeed, if α >
2, we rescale u ( t, x ) = ǫU ( ǫ t, x ) with e − πK δ = ν = ǫ α − . Then u solves (1 .
5) with u (0 , x ) = ǫ ( e ix + δ ) and k u ( t δ ǫ ) k H = ǫ k U ( t δ ) k H ≃ ǫδ ≃ ǫ p ( α − | ln ǫ | ≫ ǫ, t δ ǫ ≃ | ln ǫ | ǫ ≪ ǫ β , for all β >
0. However, this method does not work in the critical case α = 2. If u solves i∂ t u + ǫ ∂ x u = Π( | u | u ) , u (0 , x ) = ǫ ( e ix + δ ) , after rescaling U ( t, x ) = ǫ − u ( ǫ − t, x ), we get equation (1 .
10) with ν = 1, leading to (1 .
1) with initialdata U (0 , x ) = e ix + δ . Theorem . Theorem . t ∈ R k u ( t ) k H = O ( ǫ ) , sup | t |≤ d ,sǫ δ k u ( t ) k H s = O ( ǫ ) , ∀ < δ ≪ , ∀ < ǫ < , for every s > . The problem of the optimal time interval in the case α = 2 of Theorem . Section
2, we recall some basic facts of the cubic Szeg˝o equationand its consequences. In
Section
3, we study long time behavior for (1 .
5) with small data and prove
Theorem . Theorem .
3. In
Section
4, we study the orbital stability of the plane waves e m for (1 .
5) for every m ∈ N and give the proof of Theorem . Proposition . Theorem .
8. Wecompare the NLS equation and the NLS-Szeg˝o equation in
Section Acknowledgments
The author would like to express his gratitude towards Patrick G´erard for his deep insight, generousadvice and continuous encouragement. He also would like to thank Jean-Marc Delort, Benoit Gr´ebert,Sandrine Grellier and Thomas Kappeler for useful discussions.
In this section, we recall some results of the cubic Szeg˝o equation i∂ t V = Π( | V | V ) , V (0 , · ) = V . (2.1) Given V ∈ H + , the Hankel operator H V : L → L is defined by H V ( h ) = Π( V ¯ h ) . Given b ∈ L ∞ ( S ), the Toeplitz operator T b : L → L is defined by T b ( h ) = Π( bh ) . Theorem 2.1. (G´erard–Grellier [12] ) Set V ∈ C ( R ; H s + ) for some s > . Then V solves the cubic Szeg˝oequation if and only if H V satisfies the following evolutive equation ∂ t H V = [ B V , H V ] . (2.2) where B V := i H V − iT | V | . In other words, ( L V , B V ) is a Lax pair for the cubic Szeg˝o equation. .
2) yields that the spectrum of the Hankel operator H V is invariant under the flow of thecubic Szeg˝o equation. Thus the quantity Tr | H v | is conserved. A theorem of Peller ([25] Theorem 2 p.454) states that k V k B , ≃ Tr | H V | . Using the embedding theorem H s ֒ → B , ֒ → L ∞ , for any s >
1, we have the following L ∞ estimate ofthe Szeg˝o flow. Corollary 2.2. (G´erard–Grellier [12] ) Assume V ∈ H s + for some s > , then we have sup t ∈ R k V ( t ) k L ∞ . s k V k H s The following theorem indicates its chaotic long time behavior with turbulence phenomenon for generalinitial data.
Theorem 2.3. (G´erard–Grellier [15 , ) 1.There exists a G δ − dense set U ⊂ C ∞ ( S ) T L such that if V ∈ U , then there exist two sequences ( t n ) n ∈ N and ( t ′ n ) n ∈ N tending to infinity such that ( lim n → + ∞ k V ( t n ) k Hs | t n | p = + ∞ , ∀ s > , ∀ p ≥ , lim n → + ∞ V ( t ′ n ) = V . V ∈ H + , the mapping t ∈ R V ( t ) ∈ H + is almost periodic. In order to prove the optimality of the case α >
Theorem .
3, we compare the solution u of theNLS-Szeg˝o with small dispersion to the solution of the cubic Szeg˝o equation with some special initialdata. Set V = V δ := δ + e ix , we denote V δ the solution of (2 . V δ ( t, x ) = a δ ( t ) e ix + b δ ( t )1 − p δ ( t ) e ix , (2.3)where a δ ( t ) = e − it (1+ δ ) , b δ ( t ) = e − it (1+ δ ) ( δ cos( ωt ) − i δ √ δ sin( ωt )) ,p δ ( t ) = − i √ δ sin( ωt ) e − itδ , ω = δ r δ . Proposition 2.4. (G´erard–Grellier [12 , , ) For < δ ≪ , set t δ := π ω = πδ √ δ ∼ π δ . Let V δ bethe solution of (2 . with V δ (0 , x ) = e ix + δ , then we have the following estimate k V δ ( t ) k H s . s k V δ ( t δ ) k H s ≃ s δ s − , ∀ t ∈ [0 , t δ ] . for every s > . roof. Expanding formula (2 .
3) as Fourier series, we have k V δ ( t ) k H s ≃ s | a δ ( t ) + b δ ( t ) p δ ( t ) | (1 − | p δ ( t ) | ) s +1 = k V δ ( t ) k H (1 − | p δ ( t ) | ) s − with k V δ ( t ) k H = | a δ ( t ) + b δ ( t ) p δ ( t ) | (1 − | p δ ( t ) | ) = k V δ (0) k H = 1 . By the explicit formula of p δ , we have k V δ ( t ) k H s ≃ s (cid:18) δ ( ωt ) + δ (cid:19) s − ≤ k V δ ( t δ ) k H s ≃ C s δ s − with t δ := π ω = πδ √ δ . α ≥ For all s > , consider the NLS-Szeg˝o equation with small dispersion and small data. i∂ t u + ǫ α ∂ x u = Π( | u | u ) , k u (0) k H s = ǫ, < ǫ < , α ≥ . (3.1)At first, we give the proof of the time interval I αǫ = [ − a s ǫ , a s ǫ ] in the case α ≥ Theorem .
3, whichis based on a bootstrap argument. Then we prove the maximality of I αǫ . Let a, b, T > , m > and M : [0 , T ] −→ R + be a continuous function satisfying M ( τ ) ≤ a + bM ( τ ) m , for all τ ∈ [0 , T ] Assume that ( mb ) m − M (0) ≤ and ( mb ) m − a ≤ m − m . Then M ( τ ) ≤ mm − a for all τ ∈ [0 , T ] .Proof. The function f m : z ∈ R + z − bz m attains its maximum at the critical point z c = ( mb ) − m − . f m ( z c ) = m − m ( mb ) − m − . Since a ≤ max z ≥ f m ( z ) = f m ( z c ), there exists z − ≤ z c ≤ z + such that { z ≥ f m ( z ) ≤ a } = [0 , z − ] ∪ [ z + , + ∞ [and f m ( z ± ) = a . Since f m ( M ( τ )) ≤ a , ∀ ≤ τ ≤ T and M (0) ≤ z c , we have M ([0 , T ]) ⊂ [0 , z − ]. By theconcavity of f m on [0 , + ∞ [, we have f m ( z ) ≥ f m ( z c ) z c z for all z ∈ [0 , z c ]. Consequently, M ( t ) ≤ z − ≤ mm − a ,for all 0 ≤ t ≤ T . 8 roof of of estimate (1 . in the case α > . For all α ≥ ǫ ∈ (0 ,
1) fixed, we rescale u as u = ǫµ ,equation (3 .
1) becomes ( i∂ t µ + ǫ α ∂ x µ = ǫ Π( | µ | µ ) , k µ (0) k H s = 1 . (3.2)Duhamel’s formula of equation (3 .
2) gives the following estimate:sup ≤ τ ≤ t k µ ( τ ) k H s ≤ k µ (0) k H s + C s ǫ t sup ≤ τ ≤ t k µ ( τ ) k H s (3.3)Here C s denotes the Sobolev constant in the inequality k| µ | µ k H s ≤ C s k µ k H s . We choose a s = C s andthe following estimate holds sup | t |≤ asǫ k u ( t ) k H s ≤ ǫ, ∀ α ≥ , (3.4)by using Lemma . m = 3, T = a s ǫ , a = M (0) = 1, b = C s ǫ T and M ( t ) = sup ≤ τ ≤ t k µ ( τ ) k H s . α > I αǫ in which estimate (3 .
4) holds, we set u (0 , x ) = ǫ ( e ix + δ ) and rescale u ( t, x ) = ǫU ( ǫ t, x ). Then, we have i∂ t U + ν ∂ x U = Π( | U | U ) , U (0 , x ) = e ix + δ, where ν := ǫ α − . Since the optimality is a consequence of Theorem .
9, we prove at first
Theorem . U to the solution of the cubic Szeg˝o equation with the same initial data, i∂ t V = Π( | V | V ) , V (0 , x ) = e ix + δ. Proof of
Theorem . . We shall estimate their difference r ( t, x ) := U ( t, x ) − V ( t, x ), which satisfies thefollowing equation i∂ t r + ν ∂ x r = − ν ∂ x V + Π( V r + 2 | V | r ) + Q ( r ) , r (0) = 0 , (3.5)with Q ( r ) := Π( V r + 2 V | r | + | r | r ). Thus, we can calculate the derivative of k r ( t ) k H , ∂ t k r ( t ) k H = ∂ t k r ( t ) k L + ∂ t k ∂ x r ( t ) k L =2Im h i∂ t r ( t ) , r ( t ) i L + 2Im h ∂ x ( i∂ t r ( t )) , ∂ x r ( t ) i L =2Im Z S ν ∂ x V ∂ x r + V r − V | r | r + 2Im Z S − ν ∂ x V ∂ x r + V ( ∂ x r ) + 2 V ∂ x V r∂ x r + 4Re( V ∂ x V ) r∂ x r + 2Im Z S ∂ x V r ∂ x r + 2 V r | ∂ x r | + 2 ∂ x V | r | ∂ x r + 4 V Re( r∂ x r ) ∂ x r + r ( ∂ x r ) . (cid:12)(cid:12)(cid:12) ∂ t k r ( t ) k H (cid:12)(cid:12)(cid:12) ≤ ν k ∂ x r k L ( k ∂ x V k L + k ∂ x V k L ) + 2 k V k L ∞ k r k H + 2 k V k L ∞ k r k L ∞ k r k L + 12 k V k L ∞ k r k L ∞ k ∂ x V k L k ∂ x r k L + 6 k r k L ∞ k ∂ x V k L k ∂ x r k L + 12 k V k L ∞ k r k L ∞ k ∂ x r k L + 2 k r k L ∞ k ∂ x r k L ≤ ν (2 k ∂ x r k L + k ∂ x V k L + k ∂ x V k L ) + 2 k V k L ∞ k r k H + 12 k V k L ∞ k ∂ x V k L k r k L ∞ k ∂ x r k L + O ( k r k H ) . The L ∞ estimate of V is given by Corollary . H s estimate of V is given by Proposition . s > . Thus, we have M ∞ := sup <δ< sup t ∈ R k V ( t ) k L ∞ < + ∞ , and there exist C , C > k ∂ x V ( t ) k L ≤ C δ , k ∂ x V ( t ) k L ≤ C δ , for all 0 ≤ t ≤ t δ = πδ √ δ . We use a bootstrap argument to deal with the term O ( k r k H ). Set T := sup { t > ≤ τ ≤ t k r ( τ ) k H ≤ } , then we have sup ≤ t ≤ T k r ( t ) k L ∞ ≤ C sup ≤ t ≤ T k r ( t ) k H ≤ C, where C denotes the Sobolev constant. Consequently, for all 0 ≤ t ≤ min( T, t δ ), we have (cid:12)(cid:12)(cid:12) ∂ t k r ( t ) k H (cid:12)(cid:12)(cid:12) ≤ ν ( k ∂ x V k L + k ∂ x V k L )+ k r k H (2 ν + 2 k V k L ∞ + 12 C k V k L ∞ k ∂ x V k L + 6 C k r k L ∞ k ∂ x V k L + 12 k V k L ∞ k r k L ∞ + 2 k r k L ∞ ) ≤ ν ( C δ + C δ ) + (2 + 2 M ∞ + 12 CM ∞ + 2 C + (12 CM ∞ + 6 C ) C δ ) k r k H ≤ K ( ν δ + k r ( t ) k H δ ) , ∀ < δ, ν < , with K := max( C + C , M ∞ + 12 CM ∞ + 2 C + (12 CM ∞ + 6 C ) C ). We set ν = ǫ α − = e − πK δ ⇐⇒ δ = q πK ( α − | ln ǫ | . Using Gr¨onwall’s inequality, we deduce that k r ( t ) k H ≤ ν δ e πK δ = δ − e − πK δ ≪ ≪ δ − , ∀ ≤ t ≤ t δ , ∀ < δ ≪ . Since k V ( t δ ) k H ≃ δ by Theorem .
4, we have k U ( t δ ) k H = k V ( t δ ) + r ( t δ ) k H ≃ δ .10ix α >
2, for every 0 < ǫ ≪
1, we set δ = δ α,ǫ := q πK ( α − | ln ǫ | ≪ , T α,ǫ := t δ ǫ = π ǫ δ q δ ≃ √ ( α − | ln ǫ | ǫ . Then we have k u ( T α,ǫ ) k H ≃ ǫ p ( α − | ln ǫ | ≫ ǫ , while u (0 , x ) = ǫ ( e ix + δ ). Then the optimality of I αǫ = [ − a s ǫ , a s ǫ ] is obtained. ≤ α < We assume at first that u (0) ∈ C ∞ + so that the energy functional of (3 . E α,ǫ ( u ) = ǫ α k ∂ x u k L + 14 k u k L is well defined. For general initial data u (0) ∈ H s + , if s ∈ ( , C ∞ + = H s + .We rescale u ( t, x ) ǫ − α u ( − ǫ α t, x ), then the equation (3 .
1) is reduced to the case α = 0. It suffices toprove the following estimate sup | t |≤ asǫ k u ( t ) k H s = O ( ǫ )if u solves i∂ t u + ∂ x u = Π( | u | u ) with k u (0) k H s = ǫ . The study of the resonant set of the NLS-Szeg˝o equation is necessary before Birkhoff normal form trans-formation. We refer to Eliasson–Kuksin [6] to see the analysis of the resonances for a more general nonlinear term and KAM theorem for the NLS equation.We use again the change of variable u = ǫµ and we rewrite Duhamel’s formula of µ with η ( t ) = P k ≥ η k ( t ) e ikx := e − it∂ x µ ( t ). Then we have η k ( t ) = µ ( t ) − iǫ X k − k + k − k =0 Z t e − iτ ( k − k + k − k ) η k ( τ ) η k ( τ ) η k ( τ )d τ, for all k ≥
0. Recall the classical identification of the resonant set ( k − k + k − k = 0 k − k + k − k = 0 . ⇐⇒ ( k = k k = k or ( k = k k = k . In order to cancel all the resonances, we apply the transformation v ( t ) := e itǫ k µ (0) k L µ ( t ). As k µ k L isa conservation law, we have i∂ t v ( t ) + ∂ x v ( t ) = ǫ (cid:2) Π( | v ( t ) | v ( t )) − k v ( t ) k L v ( t ) (cid:3) , ∀ t ∈ R . (3.6)The equation (3 .
6) can be seen as the Hamiltonian system with respect to the energy function H ǫ ( v ) = 12 k ∂ x v k L + ǫ (cid:0) k v k L − k v k L (cid:1) =: H ( v ) + ǫ R ( v ) . (3.7)Then we have R ( v ) = P k − k + k − k =0 k − k + k − k =0 v k v k v k v k − P k ≥ | v k | ! .11 .2.2 The Birkhoff normal form Equation (3 .
6) is transferred to another Hamiltonian equation which is closer to the linear Schr¨odingerequation by Birkhoff normal form method. We try to find a symplectomorphism Ψ ǫ such that the energyfunctional H ǫ is reduced to the Hamiltonian H ǫ ◦ Ψ ǫ ( v ) = H ( v ) + ǫ ˜ R ( v ) + O ( ǫ − α ) , where ˜ R ( v ) = − P k ≥ | v k | . Ψ ǫ is chosen as the value at time 1 of the Hamiltonian flow of some energy ǫ F .We fix the value s > . Recall that, given a smooth real valued function H , we denote X H the Hamiltonianvector field, i.e, d H ( v )( h ) = ω ( h, X H ( v )) . Given two smooth real-valued functions F and G on H s + , their Poisson bracket { F, G } is defined by { F, G } ( v ) := ω ( X F ( v ) , X G ( v )) = 2 i X k ≥ ( ∂ v k F ∂ v k G − ∂ v k G∂ v k F ) ( v ) , (3.8)for all v = P k ≥ v k e ikx ∈ H s + . In particular, if F and G are respectively homogeneous of order p and q ,then their Poisson bracket is homogeneous of order p + q − Lemma 3.2.
Set F ( v ) := P k − k + k − k =0 f k ,k ,k ,k v k v k v k v k , with the coefficients f k ,k ,k ,k = ( i ( k − k + k − k ) , if k − k + k − k = 0 , , otherwise . Thus, F is real-valued and its Hamiltonian field X F is smooth on H s + such that { F, H } + R = ˜ R andthe following estimates hold. ( k X F ( v ) k H s . s k v k H s , k d X F ( v ) k B ( H s ) . s k v k H s , for all v ∈ H s + .Proof. F well defined because sup ( k ,k ,k ,k ) ∈ Z | f k ,k ,k ,k | ≤ , the Sobolev embedding yields that | F ( v ) | ≤ (cid:16)P k ≥ | ˆ u ( k ) | (cid:17) . s k u k H s . The Young’s convolution inequality l ∗ l ∗ l ֒ → l implies that k X F ( v ) k H s . s k v k H s and k d X F ( v ) k B ( H s ) . s k v k H s . Using (3 .
8) and the definition of f k ,k ,k ,k , wehave { F, H } ( v ) = i X k − k + k − k =0 ( k − k + k − k ) f k ,k ,k ,k v k v k v k v k = − X k − k + k − k =0 k − k + k − k =0 v k v k v k v k = − R ( v ) + ˜ R ( v ) . χ σ := exp( ǫ σX F ) the Hamiltonian flow of ǫ F , i.e.,dd σ χ σ ( u ) = ǫ X F ( χ σ ( u )) , χ ( u ) = u. We perform the canonical transformation Ψ ǫ := χ = exp( ǫ X F ). The next lemma will prove the localexistence of χ σ , for | σ | ≤ v and Ψ − ǫ ( v ) Lemma 3.3.
For s > , there exist two constants ρ s , C s > such that for all v ∈ H s + , if ǫ k v k H s ≤ ρ s ,then χ σ ( v ) is well defined on the interval [ − , and the following estimates hold: sup σ ∈ [ − , k χ σ ( v ) k H s ≤ k v k H s , sup σ ∈ [ − , k χ σ ( v ) − v k H s ≤ C s ǫ k v k H s , k d χ σ ( v ) k B ( H s ) ≤ exp( C s ǫ k v k H s | σ | ) , ∀ σ ∈ [ − , . Proof.
The inequality k d X F ( v ) k B ( H s ) ≤ C s k v k H s implies that the Lipschitz coefficient of the mapping v ǫ X F ( v ) is bounded by C s ǫ k v k H s ≤ C s ρ s . If ρ s is sufficiently small, then the Hamiltionian flow( σ, v ) χ σ ( v ) exists on the maximal interval ( − σ ∗ , σ ∗ ), by the Picard-Lindel¨of theorem. Assume that σ ∗ <
1, then
Lemma . χ σ ( v ) = v + ǫ Z σ X F ( χ τ ( v ))d τ, ∀ ≤ σ < σ ∗ . (3.9)yield that sup ≤ τ ≤ σ k χ τ ( v ) k H s ≤ k v k H s + C s σǫ sup ≤ τ ≤ σ k χ τ ( v ) k H s , ∀ ≤ σ < σ ∗ < . By Lemma . M ( t ) = sup ≤ τ ≤ t k χ τ ( v ) k H s , m = 3, a = M (0) = k v k H s and b = C s ǫ , we havesup | σ |≤ σ ∗ k χ σ ( v ) k H s ≤ k v k H s , if ǫ k v k H s ≤ √ C s . This is a contradiction to the blow-up criterion. Hence σ ∗ ≥
1, and we havesup | σ |≤ k χ σ ( v ) k H s ≤ k v k H s , if ǫ k v k H s ≤ ρ s := √ C s .Since k X F ( v ) k H s ≤ C s k v k H s , for all σ ∈ [ − , k χ σ ( v ) − v k H s ≤ | σ | ǫ sup ≤ t ≤| σ | k X F ( χ t ( v )) k H s ≤ C s ǫ sup ≤ t ≤| σ | k χ t ( v ) k H s ≤ C s ǫ k v k H s . if ǫ k v k H s ≤ ρ s . We differentiate equation (3 .
9) and use again
Lemma . k d χ σ ( u ) k B ( H s ) = k Id H s + ǫ Z σ d X F ( χ t ( u ))d χ t ( u )d t k B ( H s ) ≤ ǫ (cid:12)(cid:12)(cid:12) Z σ k d X F ( χ t ( v )) k B ( H s ) k d χ t ( v ) k B ( H s ) d t (cid:12)(cid:12)(cid:12) ≤ C s ǫ k v k H s (cid:12)(cid:12)(cid:12) Z σ k d χ t ( v ) k B ( H s ) d t (cid:12)(cid:12)(cid:12) ≤ e C s ǫ | σ |k v k Hs , ∀ σ ∈ [ − , . ǫ = χ . The normal form of the energy H ǫ is given below. Lemma 3.4.
For s > , there exists a smooth mapping Y : H s + −→ H s + and a constant C ′ s > suchthat ( X H ǫ ◦ Ψ ǫ = X H + ǫ X ˜ R + ǫ Y, k Y ( v ) k H s ≤ C ′ s k v k H s , for all v ∈ H s + such that ǫ k v k H s ≤ ρ s . Let us set w ( t ) := Ψ − ǫ ( v ( t )) , then we have (cid:12)(cid:12)(cid:12) dd t k w ( t ) k H s (cid:12)(cid:12)(cid:12) ≤ C ′ s ǫ k w ( t ) k H s , (3.10) if ǫ k w ( t ) k H s ≤ ρ s .Proof. We expand the energy H ǫ ◦ Ψ ǫ = H ǫ ◦ χ with Taylor’s formula at time σ = 1 around 0. Since χ = Id H s + , one gets H ǫ ◦ χ = H ◦ χ + ǫ R ◦ χ = (cid:18) H + dd σ [ H ◦ χ σ ] | σ =0 + Z (1 − σ ) d d σ [ H ◦ χ σ ]d σ (cid:19) + ǫ R + ǫ Z dd σ [ R ◦ χ σ ]d σ = H + ǫ [ { F, H } + R ] + ǫ Z (1 − σ ) { F, { F, H }} ◦ χ σ + { F, R } ◦ χ σ d σ = H + ǫ ˜ R + ǫ Z h (1 − σ ) { F, ˜ R } + σ { F, R } i ◦ χ σ d σ We set G ( σ ) := (1 − σ ) { F, ˜ R } + σ { F, R } , ∀ σ ∈ [0 , X { F,R } and X { F, ˜ R } ( u ) are homogeneous ofdegree 5 with uniformly bounded coefficients, we have k X G ( σ ) ( v ) k H s ≤ (1 − σ ) k X { F, ˜ R } ( v ) k H s + σ k X { F,R } ( v ) k H s . s k v k H s , ∀ v ∈ H s + , By the chain rule of Hamiltonian vector fields: X G ( σ ) ◦ χ σ ( v ) = d χ − σ ( χ σ ( v )) ◦ X G ( σ ) ( χ σ ( v )) , ∀ v ∈ H s + , ∀ σ ∈ [0 , , (3.11)and Lemma .
3, we have k X G ( σ ) ◦ χ σ ( v ) k H s ≤ k d χ − σ ( χ σ ( v )) k B ( H s ) k X G ( σ ) ( χ σ ( v )) k H s . s e C s ǫ k v k Hs k χ σ ( v ) k H s . s k v k H s , for all v ∈ H s + such that ǫ k v k H s ≤ ρ s . Thus we define Y := R X G ( σ ) ◦ χ σ d σ and we have X H ǫ ◦ Ψ ǫ = X H + ǫ X ˜ R + ǫ Y. If ǫ k v k H s ≤ ρ s , then k Y ( v ) k H s . s k v k H s . 14ince \ X ˜ R ( w )( k ) = − i | w k | w k , ∀ k ≥ w ( t ) = Ψ − ǫ ( v ( t )), we have the following infinite dimensionalHamiltonian system on the Fourier modes: i∂ t w k ( t ) − k w k ( t ) − ǫ | w k ( t ) | w k ( t ) = iǫ \ Y ( w ( t ))( k ) , ∀ k ≥ . (3.12)If ǫ k w ( t ) k H s ≤ ρ s , then we have (cid:12)(cid:12)(cid:12) ∂ t k w ( t ) k H s (cid:12)(cid:12)(cid:12) ≤ ǫ k Y ( w ( t )) k H s k w ( t ) k H s . s ǫ k w ( t ) k H s . ≤ α < Proof.
Recall that w ( t ) = χ − ( v ( t )) and k v (0) k H s = 1. Lemma . k v (0) − w (0) k H s = k v (0) − χ − ( v (0)) k H s ≤ C s ǫ k v (0) k H s ≤ C s . Set K s := 3( C s + 1). Then k w (0) k H s ≤ K s . We define ǫ s := min (cid:18) m s K s , p C s K s (cid:19) and T := sup { t ≥ ≤ τ ≤ t k v ( τ ) k H s ≤ K s } . For all ǫ ∈ (0 , ǫ s ) and t ∈ [0 , T ], we have ǫ k v ( t ) k H s ≤ m s . Hence Lemma . k w ( t ) k H s ≤k v ( t ) k H s + k χ − ( v ( t )) − v ( t ) k H s ≤k v ( t ) k H s + C s ǫ k v ( t ) k H s ≤ K s + 8 C s K s ǫ ≤ K s . So we have ǫ sup ≤ t ≤ T k w ( t ) k H s ≤ m s and (cid:12)(cid:12)(cid:12) dd t k w ( t ) k H s (cid:12)(cid:12)(cid:12) ≤ C ′ s ǫ k w ( t ) k H s , by Lemma .
4. Set a s := K s C ′ s . We can precise the estimate of k w ( t ) k H s by limiting | t | ≤ a s ǫ − : k w ( t ) k H s ≤k w (0) k H s + C ′ s | t | ǫ k w ( t ) k H s ≤ K s C ′ s K s | t | ǫ ≤ K s , for all 0 ≤ t ≤ min( T, a s ǫ ). Then we have k v ( t ) k H s ≤ k w ( t ) k H s + k χ ( w ( t )) − w ( t ) k H s ≤ k w ( t ) k H s + C s ǫ k w ( t ) k H s ≤ K s for all t ∈ [0 , a s ǫ ]. Consequently, we havesup ≤ t ≤ asǫ k u ( t ) k H s = ǫ sup ≤ t ≤ asǫ k v ( t ) k H s ≤ K s ǫ. In the case t <
0, we use the same procedure and we replace t by − t .15 .2.4 The open problem of optimality Let u be the solution of the NLS-Szeg˝o equation i∂ t u + ∂ x u = Π( | u | u ) , k u (0 , · ) k H s = ǫ. (3.13)Recall that H ǫ = H + ǫ R is the energy functional of the equation (3 .
13) with H ( v ) = 12 k ∂ x v k L , R ( v ) = 14 ( k v k L − k v k L ) .χ σ = exp( ǫ σX F ), with F ( v ) := P k − k + k − k =0 f k ,k ,k ,k v k v k v k v k , with the coefficients f k ,k ,k ,k = ( i ( k − k + k − k ) , if k − k + k − k = 0 , , otherwise . We recall also that ˜ R ( v ) = { F, H } ( v ) + R ( v ) = − P k ≥ | v k | .In order to get a longer time interval in which the solution is uniformly bounded by O ( ǫ ), we expand theHamiltonian H ǫ ◦ χ by using the Taylor expansion of higher order to see whether the resonances can becancelled by the Birkhoff normal form method. H ǫ ◦ χ = H ◦ χ + ǫ R ◦ χ = H + ∂ σ ( H ◦ χ σ ) (cid:12)(cid:12) σ =0 + 12 ∂ σ ( H ◦ χ σ ) (cid:12)(cid:12) σ =0 + 12 Z (1 − σ ) ∂ σ ( H ◦ χ σ )d σ + ǫ (cid:18) R + ∂ σ ( R ◦ χ σ ) (cid:12)(cid:12) σ =0 + Z (1 − σ ) ∂ σ ( H ◦ χ σ )d σ (cid:19) = H + ǫ ˜ R + ǫ { F, R + ˜ R } + ǫ Z (1 − σ ) { F, { F, (1 − σ ) ˜ R + (1 + σ ) R }} ◦ χ σ d σ We try to cancel the term ǫ { F, R + ˜ R } by using a canonical transform to H ǫ = H ǫ ◦ χ with the followingfunctional G ( v ) = X k − k + k − k + k − k =0 g k ,k ,k ,k ,k ,k v k v k v k v k v k v k . We want to solve the homological equation { G, H } + { F, R + ˜ R } = 0. We can calculate that12 { F, R + ˜ R } ( v )=2Im X k − k + k − k + k − k =0 k i ,k := k − k + k ≥ f k ,k ,k ,k v k v k v k v k v k v k − X k − k + k − k + k − k =0 k i ,k := k − k + k ≥ ,k = k f k ,k ,k ,k v k v k v k v k v k v k − X k − k + k − k + k − k =0 k i ,k := k − k + k ≥ ,k = k = k f k ,k ,k ,k v k v k v k v k v k v k
16n the first term of the preceding formula, there is a resonance set k − k + k − k + k − k = 0 thatcannot be cancelled by the other two terms. We can see Gr´ebert–Thomann [20] and Haus–Procesi [21]for instance to analyse the resonant set for 6 indices for the quintic NLS equation. { G, H } ( v )= i X k − k + k − k + k − k =0 ( k − k + k − k + k − k ) g k ,k ,k ,k ,k ,k v k v k v k v k v k v k . Thus the resonant subset ( k − k + k − k + k − k = 0 k − k + k − k + k − k = 0should be cancelled before the Birkhoff normal form transform, just like the step µ v = e itǫ k µ (0) k L µ ( t ),which can cancel all the resonances ( k − k + k − k = 0 k − k + k − k = 0before we do the canonical transformation H ǫ ◦ χ . We only know that f k ,k ,k ,k − k + k = f k ,k ,k ,k − k + k if ( k − k + k − k + k − k = 0 k − k + k − k + k − k = 0 . This resonant subset contains the case k = k . The optimality of the time interval for the case 0 ≤ α < m Consider the following NLS-Szeg˝o equation i∂ t u + ǫ α ∂ x u = Π( | u | u ) , < ǫ < , ≤ α ≤ . (4.1)We shall prove at first H -orbital stability of the traveling waves e m , for all m ∈ N . Then, we study theirlong time H s -stability, for all s ≥ . We follow the idea of using conserved quantities mentioned in Gallay–Haragus [9] for equation (4 . Proof.
For all m ∈ N , 0 < ǫ < ≤ α ≤
2, we denote u (0 , x ) = e imx + ǫf ( x ) with k f k H ≤
1. TheNLS-Szeg˝o equation has three conservation laws: Q ( u ( t )) := k u ( t ) k L = k u (0) k L ; P ( u ( t )) := ( Du ( t ) , u ( t )) = P ( u (0)); E α,ǫ ( u ( t )) := ǫ α k ∂ x u ( t ) k L + k u ( t ) k L = E α,ǫ ( u (0)) , D = − i∂ x and ( u, v ) := Re R S ¯ uv . Thus the following quantity is conserved, ǫ α k Du ( t ) − mu ( t ) k L + 14 k| u ( t ) | − k L = E α,ǫ ( u ( t )) − ǫ α mP ( u ( t )) + | m | ǫ α − Q ( u ( t )) + 14= ǫ Z S | Re f ( x ) e − imx | d x + ǫ α k Df − mf k L + ǫ Z S | f ( x ) | Re( f ( x ) e − imx )d x + ǫ k f k L . m ǫ . Then, we have sup t ∈ R k Du ( t ) − mu ( t ) k L . m ǫ − α . Recall that e m ( x ) = e imx , then the following estimateholds, k u ( t ) − u m ( t ) e m k H = X n = m (1 + n ) | u n ( t ) | . m k Du ( t ) − mu ( t ) k L . m ǫ − α . We have inf θ ∈ R k u ( t ) − u m (0) e iθ e m k H = k u ( t ) − u m (0) e i (arg u m ( t ) − arg u m (0)) e m k H =(1 + m ) (cid:12)(cid:12)(cid:12) | u m ( t ) | − | u m (0) | (cid:12)(cid:12)(cid:12) + k u ( t ) − u m ( t ) e m k H and by the conservation of k u ( t ) k L , we have (cid:12)(cid:12)(cid:12) | u m ( t ) | − | u m (0) | (cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12) | u m ( t ) | − | u m (0) | (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) X n = m | u n (0) | − X n = m | u n ( t ) | (cid:12)(cid:12)(cid:12) = max( k u (0) − u m (0) e m k L , k u ( t ) − u m ( t ) e m k L ) . m ǫ − α . Thus sup t ∈ R k u ( t ) − u m (0) e i (arg u m ( t ) − arg u m (0)) e m k H . m ǫ − α . The proof can be finished by u m (0) =1 + ǫf m = 1 + O ( ǫ ).The preceding theorem also holds for the defocusing NLS equation on T d , for d = 1 , , T = R / Z ≃ S .(see Gallay–Haragus [9 , .
3] fora detailed analysis of the stability of plane waves.
Remark 4.1.
Obtaining the estimate sup t ∈ R k u ( t ) − u m ( t ) e m k L ∞ . m ǫ − α by the Sobolev embedding H ( S ) ֒ → L ∞ , we can also proceed by using the following estimate, which is uniform on x and t , | u ( t, x ) | − | u m ( t ) | − O m ( ǫ − α ) . Integrating the preceding term with respect to x , we have || u m ( t ) | − | ≤ k| u ( t ) | − k L + kO m ( ǫ − α ) k L . m ǫ − α Thus sup t ∈ R | u m ( t ) | = 1 + O m ( ǫ − α ) and u m ( t ) = e i arg u m ( t ) + O m ( ǫ − α ) . Then we have sup t ∈ R k u ( t ) − e i arg u m ( t ) e m k H . m ǫ − α . Recall that if z = 1 + O ( ǫ ) then e i arg z = 1 + O ( ǫ ) . .2 Long time H s -stability For every s ≥
1, we suppose that k u (0) − e m k H s ≤ ǫ . Thanks to the estimatesup t ∈ R k u ( t ) − e i arg u m ( t ) e m k H . m ǫ − α , we change the variable u v = v m,α,ǫ ( t, x ) = P n ≥ v n ( t ) e inx ∈ C ∞ ( R × S ) such that u ( t, x ) = e i arg u m ( t ) ( e imx + ǫ − α v ( t, x )) (4.2)to study H s -stability of plane waves e m and we have I m := sup ≤ α ≤ sup <ǫ< sup t ∈ R k v ( t ) k H < + ∞ . (4.3) Proposition 4.2.
For every s ≥ , m ∈ N , ǫ ∈ (0 , and α ∈ [0 , , if u is smooth and solves (4 . with u (0 , x ) = e imx + ǫf ( x ) and k f k H s ≤ , v is defined by formula (4 . , then we have v m ( t ) ∈ R , ∀ t ∈ R , sup t ∈ R | v m ( t ) | . m ǫ min( α , − α ) , sup t ∈ R | ∂ t v m ( t ) | . m ǫ − α , k v (0) k H s . m,s ǫ α . Moreover, there exists a smooth function ϕ = ϕ m : R → R / π Z ≃ S and ǫ ∗ m ∈ (0 , such that for every < − < , we have ( arg u m ( t ) = − (1 + m ǫ α ) t + ǫ min(1 , − α ) ϕ ( t ) , sup ≤ α ≤ − sup <ǫ<ǫ ∗ m sup t ∈ R | ϕ ′ ( t ) | < + ∞ . The parameter v satisfies the following equation i∂ t v + ǫ α ∂ x v − H e imx ( v ) − (1 − m ǫ α + ǫ min(1 , − α ) ϕ ′ ( t )) v = ǫ min( α , − α ) ϕ ′ ( t ) e imx + ǫ − α Π( e − imx v + 2 e imx | v | ) + ǫ − α Π( | v | v ) , (4.4) where H e imx ( v ) := Π[ e imx ¯ v ] denotes the Hankel operator.Proof. Since u m ( t ) = e i arg u m ( t ) (1 + ǫ − α v m ( t )), we have 1 + ǫ − α v m ( t ) = | u m ( t ) | ∈ R . So v m ( t ) ∈ R ,for all t ∈ R . By using the conservation law k · k L and estimate (4 . ǫ Re f m + ǫ = k u (0) k L = k u ( t ) k L = 1 + 2 ǫ − α v m ( t ) + ǫ − α k v ( t ) k L , which yields that sup t ∈ R | v m ( t ) | . m ǫ min( α , − α ) . Recall that u (0 , x ) = X n ≥ u n (0) e inx = e imx + ǫf ( x ) . Then we have u m (0) = 1 + ǫf m = 1 + O ( ǫ ) and | e i arg u m (0) − | . ǫ . Thus we have ǫ − α k v (0) k H s . (1 + m ) s | e i arg u m (0) − | + ǫ k f k H s . m,s ǫ.
19e define θ ( t ) := arg( u m ( t )). Combing the following two formulas i∂ t u + ǫ α ∂ x u = e iθ ( t ) (cid:2) ǫ − α ( i∂ t v + ǫ α ∂ x v − θ ′ ( t ) v ) − ( m ǫ α + θ ′ ( t )) e imx (cid:3) Π[ | u | u ] = e iθ ( t ) h e imx + ǫ − α (2 v + Π( e imx v )) + ǫ − α Π( e − imx v + 2 e imx | v | ) + ǫ − α ) Π( | v | v ) i we obtain that ǫ − α [ i∂ t v + ǫ α ∂ x v − H e imx ( v ) − (2 + θ ′ ( t )) v ]=(1 + m ǫ α + θ ′ ( t )) e imx + ǫ − α Π( e − imx v + 2 e imx | v | ) + ǫ − α ) Π( | v | v ) , (4.5)where H e imx ( v ) := Π[ e imx ¯ v ] denotes the Hankel operator. The Fourier mode v m ( t ) satisfies the followingequation ǫ − α h i∂ t v m ( t ) − m ǫ α v m ( t ) − v m ( t ) − (2 + θ ′ ( t )) v m ( t ) i =1 + m ǫ α + θ ′ ( t ) + ǫ − α Π( e − imx v + 2 e imx | v | ) m ( t ) + ǫ − α ) Π( | v | v ) m ( t ) . Estimate (4 .
3) yields thatsup t ∈ R | ǫ − α Π( e − imx v + 2 e imx | v | ) m ( t ) + ǫ − α ) Π( | v | v ) m ( t ) | . m ǫ − α . Thus, we have ǫ − α (cid:2) i∂ t v m ( t ) − (3 + m ǫ α + θ ′ ( t )) v m ( t ) (cid:3) = 1 + m ǫ α + θ ′ ( t ) + O m ( ǫ − α ) . (4.6)The imaginary part and the real part of (4 .
6) give respectively the two following estimates:sup t ∈ R | ∂ t v m ( t ) | . m ǫ − α ;1 + m ǫ α + θ ′ ( t ) = − ǫ − α v m ( t ) + O m ( ǫ − α )1 + ǫ − α v m ( t ) = O m ( ǫ min(1 , − α ) )1 + O m ( ǫ min(1 , − α ) ) = O m ( ǫ min(1 , − α ) ) . for all 0 < ǫ ≪
1. Then we define ϕ ( t ) := (1+ m ǫ α ) t + θ ( t ) ǫ min(1 , − α ) . Consequently, there exists ǫ ∗ m ∈ (0 ,
1) such that ( arg u m ( t ) = − (1 + m ǫ α ) t + ǫ min(1 , − α ) ϕ ( t )sup ≤ α ≤ − sup <ǫ<ǫ ∗ m sup t ∈ R | ϕ ′ ( t ) | < + ∞ . We replace θ ′ ( t ) by − − m ǫ α + ǫ min(1 , − α ) ϕ ′ ( t ) in (4 .
5) and we obtain (4 . . n ∈ N , we define the projector P n : L → L such that P n ( X k ≥ v k e ikx ) = n X j =0 v k e ikx . Now we prove
Proposition . roof. At the beginning, we suppose that u (0) ∈ C ∞ + . In the general case u (0) ∈ H s + , the proof canbe completed by using the continuity of the flow u (0) u from H s + to C ([ − b s,m ǫ − α , b s,m ǫ − α ] , H s + ). We usethe same transformation u v as (4 . Proposition . A m,s ≥ k v (0) k H s . m,s ǫ α ≤ A m,s . By using estimate (4 . ǫ ∈ (0 , sup t ∈ R k P m ( v ( t )) k H s ≤ (1 + 4 m ) s I m We define that L m,s := max(2(1 + 4 m ) s I m , A m,s + 1) and T := sup { t > ≤ τ ≤ t k v ( τ ) k H s ≤ L m,s } . Rewrite equation (4 .
4) on Fourier modes and we have i∂ t v n − (1 + ( n − m ) ǫ α + ǫ min(1 , − α ) ϕ ′ ( t )) v n = ǫ − α [ Z ( v ( t ))] n , ∀ n ≥ m + 1 , with Z ( v ) = P n ≥ [ Z ( v )] n e inx = Π( e − imx v + 2 e imx | v | ) + ǫ − α Π( | v | v ). Then we have k Z ( v ) − P m ( Z ( v )) k H s . s k v k H s + ǫ − α k v k H s . (4.7)Then we have | ∂ t k v ( t ) − P m ( v ( t )) k H s | ≤ ǫ − α X n ≥ m +1 (1 + n ) s | v n ( t ) || [ Z ( v ( t ))] n |≤ ǫ − α k v ( t ) k H s k Z ( v ( t )) − P m ( Z ( v ( t )) k H s . s ǫ − α k v ( t ) k H s + ǫ − α k v ( t ) k H s . For all t ∈ [0 , T ], we have k v ( t ) k H s = k P m v ( t ) k H s + k v ( t ) − P m ( v ( t )) k H s ≤ (1 + 4 m ) s I m + C s ( ǫ − α k v ( t ) k H s + ǫ − α k v ( t ) k H s ) t + k v (0) k H s ≤ L m,s + 32 C s L m,s ǫ − α t + A m,s . Define b m,s = C s L m,s and we have k v ( t ) k H s ≤ L m,s , for all t ∈ [0 , b s,m ǫ − α ]. The case t < We try to improve
Proposition . v is still boundedby O (1), by using The Birkhoff normal form method. Recall the symplectic form ω ( u, v ) = Im R S uv d θ π on the energy space H and the Poisson bracket for two smooth real-valued functionals F, G : C ∞ + → R { F, G } ( v ) = 2 i X k ≥ ( ∂ v k F ∂ v k G − ∂ v k G∂ v k F ) ( v ) , for all v = P k ≥ v k e ikx ∈ C ∞ + . For all 0 ≤ α < < ǫ ≪
1, equation (4 .
4) has also the Hamiltonianformalism, which is non autonomous. Its energy functional is H m,α,ǫ ( t, v )= H m,α,ǫ ( v ) + ǫ − α H m ( v ) + ǫ − α N ( t, v ) + ǫ min( α , − α ) ϕ ′ ( t )( L m ( v ) + ǫ − α N ( v )) , H m,α,ǫ ( v ) = ǫ α k ∂ x v k L + − m ǫ α k v k L + R S Re( e imx v ) , H m ( v ) = Re( R S e − imx | v | v ) , N p ( v ) = k v k pL p , p = 2 or 4 , L m ( v ) = Re v m . We want to cancel all the high frequencies in the term H m ( v ) by composing H m,α,ǫ and the Hamiltonianflow of some auxiliary functional F m . In order to get the appropriate F m , we need to solve the homologicalsystem ( {F m , H m,α,ǫ } ( v ) + H m ( v ) = R m ( v ) {F m , L m } ( v ) = − ˜ N ( v ) := − P n ≥ m +1 | v n | such that R m depends only on finitely many Fourier modes of v . The remaining coefficient in front of ǫ − α would be R m + ϕ ′ ( t ) ǫ min( α , − α ) ( − ˜ N + N ). One can prove the following proposition.(see also Proposition . Appendix for the proof in the special case α = 0) Proposition 4.3.
For every m ∈ N , we define the following homogenous functional F m of degree : F m ( v ) = X j − l + k = mj,k,l ∈ N Re( a j,l,k v j v l v k ) , ∀ v ∈ [ n ≥ P n ( C ∞ + ) , for some a k,l,j = a j,l,k ∈ C . Then we have the following formula {F m , H m,α,ǫ } ( v ) + H m ( v ) = Re (cid:16) Reson low ( v ) + Reson ≥ m +1 ( v ) (cid:17) , where Reson low ( v )= X ≤ j,k ≤ m c j,j + k − m,k v j v j + k − m v k − X j − l + k = mj,l,k ∈ N ,l ≤ m ia j,l,k v j v k (cid:2) (1 + ( l − m ) ǫ α ) v l + v m − l (cid:3) , for some c j,j + k − m,k = c k,j + k − m,j ∈ C and Reson ≥ m +1 ( v )= X k ≥ m +1 m − X j =0 − ia m − j,m + k − j,k + (1 + 2( m − j )( k − j ) ǫ α ) ia j,k,k + m − j + 1) v j v k v k + m − j + X k ≥ m +1 m − X j =0 − k − m )( m − j ) ǫ α ) ia m − j,m + k − j,k − ia j,k,k + m − j + 1) v m − j v m + k − j v k + X k ≥ m +1 (cid:0) ia m,k,k − ia m,k,k + 1) v m | v k | + ((1 − k − m ) ǫ α ) ia k, k − m,k + 1) v k v k − m (cid:1) + X k ≥ m +1 X j ≥ k +1 − j − m )( k − m ) ǫ α ) ia k,j + k − m,j + 1) v k v j + k − m v j . The term Reson low depends only on the small Fourier modes v , v , · · · , v m . We try to find a boundedsequence ( a j,l,k ) j − l + k = m such that Reson ≥ m +1 = 0 in order to cancel the term H m . However, the22oefficient (1 − j − m )( k − m ) ǫ α ) in front of the parameter a k,j + k − m,j may have an arbitrarily smallabsolute value if α >
0. Such sequence does not exist if ǫ − α ∈ N T [2( m + 1) , + ∞ ).We suppose that ǫ − α / ∈ Q , then Reson ≥ m +1 = 0 is equivalent to a linear system, which has a uniquesolution a m − j,m + k − j,k = a k,m + k − j, m − j = i ( k − j )( m − j )(1 − k − m )( k − j ) ǫ α ) , ∀ ≤ j ≤ m − ,a j,k,k + m − j = a k + m − j,k,j = i ( m − k )( m − j )(1 − k − m )( k − j ) ǫ α ) , ∀ ≤ j ≤ m − ,a m,k,k = a k,k,m = i ,a k,k + j − m,j = a j,k + j − m,k = i − j − m )( k − m ) ǫ α , ∀ j ≥ m + 1 , (4.8)for all k ≥ m + 1. In the case m = 0, (4 .
8) has only the last two formulas. When α >
0, the sequence( a j,l,k ) j − l + k = m can be arbitrarily large, for 0 < ǫ ≪
1. We suppose that ǫ α is an irrational algebraicnumber of degree d ≥
2. Then we have the Liouville estimate [23] | a j,j + k − m,k | ≤ c ǫ,α (2( j − m )( k − m )) d − , ∀ j, k ≥ m + 1 , which loses the regularity of v in the estimate of X F m ( v ). It is difficult to find the same kind of estimatefor the transcendental numbers, which can preserve the regularity. So we return to the case α = 0. H s -stability in the case α = 0 For α = 0 and every m ∈ N and s ≥
1, assume that u is the smooth solution of the NLS-Szeg˝o equation i∂ t u + ∂ x u = Π( | u | u ) , u (0 , x ) = e imx + ǫf ( x ) , k f k H s ≤ , and u ( t, x ) = e i arg u m ( t ) ( e imx + ǫv ( t, x )). Then v is the solution of the following Hamiltonian equation i∂ t v + ∂ x v − H e imx ( v ) − (1 − m + ǫϕ ′ ( t )) v = ϕ ′ ( t ) e imx + ǫ Π( e − imx v + 2 e imx | v | ) + ǫ Π( | v | v ) . Its energy functional is H m,ǫ ( t, v ) = H m ( v ) + ϕ ′ ( t ) L m ( v ) + ǫ ( H m ( v ) + ϕ ′ ( t )2 N ( v )) + ǫ N ( v ) , with H m ( v ) = k ∂ x v k L + − m k v k L + R S Re( e imx v )d x, L m ( v ) = Re v m , H m ( v ) = Re( R S e − imx | v | v )d x, N p ( v ) = k v k pL p , p = 2 or 4 . We define ˜ N ( v ) := k v − P m v k L = P n ≥ m +1 | v n | and the following proposition holds. Proposition 4.4.
For every s > and m ∈ N , there exists a sequence ( a j,l,k ) j − l + k = m such that a j,l,k = a k,l,j , sup m ≥ sup j − l + k = m | a j,l,k | = and the functional F m : H s + → R , defined by F m ( v ) = X j − l + k = mj,k,l ∈ N Re( a j,l,k v j v l v k ) , ∀ v ∈ C ∞ + , atisfy that {F m , L m } = − ˜ N and R m := {F m , H m } + H m is a finite sum of the Fourier modes v , · · · , v m . Moreover, for all v, h ∈ H s + , we have k X F m ( v ) k H s . m,s k v k H s , k d X F m ( v ) h k H s . m,s k v k H s k h k H s . Proof.
For the convenience of the reader, the detailed calculus for R m = {F m , H m } + H m and formula(4 .
8) in the case α = 0 are given in Appendix . We define a j,j + k − m,k = 0, for all 0 ≤ j, k ≤ m and a n,m +1+ n, m +1 = a m +1 ,m +1+ n,n = 0, for all 0 ≤ n ≤ m −
1. Combing
Proposition . .
8) with α = 0, we have ( Reson ≥ m +1 ( v ) (cid:12)(cid:12)(cid:12) α =0 = 0 , Re (cid:16) Reson low ( v ) (cid:12)(cid:12)(cid:12) α =0 (cid:17) = R m ( v ) , {F m , L m } ( v ) = 2Im( P k ≥ a m,k,k | v k | + P j + k =2 m a j,m,k v j v k ) = − P n ≥ m +1 | v n | . By (4 .
8) with α = 0, we have | a j,j + k − m,k | ≤ , for all j, k ≥
0. By the definition of F m , we have ( \ [ X F m ( v )]( n ) = − i P k − l + n = m a k,l,n v k v l − i P k − n + l = m a k,n,l v k v l \ [d X F m ( v ) h ]( n ) = − i P k − l + n = m a k,l,n ( v k h l + v l h k ) − i P k − n + l = m a k,n,l v k h l , , ∀ n ≥ . The last two estimates are obtained by Young’s convolution inequality for l ∗ l ֒ → l . Set χ mσ := exp( ǫσX F m ) the Hamiltonian flow of ǫ F m , i.e.,dd σ χ mσ ( v ) = ǫX F m ( χ mσ ( v )) , χ m ( v ) = v. We perform the canonical transformation Ψ m,ǫ := χ m = exp( ǫX F m ). We want to reduce the energyfunctional H m,ǫ to the following norm H m,ǫ ( t ) ◦ Ψ m,ǫ = H m + ϕ ′ ( t ) L m ( v ) + ǫ (cid:18) R m + ϕ ′ ( t )( − ˜ N + N (cid:19) + O ( ǫ ) . Since R m depends only on low frequency Fourier modes v , · · · , v m , the high-frequency filtering H s normof the solution of ∂ t w ( t ) = X H m,ǫ ( t ) ◦ Ψ m,ǫ ( w ( t )) is handled by the Birkhoff normal form transformation.The estimate of k P m ( v ) k H s is given by (4 . χ mσ , for | σ | ≤ v and Ψ − m,ǫ ( v ). Lemma 4.5.
For every s > and m ∈ N , there exist two constants γ m,s , C m,s > such that for all v ∈ H s + , if ǫ k v k H s ≤ γ m,s , then χ mσ ( v ) is well defined on the interval [ − , and the following estimateshold: sup σ ∈ [ − , k χ mσ ( v ) k H s ≤ k v k H s , sup σ ∈ [ − , k χ mσ ( v ) − v k H s ≤ C m,s ǫ k v k H s , k d χ mσ ( v ) k B ( H s ) ≤ exp( C m,s ǫ k v k H s | σ | ) , ∀ σ ∈ [ − , . The proof is based on a bootstrap argument, which is similar to
Lemma .
3, given by
Lemma . m = 2. We shall perform the canonical transform below. Recall that Ψ m,ǫ = χ m .24 emma 4.6. For all s > , m ∈ N and < ǫ < ǫ ∗ m , there exists a smooth mapping Y m : R × H s + −→ H s + and a constant C ′ m,s > such that for all t ∈ R , we have X H m,ǫ ( t ) ◦ Ψ m,ǫ = X H m + ϕ ′ ( t ) X L m + ǫ (cid:18) X R m + ϕ ′ ( t )( − X ˜ N + 12 X N ) (cid:19) + ǫ Y m ( t ) , and sup t ∈ R kY m ( t, v ) k H s ≤ C ′ m,s k v k H s (1 + k v k H s ) , for all v ∈ H s + such that ǫ k v k H s ≤ γ m,s . Set w ( t ) :=Ψ − m,ǫ ( v ( t )) , then we have ∂ t w ( t ) = X H m,ǫ ( t ) ◦ Ψ m,ǫ ( w ( t )) and (cid:12)(cid:12)(cid:12) dd t k w ( t ) − P m ( w ( t )) k H s (cid:12)(cid:12)(cid:12) ≤ C ′ m,s ǫ k w ( t ) k H s (1 + k w ( t ) k H s ) , (4.9) if ǫ k w ( t ) k H s ≤ γ m,s .Proof. For every t ∈ R , we expand the energy H m,ǫ ( t ) ◦ Ψ m,ǫ = H m,ǫ ( t ) ◦ χ m with Taylor’s formula attime σ = 1 around 0. Since χ m = Id H s + , we have( H m + ϕ ′ ( t ) L m ) ◦ χ m = H m + ϕ ′ ( t ) L m + dd σ [( H m + ϕ ′ ( t ) L m ) ◦ χ mσ ] | σ =0 + Z (1 − σ ) d d σ [( H m + ϕ ′ ( t ) L m ) ◦ χ mσ ]d σ = H m + ϕ ′ ( t ) L m + ǫ {F m , H m + ϕ ′ ( t ) L m } + ǫ Z (1 − σ ) {F m , {F m , H m + ϕ ′ ( t ) L m }} ◦ χ mσ d σ and (cid:18) H m + ϕ ′ ( t )2 N (cid:19) ◦ χ m = H m + ϕ ′ ( t )2 N + Z dd σ [( H m + ϕ ′ ( t )2 N ) ◦ χ mσ ]d σ = H m + ϕ ′ ( t )2 N + ǫ Z {F m , H m + ϕ ′ ( t )2 N } ◦ χ mσ d σ. Since we have the homological system ( {F m , H m } + H m = R m {F m , L m } + ˜ N = 0 in Proposition .
4, we have H m,ǫ ( t ) ◦ χ m = H m + ϕ ′ ( t ) L m + ǫ (cid:18) {F m , H m } + H m + ϕ ′ ( t )( {F m , L m } + 12 N ) (cid:19) + ǫ (cid:20)Z {F m , (1 − σ ) {F m , H m + ϕ ′ ( t ) L m } + H m + ϕ ′ ( t )2 N } ◦ χ mσ d σ + N ◦ χ m (cid:21) = H m + ϕ ′ ( t ) L m + ǫ (cid:18) R m + ϕ ′ ( t )( − ˜ N + 12 N ) (cid:19) + ǫ (cid:20)Z G m ( t, σ ) ◦ χ mσ d σ + N ◦ χ m (cid:21) , where G m ( t, σ ) = {F m , (1 − σ ) R m + σ H m + ϕ ′ ( t )(( σ −
1) ˜ N + N ) } . We set Y m ( t, v ) := Z X G m ( t,σ ) ◦ χ mσ ( v )d σ + 14 X N ◦ χ m ( v ) , then we get X H m,ǫ ( t ) ◦ χ m = X H m + ϕ ′ ( t ) X L m + ǫ (cid:0) X R m + ϕ ′ ( t )( − X ˜ N + X N ) (cid:1) + ǫ Y m ( t ).25ince F m , H m and R m are homogeneous series of order 3 with uniformly bounded coefficients, N and˜ N are homogeneous series of order 2 with uniformly bounded coefficients, we have ( k X {F m , H m } ( v ) k H s + k X {F m , R m } ( v ) k H s . s k v k H s , k X {F m , N } ( v ) k H s + k X {F m , ˜ N } ( v ) k H s . s k v k H s , because for J m ( v ) = P j − l + k = m Re( b j,l,k v j v l v k ) with sup j − l + k = m | b j,l,k | < + ∞ , we have {F m , J m } ( v )=4Im X n ≥ ∂ v n F m ( v ) ∂ v n J m ( v )= X k + k = l + l Im(4 a k ,l ,m + l − k b l ,k ,m + k − l + a l ,l + l − m,l b k ,k + k − m,k ) v k v k v l v l + X k + k + l − l =2 m a k ,k + l − m,l b k ,l ,m + l − k − a k ,l ,m + l − k b k ,l + k − m,l ) v k v k v l v l and {F m , N } ( v ) = − P j − l + k = m a j,l,k v j v l v k ). Recall that sup <ǫ<ǫ ∗ m sup t ∈ R | ϕ ′ ( t ) | < + ∞ and X N ( v ) = − i Π( | v | v ), then we havesup ≤ σ ≤ sup t ∈ R k X G m ( t,σ ) ( v ) k H s + k X N ( v ) k H s . m,s k v k H s (1 + k v k H s ) . By using
Lemma . . v ∈ H s + such that ǫ k v k H s ≤ γ m,s , we havesup ≤ σ ≤ sup t ∈ R k X G m ( t,σ ) ◦ χ mσ ( v ) k H s ≤ sup ≤ σ ≤ sup t ∈ R k d χ m − σ ( χ mσ ( v )) k B ( H s ) k X G m ( t,σ ) ( χ mσ ( v )) k H s . m,s sup ≤ σ ≤ e C m,s ǫ k χ mσ ( v ) k Hs k χ mσ ( v ) k H s (1 + k χ mσ ( v ) k H s ) . m,s k v k H s (1 + k v k H s )and sup t ∈ R k X N ◦ χ m ( v ) k H s . m,s k v k H s (1 + k v k H s ). Then we obtain the estimate of Y m .Since w ( t ) = χ m − ( v ( t )), we have the following infinite dimensional Hamiltonian system on the Fouriermodes: i∂ t w n ( t ) = (1 + n − m − ǫϕ ′ ( t )) w n ( t ) + iǫ \ Y m ( t, w ( t ))( n ) , ∀ n ≥ m + 1 . because for all n ≥ m + 1, we have ( \ H e imx ( w ( t ))( n ) = \ X R m ( w ( t ))( n ) = \ X L m ( w ( t ))( n ) = 0 , \ X N ( w ( t ))( n ) = \ X ˜ N ( w ( t ))( n ) = − iw n ( t ) . Consequently, if ǫ k w ( t ) k H s ≤ γ m,s , then we have (cid:12)(cid:12)(cid:12) ∂ t k w ( t ) − P m ( w ( t )) k H s (cid:12)(cid:12)(cid:12) ≤ ǫ X n ≥ m +1 (1 + n ) s | \ Y m ( t, w ( t ))( n ) || w n ( t ) |≤ ǫ kY m ( t, w ( t )) k H s k w ( t ) k H s . m,s ǫ k w ( t ) k H s (1 + k w ( t ) k H s ) . .3.2 End of the proof of Theorem . . u (0) ∈ C ∞ + by the same density argument in the proofof Proposition . Proof.
For all m ∈ N and s ≥
1, we recall that u ( t, x ) = e i arg u m ( t ) ( e imx + ǫv ( t, x )), ∂ t v ( t ) = X H m,ǫ ( t ) ( v ( t ))and w ( t ) = χ m − ( v ( t )). In Proposition .
2, we have shown that there exists A m,s ≥ <ǫ< k v (0) k H s ≤ A m,s . By using (4 . <ǫ< sup t ∈ R k P m ( v ( t )) k H s ≤ (1 + 9 m ) s I m . Set K m,s := max(6 A m,s , m ) s I m ), ǫ m,s := min( ǫ ∗ m , γ m,s K m,s , C m,s K m,s ) and T m,s := sup { t ≥ ≤ τ ≤ t k v ( τ ) k H s ≤ K m,s } . We choose ǫ ∈ (0 , ǫ m,s ). Since ǫ = ǫ k v (0) k H s ≤ ǫA m,s ≤ γ m,s , Lemma . k v (0) − w (0) k H s = k v (0) − χ m − ( v (0)) k H s ≤ ǫA m,s C m,s ≤ A m,s . So we have k w (0) k H s ≤ K m,s . For all t ∈ [0 , T m,s ], we have ǫ k v ( t ) k H s ≤ γ m,s . Hence Lemma . k w ( t ) k H s ≤k v ( t ) k H s + k χ m − ( v ( t )) − v ( t ) k H s ≤k v ( t ) k H s + C m,s ǫ k v ( t ) k H s ≤ K m,s + 4 C m,s K m,s ǫ ≤ K m,s . So we have ǫ sup ≤ t ≤ T m,s k w ( t ) k H s ≤ γ m,s , which implies that (cid:12)(cid:12)(cid:12) dd t k w ( t ) − P m ( w ( t )) k H s (cid:12)(cid:12)(cid:12) ≤ C ′ m,s ǫ k w ( t ) k H s (1 + k w ( t ) k H s ) , in Lemma .
6. Set d m,s := K m,s C ′ m,s . We can obtain the following estimate: k w ( t ) − P m ( w ( t )) k H s ≤k w (0) k H s + C ′ m,s | t | ǫ sup ≤ τ ≤ T m,s k w ( τ ) k H s (1 + sup ≤ τ ≤ T m,s k w ( τ ) k H s ) ≤ K m,s C ′ m,s K m,s | t | ǫ ≤ K m,s , for all 0 ≤ t ≤ min( T m,s , d m,s ǫ ). We use Lemma . k v ( t ) k H s ≤ k w ( t ) − v ( t ) k H s + k w ( t ) − P m ( w ( t )) k H s + k P m ( v ( t )) k H s ≤ C m,s ǫ k v ( t ) k H s + 2 K m,s K m,s ≤ C m,s K m,s ǫ + 5 K m,s ≤ K m,s , t ∈ [0 , d m,s ǫ ]. In the case t <
0, we use the same procedure and we replace t by − t . Consequently,we have sup | t |≤ dm,sǫ k u ( t ) − e i ( · +arg u m ( t )) k H s = ǫ sup | t |≤ dm,sǫ k v ( t ) k H s ≤ K m,s ǫ. Although we have some similar results for the NLS equation, there are still some differences between theNLS equation and the NLS-Szeg˝o equation. We denote by u = u ( t, x ) = P n ≥ u n ( t ) e inx the solution ofthe NLS equation i∂ t u + ∂ x u = | u | u. (5.1)In Fourier modes, we have i∂ t u n = n u n + P k − k + k = n u k u k u k . Fix m ∈ Z , for every n ∈ Z , wedefine v n := u n + m e i ( m +2 nm ) t . Then k v ( t ) k L = k u ( t ) k L and we have i∂ t v n = n v n + X k − k + k = n v k v k v k . (5.2)If u is localized in the m -th Fourier mode, then v is localized on the zero mode. Thus the orbital stabilityof the traveling wave e m can be reduced to the case m = 0. In Faou–Gauckler–Lubich [8], H s -orbitalstability of plane wave solutions is established by limiting the mass of the initial data k u k H s to a cer-tain full measure subset of (0 , + ∞ ) for the defocusing cubic Schr¨odinger equation with the time interval[ − ǫ − N , ǫ − N ], for all N ≥ s ≫
1. However, this above transformation u v does not preserve the L norm for the NLS-Szeg˝o equation and formula (5 .
2) fails too.On the other hand, the approach that we use to prove
Theorem . . H m ( v ) = Re R S e − imx | v | v can not be cancelled bythe homological equation. The energy functional of the equation of v can not be reduced as H m + O ( ǫ )by using the same method in this paper. The Szeg˝o filtering to (5 .
1) makes it possible to cancel all thehigh frequency resonances in the term R m = {F m , H m } + H m . Then we use a bootstrap argument todeal with the equation ∂ t w ( t ) = X H m,ǫ ( t ) ◦ χ m ( w ( t )) after the Birkhoff normal form transformation. We give the details of the homological equation in
Proposition . .
8) and
Proposition . α = 0.For all v ∈ S n ∈ N P n ( C ∞ + ), H m ( v ) = k ∂ x v k L + − m k v k L + R S Re( e imx v )d x and F m ( v ) = P j − l + k = mj,k,l ∈ N Re( a j,l,k v j v l v k ). With the convention v n = 0, for all n <
0, we have ( ∂ v n H m ( v ) = ∂ ¯ v n H m ( v ) = n − m v n + v m − n ,∂ ¯ v n F m ( v ) = ∂ v n F m ( v ) = P k − l + n = m ¯ a k,l,n v k v l + P k − n + l = m a k,n,l v k v l . . F m and H m , {F m , H m } ( v )= − i X n ≥ ( ∂ v n F m ∂ v n H m − ∂ v n H m ∂ v n F m ) ( v )=Im( X k − l + n = mk,l,n ∈ N n − m ) a k,l,n v k v l v n + X k − n + l = mk,l,n ∈ N (1 + n − m ) a k,n,l v k v n v l )+ Im( X k − l + n = mk,l,n ∈ N ,n ≤ m a k,l,n v l v k v m − n + X k − n + l = mk,l,n ∈ N ,n ≤ m a k,n,l v k v l v m − n )=Im( A + A + A + A ) . The term A = P k − n + l = mk,l,n ∈ N ,n ≤ m a k,n,l v k v l v m − n has only finite term depending on low frequency Fouriermodes v , · · · , v m . We divide A , A , A into two parts. The first part consists of only low frequencyresonances, the second part consists of high frequency resonances. A = − X k − l + n = mk,l,n ∈ N n − m ) a k,l,n v k v l v n = A low + A ≥ m +11 , where A low consists of all the resonances v j v l v k such that j, k ≤ m , A low = − ( ⌊ m − ⌋ X j =0 2 m X k = m − j + m − X j = ⌊ m +12 ⌋ m X k = j +1 )2(2 + j + k − m ) a j,j + k − m,k v j v j + k − m v k − m − X j =0 2 m X k = j + m +1 j + ( k + m − j ) − m ) a j,k,k + m − j v j v k v k + m − j − m X k = ⌊ m +12 ⌋ k − m ) a k, k − m,k v k v k − m , and A ≥ m +11 contain other resonances v j v l v k such that at least one of j, k is strictly larger than 2 m . A ≥ m +11 = − m X j =0 X k ≥ m +1 j + ( k + m − j ) − m ) a j,k,k + m − j v j v k v k + m − j − m X j = m +1 X k ≥ m +1 j + k − m ) a j,j + k − m,k v j v j + k − m v k − X k ≥ m +1 X j ≥ k +1 k + j − m ) a k,k + j − m,j v k v k + j − m v j − X k ≥ m +1 k − m ) a k, k − m,k v k v k − m . Then, we calculate A = P k − n + l = mk,l,n ∈ N (1 + n − m ) a k,n,l v k v n v l = A low + A ≥ m +12 . A low consists of all29he resonances v k v l v n such that k, n ≤ m or l ≤ m . A low = X j − l + k = mj,l,k ∈ N ,l ≤ m (1 + l − m ) a j,l,k v j v l v k + X j − l + k = mm +1 ≤ j ≤ m − j +1 ≤ k ≤ m,l ≥ m +1 l − m ) a j,l,k v j v l v k + m X k = ⌊ m ⌋ +1 (1 + (2 k − m ) − m ) a k, k − m,k v k v k − m ; A ≥ m +12 plays the same role as A ≥ m +11 . A ≥ m +12 = m X j =0 X k ≥ m +1 k − m ) a j,k,m + k − j v j v k v m + k − j + m X j = m +1 X k ≥ m +1 j + k − m ) − m ) a j,j + k − m,k v j v j + k − m v k + X k ≥ m +1 X j ≥ k +1 k + j − m ) − m ) a k,k + j − m,n v k v k + j − m v j + X k ≥ m +1 (1 + (2 k − m ) − m ) a k, k − m,k v k v k − m . At last, A = P k − l + n = mk,l,n ∈ N ,n ≤ m a k,l,n v l v k v m − n = P j − l + k = mj,k,l,n ∈ N ,j ≤ m a l,k, m − j v k v l v j . Using the same idea,we have A = A low + A ≥ m +13 , with A low = m X j =0 2 m X k =0 a m − j,m + k − j,k v j v k v m + k − j + m X j = m +1 2 m X k =0 a m − j,k,k + j − m v j v k + j − m v k ;and A ≥ m +13 = m X j =0 X k ≥ m +1 a m − j,m + k − j,k v j v k v m + k − j + m X j = m +1 X k ≥ m +1 a m − j,k,k + j − m v j v k + j − m v k . Recall that H m ( v ) = Re( R S e − imx | v | v ) = P k − l + n = m Re( v k v l v n ). A similar calculus as in the case of A shows that H m ( v ) = Re( B low + B ≥ m +1 ) with B low =( ⌊ m − ⌋ X j =0 2 m X k = m − j + m − X j = ⌊ m +12 ⌋ m X k = j +1 )2 v j v j + k − m v k + m X k = ⌊ m +12 ⌋ v k v k − m + m − X j =0 2 m X k = j + m +1 v j v k v k + m − j , ≥ m +1 = X k ≥ m +1 m X j =0 v j v k v k + m − j + m X j = m +1 v j v j + k − m v k + v k v k − m + X j ≥ k +1 v k v k + j − m v j . At last we define Reson low ( v ) (cid:12)(cid:12)(cid:12) α =0 = − i ( A low + A low + A low + A ) + B low andReson ≥ m +1 ( v ) (cid:12)(cid:12)(cid:12) α =0 = − i ( A ≥ m +11 + A ≥ m +12 + A ≥ m +13 ) + B ≥ m +1 . Then we have {F m , H m } ( v )+ H m ( v ) = Re(Reson low ( v ) (cid:12)(cid:12)(cid:12) α =0 +Reson ≥ m +1 ( v ) (cid:12)(cid:12)(cid:12) α =0 ). Since Reson low ( v ) (cid:12)(cid:12)(cid:12) α =0 contains only finite terms and depends only on v , · · · , v m , so is R m = Re(Reson low ( v ) (cid:12)(cid:12)(cid:12) α =0 ). For highfrequency resonances, we computeReson ≥ m +1 ( v ) (cid:12)(cid:12)(cid:12) α =0 = X k ≥ m +1 m − X j =0 − ia m − j,m + k − j,k + (1 + 2( m − j )( k − j )) ia j,k,k + m − j + 1) v j v k v k + m − j + X k ≥ m +1 2 m X j = m +1 − k − m )( j − m )) ia j,j + k − m,k − ia m − j,k,k + j − m + 1) v j v j + k − m v k + X k ≥ m +1 (cid:2) ia m,k,k − ia m,k,k + 1) v m | v k | + ((1 − k − m ) ) ia k, k − m,k + 1) v k v k − m (cid:3) + X k ≥ m +1 X j ≥ k +1 − j − m )( k − m )) ia k,j + k − m,j + 1) v k v j + k − m v j . After replacing j by 2 m − j in the sum m +1 ≤ j ≤ m , we have the equivalence between Reson ≥ m +1 (cid:12)(cid:12)(cid:12) α =0 =0 and (4 .
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