Note on semiclassical analysis for nonlinear Schrödinger equations. Satoshi Masaki

Transcription

1 Note on semiclassical analysis for nonlinear Schrödinger equations Satoshi Masaki

2 Preface This article is a de facto doctoral thesis of the author. This consists of the summary of the author s research during the ph.d. course of the Kyoto university. The de jure doctoral thesis is another one which has the same contents as [58]. The author would like to express his sincere gratitude to his adviser Professor Yoshio Tsutsumi for his constant encouragement and for many valuable suggestions. Deep appreciation goes to Professor Mitsuru Ikawa for leading the author to the study of PDE s. The author would like to express his sincere thanks to Professor Rémi Carles for fruitful discussions. The part of this article consists of the joint work of he and the author. Thanks are due to Professors Kenji Nakanishi and Hideo Kubo for many helpful comments and constant support. The author is also grateful to all members of NLPDE seminar at Kyoto university. Especially, the author thanks Doctors Hiroaki Kikuchi and Masaya Maeda for reading the manuscript of this article and giving useful comments. This work is supported by the JSPS fellowship.

3 Contents Introduction 5. Introduction Several remarks on function spaces Lebesgue space Sobolev space Zhidkov space Small time WKB analysis for nonlinear Schrödinger equations. Introduction Equations and main results Two different approaches General strategy and problem Existence of the solution and the problem on the energy estimate Construction procedures of the phase function Expansion of the solution of the system Nonlinear WKB approximation Example : Local nonlinearity Cancellation Existence result Justification of WKB approximation Example : Nonlocal nonlinearities Smoothing by the nonlocal nonlinearity Existence result Justification of WKB approximation Example 3: Local and nonlocal nonlinearity Cancellation versus smoothing Existence result Justification of WKB estimate

4 3 Analysis of Classical trajectories Introduction Semiclassical analysis and Euler equation Classical trajectory Preliminary results Reduction of Euler-Poisson equations to an ODE of classical trajectories Local existence of classical solution Pointwise condition for finite-time breakdown Global existence of classical solutions to radial Euler-Poisson equations : without background Attractive case Repulsive case : n = Repulsive case : n Repulsive case 3: n = Applications Global existence of classical solutions to radial Euler-Poisson equations : existence of constant background Attractive case : n = Attractive case : n Attractive case 3: n = Repulsive case : n = Repulsive case : n Repulsive case 3: n = The zero background limit Attractive case Repulsive case Large time WKB analysis for Schrödigner-Poisson system 3 4. Introduction and main result Global existence of limit solution Auxiliary system Proof of the theorem Regularity of limit solution Choice of the function space and regularity theorem Preliminary lemma Regularity at the initial time Persistence of the regularity Local existence with slowly decaying data Lack of the decay of the phase function Modified energy estimate Existence result Large time WKB analysis Main result

5 4.5. Proof the theorem part : the zeroth order Proof the theorem part : the first order Proof the theorem part 3: higher order A Tool box 57 A. Basic inequalities A.. The Hölder inequality A.. The Sobolev inequality A..3 The Hardy-Littlewood-Sobolev inequality A. Tools for energy estimates A.. Gronwall s lemma A.. Commutator estimate

6 Notations We list the notations and collect the definitions that we use throughout this article. i) C, R, R +, Z, N denote the set of complex number, real number, nonnegative number, integer, and positive integers. ii) We denote by R n the Euclidean n-dimensional space with point x = a,..., x n ). iii) A := B means that A is defined by B. iv) Let u: R n C or R n R) is smooth. xi denotes the partial derivatives of a function u with respect to x i. We sometimes write i, for short. When n =, we denote u the derivative of a function u. Moreover, α denotes α / α x ) α n / α n x n ) for a multi-index α = α,..., α n ) N n. v) We denote by u the gradient of a function u, that is, u = u,..., n u). When n =, we use d/dx instead of. vi) stands for the Laplacian on R n, that is, = n i= /d i. When n =, we use d /dx instead of. vii) denotes the tensor product of, that is, u is the n n matrix i j u) 6i,j6n. viii) fx) = Ogx)) as x x means that fx)/gx) is bounded as x x. Moreover, fx) = ogx)) as x x means that fx)/gx) tends to zero as x x. ix) C k R n ) stands for the set of k-time differentiable function on R n, and C R n ) := k> C k R n ) for the set of infinitely differentiable function on R n. C R) is the set of infinitely differentiable function with compact support. x) Let I be an interval of R and let X be a Banach space. C k I, X) is the space of k-time continuously differentiable function from I to X. 5

7 xi) L p R n ) denotes the Banach space of measurable functions u: R n C or R n R) such that u L p R n ) < with u L p R n ) = ) p if p [, ), ux) p dx R n ess sup ux) if p. x R n We write L p if there is no risk of confusion. xii) SR n ) is the set of Schwartz function rapidly decreasing function) on R n. S R n ) is the set of tempered distributions. xiii) F denotes the Fourier transform Ff)ξ) = π) n/ R n e ix ξ fx)dx. xiv) For s R and f SR n ), we define ) a/ as ) a/ f)x) = F [+ ξ ) a/ Ff]x), and a as a f)x) = F [ ξ a Ff]x). We sometimes denote ) / by Λ. xv) For s R and p [, ], W s,p R n ) denotes the Sobolev space, that is, Banach space of functions u: R n C or R n R) equipped with the norm u W s,p R n ) := ) s/ u <. L p If there is no risk of confusion, we write W s,p. xvi) H s R n ) := W s, R n ). If there is no risk of confusion, we write H s. 6

8 Chapter Introduction. Introduction In this article, we consider the Cauchy problem of the semiclassical nonlinear Schrödinger equation iε t u ε + ε uε = N u ε )u ε ; u ε, x) = u x), NLS) where u ε = u ε t, x) is a complex-valued function on t, x) R R n. ε is a positive parameter corresponding to the scaled Planck s constant ε and N denotes the nonlinearity. We are concerned with the problem of semiclassical limit ε. The aim of this article is to describe the results about the asymptotic behavior of the solution u ε in this limit. In particular, we are interested in a phase-amplitude approximation, called WKB approximation, of the solution u ε : u ε t, x) = e i φ t,x) ε a t, x) + εa t, x) + ε a t, x) + ),..) where φ is a real-valued function and a i i =,,,... ) is a complex-valued function. The nonlinear Schrödinger equations appears in many physical contexts. For example, NLS) with the quintic nonlinearity Nu ε )u ε = u ε 4 u ε is sometimes used as a model for one-dimensional Bose-Einstein condensation in space dimension n =. When n = or n = 3, a cubic nonlinearity Nu ε )u ε = u ε u ε is usually considered. The Schrödinger-Poisson system SP) below) is studied as the fundamental equation in semiconductors appication, with b > standing for a constant background charge and λ being the reciprocal of the square of the Debye number. In Chapter, we justify the WKB approximation of the solution to NLS) with some typical nonlinearities in a time interval which is small in general) but independent of ε. For this approximation, several approaches are known. We follow the one by the pioneering works of Gérard [3] and 7

9 Grenier [34], for the NLS with local nonlinearity. It consists in using the modified Madelung transform u ε = a ε e iφε /ε. Then, it turns out that the problem boils down to the analysis of the system t a ε + φ ε )a ε + aε φ ε = i ε aε, t φ ε + φε + N a ε ) =,..) a ε, x), φ ε, x)) = A ε, Φ ). Note that u ε = a ε e i φε ε is an exact solution of NLS) if a ε, φ ε ) solves..). The main purpose of this chapter is the following two points: First is to clarify the difficulty of this method which appears in all kinds of nonlinearity. Second is to present how to overcome this difficulty with some typical examples. In this chapter, we treat the essentially) cubic nonlinear Schrödinger equation, Schrödinger-Poisson system, Hartree equation, and Hartree equation with local nonlinearity. For cubic nonlinear Schrödinger equation, we give a slightly different formulation of the proof of the result in [34], and generalized this result into the above typical equations. Though the basic strategy of the proof is the same. However, the details are quite different and so far there is no general theory to treat them at once. In Chapter 3, we turn to the analysis of classical trajectories. It is a fundamental principle in quantum mechanics that, when the time and distance scales are large enough relative to the Planck constant, the system will approximately obey the laws of classical Newtonian mechanics. The equations..) is a kind of quantum hydrodynamics equations, which is classical hydrodynamics equations with a quantum correction term. In the limit, the Euler equation for an isentropic compressible flow is formally recovered from the nonlinear Schrödinger equation. Indeed, denoting ρ := lim ε a ε and v := lim ε φ ε for a solution of..), one verifies that, at least formally, ρ, v) solves the Euler equation { t ρ + divρv) =, t v + v )v + N..3) ρ) =, which are the statements of the conservation of mass and Newton s second law, respectively. In this chapter, we analyze..3) by using the method of characteristic curves. The characteristic curve of v is called classical trajectory in the context of Schödinger equations. It is known that when the characteristic curves cross each other, the solution to this equation breaks down by a formation of singularity, a shock. This is also related to the theory of geometrical optics. The classical trajectory is an analogue of the notion of ray developed initially to describe the propagation of electro-magnetic waves, such as light. The breakdown of the solution of..3) is closely 8

10 related to the occurrence of caustics. We choose the Schrödinger-Poisson system with constant background b : iε t u ε + ε uε = λv P u ε, V P = u ε b, V P as x SP) and corresponding Euler-Poisson equations t ρ + divρv) =, t v + v )v + λ V P =, V P = ρ b, V P as x EP) as a target equation of this chapter. Under the radial symmetry, we derive the necessary and sufficient conditions that ensure the classical solution to EP) is global. In the case where b > and n = and the case where b = and n and some more case such as the presence of relaxation term) are studied precisely in [5]. We stem the missing parts and give complete descriptions of the necessary and sufficient conditions for all n and b. As a result, we will see that, under the assumption that n 3, ρ is integrable, and v decays at spatial infinity and they are radially symmetric), there is only one possible form of the initial data which admits the global solution Theorem 3.3.4). In Chapter 4, we justify the WKB approximation..) of the solution to Shcrödinger-Poisson system SP) for large time. In one dimensional case, Liu and Tadmor [46] show by applying the result [5] that, for a class of initial data admitting a global solution of EP),..) holds for an interval which depends on the parameter ε and becomes arbitrarily large as ε. In this chapter, we generalize this result to the n 3 case. This is done by a combination of results in Chapters and 3. An example of global solution to EP) is given by the results in Chapter 3. Then, using the modified version of) the analysis in Chapter, we justify..) for large time.. Several remarks on function spaces We give several remarks on the function spaces which we use throughout this article... Lebesgue space The first is the Lebesgue space L p R n ). In this article, we often use Lebesgue spaces to investigate the decay property of functions. As usual, it is defined as the Banach space of measurable functions f: R n C or R n R) such 9

11 that f L p R n ) < with f L p R n ) = ) p if p [, ), fx) p dx R n ess sup fx) if p. x R n In general, if a measurable function f is not integrable, then the one of the following holds:. There exists a bounded set Ω R n with arbitrarily small measure) such that Ω f dx =.. Ω f dx is finite for any bounded set Ω Rn. However, Ω f dx as Ω. In the first case, f has a singularity at some point. An example of such function is fx) = x n { x 6} x). In the second case, f is not integrable because the decay of f is not enough. fx) is an example. It can be said that the index p of L p R n ) indicates both the strength of the singularity and the rate of the decay. For example, x q { x 6} x) belongs to L p space if and only if q < n/p. Similarly, x r { x 6} x) belongs to L p space if and only if r > n/p. Hence, very roughly speaking, an element of L p space is a function which has a singularity of order at most O x n p +ε ) and decays at spatial infinity at least order O x n p ε ). L space is rather special: The functions in L do not necessarily decay at spatial infinity. p small large singularity at a point strong weak none decay at spatial infinity rapid slow none The Hölder inequality A.. shows that these two properties, singularity and decay, are monotone in p. To concentrate on singularities, assume that f is supported on a bounded set Ω R n. Then, we have L p Ω) L q Ω) p q) since f L p Ω) Ω p q f L q Ω) for p q by Hölder inequality. This suggests that if the singularity of f is so weak that f L q Ω) then f automatically belongs to L p Ω). The converse is not true, as the following example shows: For a bounded set Ω and p < q, x n p+q Ω x) L p Ω)\L q Ω) since n/p > n/p+q) > n/q. To concentrate on the decay property we now assume g is bounded. In this case, q p g L p g L g q p q L for p q,

12 and so L p R n ) L R n )) L q R n ) L R n )) for p q. This suggests that if the decay of f is so rapid that f L q R n ) then this decay is enough for being f L p R n ). The converse is false, as the following example shows: For p > q, x n p+q { x >} x) L p R n ) \ L q R n ) since n/p < n/p + q) < n/q. Tail estimate Let p < q. Take a function f L q L with f L p. If there exists some function g such that f g L p L then g is an approximation of the tail part of f in such a sense that the decay of f g is faster than f itself. In this respect, we call such an estimate as a tail estimate... Sobolev space The next is the Sobolev space H s R n ) and W s,p R n ). For s R and p [, ], W s,p R n ) denotes the Banach space of functions u: R n C or R n R) equipped with the norm u W s,p R n ) := ) s/ u <. L p Moreover, H s R n ) := W s, R n ). The Sobolev embedding reveals the connection between the integrability of higher derivative and of lower derivative. Indeed, for q p <, we have f L p C n q n p f. L q It suggests the fundamental principle that the differentiation makes the singularity stronger and the decay faster. Indeed, for n q n p f L q R n ) q < p) being true, f is required to have so weak singularity that f L p R n ) holds, however, the decay is not required no more than f L p R n ). The Hardy-Littlewood-Sobolev inequality Lemma A..5) is a counterpart of the Sobolev embedding in a sense. For γ, n), we have F x γ = c n,γ ξ n+γ see e.g. [65]). Therefore, the Hardy-Littlewood-Sobolev inequality can be written as n q n p f ) C f L p L q for < q < p <. Hence, it suggests the fundamental principle that the integration makes the singularity weaker and the decay slower.

13 ..3 Zhidkov space We also use the Zhidkov space X s R n ) and its modified space Y s p,qr n ). The Zhidkov space is defined as follows: For s > n/, The norm of X s is given by X s R n ) := {u L R n ) u H s R n )}. X s R n ) := L R n ) + H s R n ). The space was introduced in [74] in the case n =, and its study was generalized to the multidimensional case in [6] see also [5, 5]). In general, a function in the Zhidkov space does not have spatial decay at all. For example, constant function fx) belongs to X s R n ), while it does not belongs to any Lebesgue spaces or Sobolev spaces. However, if n 3 then we see from Lemma.. below that for all f X s R n ) there exists a constant C such that f C L n/n ) L. This is a kind of tail estimate. Recall that f itself belongs to L p only if p =. In Chapter 4, we will use the modified one Yp,qR s n ). For n 3, s > n/ +, p [, ], and q [, ], we define a function space Yp,qR s n ) by with norm Y s p,qr n ) = C Rn ) Y s p,q Rn ) Y s p,q R n ) := L p R n ) + L q R n ) + H s R n ). The indices p and q indicate the decay rate of the function and its first derivative, respectively. This is a generalized version of X s R n ) if n 3. Y, s Rn ) is almost equal to X s R n ). The difference is the fact that all functions in Y, s Rn ) decays at spacial infinity. However, as noted above, for all f X s R n ), there exists a constant C such that f C Y, s Rn ). We also deduce form Lemma.. that Y, s Rn ) = Y, s Rn ) Y s, Rn ), where = n/n ). We discuss this space again in Section 4.3..

14 Chapter Small time WKB analysis for nonlinear Schrödinger equations. Introduction.. Equations and main results In this chapter, we consider the asymptotic behavior of the solution to the Cauchy problem iε t u ε + ε uε = N u ε )u ε ; u ε, x) = A ε x) expiφ x)/ε)...) In particular, the purpose of this chapter is to give a WKB-type approximation u ε t, x) e i φt,x) ε b t, x) + εb t, x) + ε b t, x) + )..) in a time interval [, T ] which is small in general but independent of ε. We construct suitable phase φ and amplitude b i, and provide a pointwise description of u ε as ε in the following cases: Defocusing nonlinearity which is cubic at the origin: N u ε ) = f u ε ) with f : R + R + satisfying f, f >, and f) =. Focusing or defocusing nonlocal nonlinearity: N u ε ) = ± ) u ε or N u ε ) = ± x γ u ε ). We also analyze the case where the nonlinearity is the sum of the above two types. More precisely, the target equations of this chapter are the following: 3

15 . The defocusing essentially) cubic nonlinear Schrödinger equation iε t u ε + ε uε = f u ε )u ε ; u ε, x) = A ε x) expiφ x)/ε) CNLS) with f : R + R + satisfying f, f y) >, and f) =. This is treated in [34] for generalized or other types of local nonlinearities, see [3, 4, 3,, 3, 8, 44]).. The Schrödinger-Poisson system without background iε t u ε + ε uε = λvpu ε ε, VP ε = u ε, VP ε as x, u ε, x) = A ε x) expiφ x)/ε), SP) where λ = ±. For this equation, see [5, 43, 46, 7, 73]. 3. The Hartree equation iε t u ε + ε uε = λ x γ u ε )u ε ; u ε, x) = A ε x) expiφ x)/ε), where λ = ±. Hartree equation is treated in [5]. 4. The nonlinear Schrödinger equation with local nonlinearity and nonlocal nonlinearity iε t u ε + ε uε = f u ε ) + λ x γ u ε )u ε, L-NL) u ε, x) = A ε x) expiφ x)/ε) H) with f : R + R + satisfying f, f y) >, and f) =, and λ = ± =. We reformulate and generalize the previous results. Especially, we would like to relax the decay condition on the initial phase Φ. This is due to the fact that Φ is not necessarily to be bounded for being u ε ) = A ε eiφ /ε H s. From this respect, in the following theorems, we try to make our assumptions on Φ so close to the one Φ is bounded without spatial decay) as possible. The following are the main results of this chapter. Theorem.. [34], WKB analysis for CNLS)). Let f C R + : R + ) with f) = and f >. Let k be an integer and s > n/ + k + 4 be a real number. Assume that Φ X s+, and that A ε writes A ε = k ε j A j + oε k ) in H s..3) j= 4

16 for ε [, ]. Then, there exist a existence of time T > independent of ε and a solution u ε C[, T ]; H s ) of CNLS). There also exists φ C[, T ]; X s+ ) and β j H s j such that u ε = e i φ ε β + εβ + + ε k β k + oε k )) in C[, T ]; H s k )...4) Furthermore, φ satisfies φ t, x) Φ x) W s, R n ). Theorem.. WKB analysis for SP)). Let n 3 and λ R. Let k be a positive integer and let s > n/ + k + 3 be a real number. Assume that Φ C k+5 with Φ H s, and that A ε writes..3). for ε [, ]. Then, there exist a existence of time T > independent of ε and a solution u ε C[, T ]; H s ) of SP). There also exists φ C[, T ]; C k+5 ) and β j H s j such that..4) holds. Furthermore, there uniquely exists a constant c such that Φ c as x and φ satisfies φ t, x) Φ x c t) L n n + L )R n ), φ t, x) Φ x) + t Φ x c s) ds L n n + L )R n ). Theorem..3 WKB analysis for H)). Let n 3 and λ R. Let γ be a positive number with n/ < γ n. Let k be a positive integer and let s > n/ + k + 3 be a real number. Assume that Φ C k+5 with Φ H s, and that A ε writes..3). or ε [, ]. Then, there exist a existence of time T > independent of ε and a solution u ε C[, T ]; H s ) of H). There also exists φ C[, T ]; C k+5 ) and β j H s j such that..4) holds. Furthermore, there uniquely exists a constant c such that Φ c as x and φ satisfies φ t, x) Φ x c t) L n γ+ + L )R n ), φ t, x) Φ x) + t Φ x c s) ds L n γ + L )R n ). Theorem..4 WKB analysis for L-NL)). Let n and λ R. Let f C R + : R + ) with f) = and f >. Let λ R and let γ be a positive number with n/ < γ n. Let k be a positive integer and let s > n/ + k + 4 be a real number. Assume that Φ X s+ and A ε writes..3). for ε [, ]. Then, there exist a existence of time T > independent of ε and a solution u ε C[, T ]; H s ) of L-NL). There also exists φ C[, T ]; X s+ ) and β j H s j such that..4) holds. Furthermore, φ satisfies φ t, x) Φ x) L n γ + L )R n ). 5

17 Remark..5. The assumption on the phase function Φ reflects the shape of nonlinearity. Under the assumption in above theorems, Φ does not necessarily decay at spatial infinity and moreover is not necessarily bounded, in general. In Theorems.. and..4, we assume Φ X s+. In this case, Φ is always bounded but does not necessarily decay. For the n 3 case, Lemma.. below shows the existence of the constant c R such that Φ c L. On the other hand, the assumption for Theorems.. and..3 is Φ C k+5 and Φ H s. This class is much larger than X s+ R n ). Especially, this Φ can tend to infinity as x. Lemma.. below shows that there exists a constant c R n such that Φ c L since n 3, and moreover that there exists a constant c such that Φ c c x L if n 5, where = ) = n/n 4). Nevertheless, in all above theorems, we can construct a function P t, x) explicitly given only by Φ such that φ t) P t) decays at spacial infinity as long as solution exists. The decay rate of this difference also reflects the shape of nonlinearity. One of the most remarkable difference between Theorems.. and..4 is this point. Remark..6. We need f > in Theorems.. and..4 because the quantity /f appears when we estimate the energy. If we try to treat more general nonlinearity such as quintic nonlinearity fy) = y then what prevents us is the fact that f ) =. We refer to [3, 4, ] for such generalized local nonlinearities. Remark..7. It is remarked in [, 6] that the above WKB analysis and a geometrical transform can help understand the behavior of a wave function near a focal point, in a supercritical régime see also [,, 3, 4, 6, 6, 6])... Two different approaches We now address an outline for the method to justify the WKB approximation..) see also [3, 8, 66]). One approach to obtain a WKB-type estimate is to use Madelung s transform u ε t, x) = ρ ε t, x)e i Sε t,x) ε. Plugging this to..) and separating real and imaginary part, we find the quantum Euler equation t ρ ε + divρ ε S ε ) =, t S ε + S ε ) S ε + N ρ ε ) = ε ) ρ ε, ρ ε ρ ε, x), S ε, x)) = A ε, Φ + ε arg A ε ))...5) 6

18 The term ε ρ ε / ρ ε ) is called quantum pressure. The equations..5) represent a fluid dynamics formulation of the..) and are known as Madelung s fluid equations [48, 49]. Taking ε, we obtain, at least formally, the compressible Euler equation t ρ + divρv) =, t v + v )v + N ρ) =,..6) ρ, x), v, x)) = A, Φ ), where ρ = lim ε ρ ε, v = lim ε S ε, and A = lim ε A ε. method, the convergence of the quadratic quantities With this u ε ρ, ε Imu ε u ε ) ρv as ε is proved in several situations. The Wigner measure is one of the strong tool for justifying this limit For this limit and the Wigner measure, consult [3, 9, 3, 37, 38, 45, 55, 56, 57, 7, 73] and references therein. Though this convergence suggests that the solution u ε may have the asymptotics of the form u ε = e is/ε ρ + o)), it is not satisfactory. In particular, the argument of the solution u ε is not clear. In fact, the asymptotics e is/ε ρ + o)) is not true. The leading order term of the amplitude of the approximate solution cannot be expected to be real-valued, even if so is this at the initial time. Another way to justify..) is to employ a modified Madelung transform u ε = a ε e i φε ε..7) and consider the system t a ε + φ ε )a ε + aε φ ε = i ε aε, t φ ε + φε + N a ε ) =, a ε, x), φ ε, x)) = A ε, Φ )...8) It is essential that a ε takes complex value. and so, S ε φ ε, in general. If we know this system has a solution a ε, φ ε ) and the solution can be expanded as a ε = a + εa + ε a +, φ ε = φ + εφ + ε φ +, then, by means of..7), we obtain WKB type estimate..) with b = a e iφ. The explicit formulae of higher order terms of amplitude b i are given in Section..4. This method is first applied to CNLS) within the framework of analytic function spaces in [3] and of Sobolev spaces [34] for CNLS). We remark that a, φ ) solves the compressible Euler equation..6). 7

19 In this chapter, we use the second method to justify the WKB approximation..) for CNLS), SP), H), and L-NL) in a time interval which is small but independent of ε. We first clarify the difficulty and illustrate the general strategy of the proof in Section.. Then, we consider CNLS) in Section.3. We give a slightly different proof from [34] which is based on the modified energy method. Section.4 is devoted to the study of the equation with nonlocal nonlinearities, SP) and H). In final Section.5, we treat L-NL).. General strategy and problem In this paragraph, we show an outline for obtaining the WKB type approximation..) of the solution of..). No rigorous result is given through this Section., although we make some observation with calculations which we use in later sections. We follow the approach by Grenier [34] the second one introduced in Section..) and work with a data in Sobolev space: We apply the modified Madelung transform..7) to..) and consider the system..8) for amplitude a ε and phase φ ε. Let us introduce a new variable v ε = φ ε. Differentiating the second equation of..8), we find t a ε + v ε )a ε + aε v ε = i ε aε, t v ε + v ε )v ε + N a ε ) =, a ε, x), v ε, x)) = A ε, Φ ). SHS) The main point of Grenier s idea is that this system can be regarded as a symmetric hyperbolic system with perturbation. We call this system as SHS) in this respect. In this section, we give a general strategy to show that SHS) admits a solution a ε, v ε ) Section..); that the solution a ε, φ ε ) of..8) can be constructed from a ε, v ε ) Section..); and that the solution is expanded in powers of ε Section..3). Once we obtain this expansion of a ε, φ ε ), the WKB approximation..) is an immediate consequence Section..4)... Existence of the solution and the problem on the energy estimate Our first step is to show that SHS) has a solution. We try to obtain a solution a ε, v ε ) in the class C[, T ]; H s ) ). The main part of the proof is a priori estimate. Hence, we shall detail this part only. For other parts of proof, see [5, 5, 68]. Let us go along the classical energy method. Consider the energy Et) := a ε H s + vε H s. 8

20 As a matter of fact, we cannot close the energy estimate with this energy. The purpose of this section is only to reveal what is wrong with this energy. In the concrete examples below, we modify this energy. These modifications are considered in Sections.3.,.4., and.5.. Let us proceed with the standard energy argument as further as we can. Estimates in this section are quoted sometimes in forthcoming sections. We use the convention for the inner product in L : f, g L = fgdx. R n We also denote ) / by Λ. Take s. From the first line of SHS), we have d dt aε H s = Re Λs t a ε, Λ s a ε L = Re Λ s v ε )a ε ), Λ s a ε L Re Λ s a ε v ε ), Λ s a ε L + Re iελ s a ε, Λ s a ε L =: I + I + I 3...) By integration by parts, we see I 3 = Reiε a ε Hs) =...) This fact is one of the most remarkable point of modified Madelung s transform..7). It is very contrast with the fact that treatment of quantum pressure term often needs some care such as ρ ε > ) when we employ Madlung s transform and work with..5) see [7]). Moreover, I is a good term: It writes I = Re v ε )Λ s a ε, Λ s a ε L Re [Λ s, v ε ]a ε, Λ s a ε L = Re v ε )Λ s a ε, Λ s a ε L Re [Λ s, v ε ]a ε, Λ s a ε L by integration parts. By the Hölder inequality and the commutator estimate Lemma A..), we obtain I C v ε L a ε H s + vε H s a ε L a ε H s)...3) Therefore, if we set s > n/ + then the right hand side is bounded by C a ε H s + v ε H s) 3. On the other hand, I is rather bad in such a sense that it requires the bound of s + )-th derivative of v ε. Indeed, extracting the main part, we have I = Re a ε Λ s v ε, Λ s a ε L Re [Λ s, a ε ] v ε, Λ s a ε L. 9

21 Remark that integration by parts does not work so well as in the estimate of I. Therefore, we estimate as I v ε H s a ε L a ε H s + C a ε L v ε H s + v ε L a ε H s ) a ε H s...4) Thus, we need s + )-th derivative of v ε to be bounded in a suitable sense. For s > n/ +, the right hand side is bounded by C a ε H s + v ε H s+) 3. Alternatively, we have I + Re a ε Λ s v ε, Λ s a ε L C a ε L v ε H s + v ε L a ε H s ) a ε H s...5) If we are able to remove the bad part, then we only need s-time derivative of v ε. We will see later that this fact is the key for CNLS) case. Now let us turn to the estimate of v ε. We estimate H s norm of v ε, although the estimate actually required in..4) is the H s norm of v ε. The estimate of H s norm of v ε is an easy modification. From the second line of SHS), we find d dt vε H s = Re Λs t v ε, Λ s v ε L = Re Λ s v ε )v ε ), Λ s v ε L Re Λ s N a ε ), Λ s v ε L =: I 4 + I 5...6) I 4 is also a good term. The estimate of I 4 is similar to that of I : Since I 4 = Re v ε )Λ s v ε, Λ s v ε L Re [Λ s, v ε ]v ε, Λ s v ε L = Re v ε )Λ s v ε, Λ s v ε L Re [Λ s, v ε ]v ε, Λ s v ε L by integration by parts, the Hölder and the commutator estimate Lemma A..) yield I 4 C v ε L v ε H s...7) I 5 is the nonlinear term. The treatment of this term is the main difficulty. A straight forward calculation does not give any more than I 5 C N a ε ) H s v ε H s...8) Notice that the s+)-time derivative of the nonlinear term N a ε ) appears. Our naive hope is that this term might be bounded by the derivative of order s ) for a ε, which may enable us to close the energy estimate and obtain an estimate like d dt a H s + vε H s+) C a H s + vε H s+).

22 However, of course it is impossible in general. In particular, when we consider the local nonlinearity such as N a ε ) = f a ε ) with some smooth function f, we have N a ε ) = f a ε ) Rea ε a ε ), whose H s norm seems not to be bounded by H s norm of a ε. So, we need to make some trick. At the end of this section, we summarize the problem which we observed in this section: d. The estimate of dt aε H s s + ) for v ε through I. requires the bound of derivative of order d. The estimate of dt vε H s requires the bound of derivative of order s + ) for the nonlinearityn a ε ) through I 5. In forthcoming examples, we will see how to get over this difficulty and close the energy estimate. The required technique strongly depends on the nonlinearity N a ε ). In the local nonlinearity case, the key is the cancellation Section.3.), which is another formation of symmetrizability of SHS). On the other hand, when the nonlinearity is nonlocal, we use the smoothing property of the nonlinearity Section.4.)... Construction procedures of the phase function Once a solution a ε, v ε ) of SHS) is known, we can reconstruct a solution a ε, φ ε ) of..8). In this section, we discuss this integration procedures which we use to define φ ε from v ε so that φ ε = v ε and φ ε, x) = Φ x). There are at least three possible ways to do this. The Poicaré lemma The first is the Poicaré lemma. We suppose that a solution a ε, v ε ) of SHS) and the initial data φ ε, x) = Φ x) are known. If v ε is irrotational, that is, if v ε =, then there exist a function φ ε such that φ ε = v ε. At this step, there is a freedom of choice of a constant: Adding an arbitrary function c = ct) of time only, we see φ ε +ct) also satisfies φ ε +ct)) = v ε. However, we can determine this function by c) = Φx) φ ε, x) and c t) = φ ε + N a ε ) t φε. Then, we see that a ε, φ ε + c) is a solution of..8). Note that the right hand side does not depend on space variable by the definition of φ ε. Let us remind that this method requires the irrotational property of v ε. Direct definition The second is to define directly by the equation. We suppose that we obtain a ε, v ε ) and the uniqueness of SHS) is known. Then, we can define φ ε from

23 its initial data by φ ε t, x) = Φ x) t ) vε s, x) + N a ε )s, x) ds. One easily checks that a ε, φ ε ) solves..8) and a ε, φ ε ) solves SHS). Then, by uniqueness, we conclude that φ ε = v ε. This method requires the uniqueness of the solution to SHS). Note that in this case the irrotational property immediately follows from the fact that v ε is given by the gradient of φ ε. The Hardy-Littlewood-Sobolev inequality The third is a consequence of the Hardy-Littlewood-Sobolev inequality, which can be found in [36, Th ] or [3, Lemma 7]: Lemma... If φ D R n ) is such that j φ L p R n ), j =,..., n for some p, n), then there exists a constant c such that φ c L q R n ), with /p = /q + /n. By this lemma, we can construct φ ε as in the first case without the irrotational property of v ε nor uniqueness of the solution to SHS). However, for this method, we need the decay property of v ε in the sense that it must belong L p R n ) for some p < n. Especially, this method is difficult to apply when n = since the property v ε L R n ) is not sufficient: In space dimension n =, consider a function fx, x ) = log + logx + x ) ). One can check that f H, while f L. The first two methods are intended for giving φ ε from its first derivative v ε = φ ε. We will see later that, in some case, not the first derivative v ε = φ ε but the second derivative v ε = φ ε is first given as a source see the proof of Theorem.4. below). The third can also be used to construct v ε from v ε...3 Expansion of the solution of the system We have discussed the existence of the solution to..8) in the preceding two sections. As mentioned in Section.., to obtain the WKB-type approximation..) it suffices to expand the solution of..8) in powers of ε: a ε = a +εa + +ε k a k +oε k ), φ ε = φ +εφ + +ε k φ k +oε k )...9) In this section, we turn to the method to obtain this expansion and illustrate a scheme for the justification of..9). It turns out that a, φ ) solves a system..8) with ε = and a i, φ i ) solves a i-th linearized system of..8), and that the existence result for..8) can again be used to solve these systems and determine approximate solutions.

24 Let us describe our observation. We suppose that this expansion is given at the initial time, that is, there exists an integer k such that A ε = A + εa + + ε k A k + oε k ) in H s...) Then, letting ε = in..8), we obtain t a + φ )a + a φ =, t φ + φ + N a ) =, a, x), φ, x)) = A, Φ )...) This system can be solved exactly the same way as in the case of..8), and moreover the existence time T can be chosen the same. This follows from the fact that the existence time T of solution a ε, φ ε ) of..8) depends only on the size of the initial data: If the initial data is bounded uniformly in ε then T can be independent of ε. Thus, we obtain a, φ ) := a ε, φ ε ) ε=. The zeroth order We first prove that a ε, φ ε ) converges to a, φ ) as ε in a suitable sense. This convergence immediately provides a ε = a + o), φ ε = φ + o), which is..9) with k =. The proof of this convergence again relies on the energy method. For example, we estimate time derivative of a ε a H s + v ε v H s. At this step, again the problem is how to close the energy estimate as in Section..). The first order We next put b ε := aε a )/ε, ψ ε := φε φ )/ε. Then the system for b ε, ψε ) is very similar to the system..8) which aε, φ ε ) solves. Indeed, that system becomes t b ε + εq b ε, ψ) ε + Q b ε, φ ) + Q a, ψ) ε = i ε bε + i a, t ψ ε + ε ψε + φ ψ ε + N aε ) N a ) =, ε ) A b ε, x), ψ, ε ε x)) = A,, ε..) where Q denotes the quadratic term Q a, φ) := φ )a + /)a φ. Note that the main quadratic part of..) is the same that of..8) up to a constant ε. Therefore, we can solve..) in the same way as in 3

25 ..8), although the existence of new linear terms and the nontrivial external force i/) a cause loss of one-time derivative and two-time derivative, respectively. Note that if..) is satisfied for k then the initial value b ε t= is uniformly bounded for ε [, ], which ensure that the existence time can be chosen independently of ε. Furthermore, it coincides with A when ε =. We therefore obtain a, φ ) := b ε, φε ) ε= which solves Here, we denote t a + Q a, φ ) + Q a, φ ) = i a, N ) = N ) a, a ) = lim ε t φ + φ φ + N ) =, a, x), φ, x)) = A, ). N a ε ) N a ) ε ) a, a = b ε ε= )...3) Repeating the argument in the first step, we can claim that b ε, ψε ) converges to a, φ ) as ε by an energy estimate. This convergence implies a ε = a + εa + oε), φ ε = φ + εφ + oε), which is..9) with k =. The l-th order We use an induction argument. For l k, we put b ε l := aε l j= εj a j )/ε l, ψl ε := φ ε l j= εj φ j )/ε l. Then the system for b ε l, ψε l ) is also similar to..8): t b ε l + εl Q b ε l, ψε l ) + Q b ε l, φ ) l + Q a, ψl ε ) + Q a i, φ l i ) = i ε bε l + i a l, i= t ψl ε + εl ψε l + φ ψl ε + l φ i φ l i ) + N aε ) l j= εj N j) ε l =, i= A ε b ε l, x), ψε l, x)) = ) l j= εj A j ε l,...4) Note that the main quadratic part of..4) is still the same that of..8) up to a constant ε l. Therefore, we can solve..4) in the same way as in..8). In this step, we need the boundedness of i/) a l. Therefore, we 4

26 lose two-time derivative in each step. So long as l k, we see from..) that the initial value b ε l t= is uniformly bounded for ε [, ]. In particular, it coincides with A l when ε =. We therefore obtain a l, φ l ) := b ε l, φε l ) ε=, which solves l t a l + Q a i, φ l i ) = i a l, i= t φ l + l..5) φ i φ l i ) + N l) =, i= a l, x), φ l, x)) = A l, ), where N l) is given inductively by N ) = N a ) and N a ε N l) = N l) ) ) l j= a,..., a l ) = lim εj N j). ε The explicit form of N l) is given in the following sections see Remarks.3.6 and.4.). As in the previous steps, we can claim that b ε l, ψε l ) converges to a l, φ l ) as ε. This convergence implies a ε = l ε j a j + oε l ), φ ε = j= which is..9) with k = l...4 Nonlinear WKB approximation ε l l ε j φ j + oε l ), We finally give a WKB type approximation..) of the solution..). Let us start our observation at the step where we obtain an ε-power expansion of the solution a ε, φ ε ) to the system..8) such as a ε = a + εa + + ε k a k + oε k ), φ ε = φ + εφ + + ε k φ k + oε k ) in a suitable topology, say in C[, T ]; H s ) with s > n/+, for some integer k. Recall that if a ε, φ ε ) solves..8), then u ε = a ε e i φε ε is an exact solution to..). Now, we plug the above expansion to u ε to have u ε = a + + ε k a k + oε k )) exp i φ ) ε + iφ + + ε k iφ k + oε k ) in C[, T ]; H s ), which is written as a WKB type approximation u ε = e i φ ε β + εβ + + ε k β k + oε k ))..6) j= 5

27 in C[, T ]; H s ). Remark that the leading order of the amplitude of u ε is not a but β = a e iφ. The important thing is that the ε -order term φ have some influence on the leading order, and so that the ε -term of the initial amplitude A is not negligible when we try to obtain the correct WKB estimate. This fact leads us to some instability results [, 3, 5, 69] by the approach initiated in [8, 9,,, 4, 4]. We conclude this section with giving the explicit formulae of β j for j. Notation... For a positive integer k, a set of positive integers P is called a partition of k if P k l= { } α N l α α... α l, α + + α l = k. For a partition P of k, let P be the integer L for which P N L holds. Moreover, for a partition P of k, denote the components of P by P l l =,,..., P ). Then, the amplitude β j j =,,..., k ) in..6) is given by P P ) β j = e iφ i P a φ +Pl + i a P φ +Pl...7) P :partition of j l= We see from the trivial partition {j} of j that β j contains φ j+ in its definition. Therefore, we cannot define β k from the source {a i, φ i )} 6i6k. Note that β j H s as long as {a i, φ i )} 6i6j+ is in H s H s..3 Example : Local nonlinearity l= In this section, we consider CNLS). Then, the system..8) is t a ε + φ ε )a ε + aε φ ε = i ε aε, t φ ε + φε + f a ε ) =, a ε, x), φ ε, x)) = A ε, Φ ). We introduce new unknown v ε := φ ε and consider t a ε + v ε )a ε + aε v ε = i ε aε, t v ε + v ε )v ε + f a ε ) =, a ε, x), v ε, x)) = A ε, Φ ),.3.).3.) which corresponds to the system SHS). 6

28 .3. Cancellation For the model case fy) = y, let us first observe how we overcome the difficulty in obtaining an energy estimate listed in Section... We keep the notation I i i =,,..., 5) in Section... Using N a ε ) = a ε, we have I 5 = Re Λ s Rea ε a ε )), Λ s v ε L = 4 Re a ε Λ s a ε, Λ s v ε L 4 Re [Λ s, a ε ] a ε, Λ s v ε L. The first term of the right hand side is a bad term because it contains s + )-time derivative of a ε. We now apply the integration by parts to obtain I 5 = 4 Re a ε Λ s v ε, Λ s a ε L + 4 Re a ε Λ s a ε, Λ s v ε L 4 Re [Λ s, a ε ] a ε, Λ s v ε L. This still contains a bad term and use of the Hölder inequality and commutator estimate Lemma A..) yield I 5 C a ε L v ε H s a ε H s + C a ε L a ε H s v ε H s..3.3) So far, it seems to be impossible to close the estimate. Indeed, plugging..3),..4),..),..7), and.3.3) to..) and..6), we obtain a bad estimate d dt aε H s + vε H s) C aε H s+ + vε H s+) 3. However, one trick solves all problems at once. The remarkable fact is that the bad term of I 5 is the same as that of I with different sign. In particular, we deduce that I 5 4 Re a ε Λ s v ε, Λ s a ε L C a ε L a ε H s v ε H s.3.4) and combining this estimate with..5) causes a cancellation: I + 4 I 5 I + Re a ε Λ s v ε, Λ s a ε L + 4 I 5 4 Re a ε Λ s v ε, Λ s a ε L C a ε L v ε H s + v ε L a ε H s) a ε H s. Namely, the sum of two terms which contain bad part becomes good, and so we conclude that d a ε Hs dt + 4 ) vε Hs C a ε Hs + 4 ) 3 vε Hs..3.5) This cancellation is the heart in the case of local nonlinearity. Therefore, the sign of nonlinearity is essential, and this argument does not apply to the focusing case fy) = y. In the focusing case, we need analyticity of the data [3, 3, 69]). In the original proof in [34], we construct a symmetrizer. Our cancellation can be regarded as another formulation of symmetrizability. 7

29 .3. Existence result We now state the result about the existence of the solution to.3.). Theorem.3. Grenier [34]). Let f C R + : R + ) with f) = and f >. Let s > n/ +. Assume that Φ X s+, and that A ε is uniformly bounded in H s for ε [, ]. Then, there exist T > independent of ε [, ] and s > n/+, and u ε = a ε e iφε /ε solution to CNLS) on [, T ] for ε, ]. Moreover, a ε and φ ε are the unique solution to.3.) which are bounded in C[, T ]; H s ) and C[, T ]; X s+ ), respectively, uniformly in ε [, ]. Moreover, φ ε Φ is bounded in C[, T ]; W s, ) uniformly in ε [, ]. This result is extended to NLS with more general local nonlinearities in [3, 4,, 3, 8, 44] see also [3]). Remark.3.. It is obvious from the following proof that if Φ L p R n ) for some p then φ ε C[, T ]; L p R n )) for the same p. In particular, if Φ H s+ is assumed, as in the original proof in [34], then φ ε C[, T ]; H s+ ). Proof. The strategy is the same as in the case fy) = y. We derive the cancellation of bad terms. For this purpose, set the energy Et) as Et) := a ε H s + 4f a ε ) Λs v ε, Λ s v ε Since A ε and Φ are bounded in H s L, there exists C independent of ε [, ] such that E) / C. So long as a ε H s C, it holds that a ε L C and so there exist m and M such that L. < m inf y [,4C ] f y) sup y [,4C ] f M <. y) With these constants, it holds that m 4 vε H s 4f a ε ) Λs v ε, Λ s v ε M L 4 vε H s. We estimate d dtet). However, since the estimate of the time derivative of the Sobolev norm of a ε is the same as in Section., we omit the detail. We estimate the time derivative of the second term of Et): d dt 4f Λs v ε, Λ s v ε ) = f Λs t v ε, Λ s v ε + t 4f Λ s v ε, Λ s v ε = f Λs v ε )v ε ), Λ s v ε f Λs f), Λ s v ε ) + t 4f Λ s v ε, Λ s v ε =: Ĩ4 + Ĩ5 + Ĩ6..3.6) 8

30 The estimate of Ĩ4 is similar to..7): Ĩ 4 = Re f vε )Λ s v ε, Λ s v ε Re L f [Λs, v ε ]v ε, Λ s v ε = Re f vε )Λ s v ε, Λ s v ε + Re ) L f v ε )Λ s v ε, Λ s v ε Re f [Λs, v ε ]v ε, Λ s v ε Since there exists a constant C = Cf, C ) such that ) f y) a f a ε ) sup ε L y [,4C ] f y)) CC L a ε L, we see from..7) that L. Ĩ4 CM v ε L + C a ε L ) v ε H s..3.7) We next estimate Ĩ5. An elementary calculation shows Ĩ 5 = f Λs f Rea ε a ε )), Λ s v ε = 4 I 5 Re f [Λs, f ]a ε a ε, Λ s v ε, where I 5 is introduced in Section.3.. Therefore, by.3.4), we have Ĩ5 Re a ε Λ s v ε, Λ s a ε 4 I 5 4 Re a ε Λ s v ε, Λ s a ε ε + f [Λs, f ]a ε a ε, Λ s v C a ε L a ε H s v ε H s + f [Λs, f ]a ε a ε, Λ s v ε.3.8) By the commutator estimate, we have f [Λs, f ]a ε a ε, Λ s v ε CM f L a ε a ε Hs + f H s a ε a ε L ) v ε H s. Here, we apply the estimate of composite function Lemma A..4) to obtain f ) a H s C + a ε L ) s sup f k) y) ε y [,4C ],k [, s +] H s C + 4C ) s C a ε H s, L L 9

31 where s denotes the smallest integer bigger than or equal to s. Combining these estimates to.3.8), we find Ĩ5 Re a ε Λ s v ε, Λ s a ε C a ε W, a ε H s v ε H s..3.9) We finally estimate Ĩ6. Using the first equation of.3.), we have ) ) f y) t t f sup y [,4C ] f y)) a ε L We hence obtain L CC a ε W, v ε W, + ε a ε L ). Ĩ6 C a ε W, v ε W, + ε a ε L ) v ε H s..3.) The assumption s > n/ + comes from this point. We suppose this to ensure the Sobolev embedding a ε L C a ε H s C a ε H s. We also note that the term Ĩ6 does not appear if we assume f is a constant as we have seen in Section.3.. In this case we need only s > n/ +. We summarize estimates..),..3)..5),..),.3.6),.3.7),.3.9), and.3.). Then, d dt Et) I i + Ĩ j i=,,3 j=4,5,6 I + I + Re a ε Λ s v ε, Λ s a ε L + + Ĩ4 + Ĩ5 Re a ε Λ s v ε, Λ s a ε L + Ĩ6 C a ε W, + v ε W, + a ε W, v ε W, + ε a ε L ) a ε H s + vε H s) CEt) 3 + Et) ). Therefore, by the Gronwall lemma, for any δ > there exists T = T δ) > such that Et) 4E) holds for t [, T ]. For t [, T ], it holds that a ε H s Et) / E) / C, which ensures the above estimates. We obtain a priori estimate. Uniqueness and construction of φ ε The uniqueness of a ε, v ε ) also follows from the energy estimate. Let a ε, vε ) and a ε, vε ) be two solutions of.3.) bounded in C[, T ]; Hs ). Then, 3

32 denoting d ε a, d ε v) = a ε aε, vε vε ), we have t d ε v + d ε v )v ε + v ε )d ε v + f a ε ) Red ε a a ε + a ε dε a) t d ε a + d ε v )a ε + v ε )d ε a + dε a v ε + aε d ε v = i ε dε a, +d ε aa ε + aε d ε a) f a ε + θ a ε a ε ))dθ a ε =, d ε a, x), d ε v, x)) =, ). The bad terms are /)a ε dε v and f a ε ) Rea ε dε a) because the others do not include any derivative on d ε a, d ε v). To handle these term by cancellation, we consider E d t) := d ε a L + 4f a ε ) dε v, d ε v Then, mimicking the estimate for Et), we obtain d dt E dt) C a i H s, v i H s)e d t). Therefore, by Gronwall s lemma, E d t) = for t [, T ] follows from E d ) =, and so a, v ) = a, v ) holds. Once the uniqueness of.3.) is known, along the argument in Section.., we can construct φ ε directly by t ) φ ε = Φ vε s) + f a ε s) ) ds. Then, a ε, φ ε ) is a unique solution of.3.). Since a ε, v ε L L and f) =, we see the second term belongs to L L. Therefore, if Φ belongs to L p for some p [, ] then so is φ ε, which ensures φ ε X s+ and completes the proof. Remark.3. also follows..3.3 Justification of WKB approximation We now prove the WKB approximation of the solution to.3.). Theorem.3.3. Let f satisfy the same assumption as in Theorem.3.. Let k be an integer and s > n/ + k + 4 be a real number. Assume that Φ X s+, and that A ε writes L. k A ε = ε j A j + oε k ) j= in H s 3

33 for ε [, ]. Then, the unique solution a ε, φ ε ) of.3.) has the following expansion: a ε = φ ε = k ε j a j + oε k ) in C[, T ]; H s k ), j= k ε j φ j + oε k ) in C[, T ]; H s k ). j=.3.) Remark.3.4. Theorem.. immediately follows from.3.) by an argument in Section..4. Indeed, we obtain u ε = e i φ ε β + εβ + + ε k β k + oε k )) in C[, T ]; H s k ), where β = a e iφ and β j is given by the formula..7). Remark.3.5. Recall that φ ε φ = φ ε Φ ) φ Φ ) W s, while φ ε and φ belong to X s+ and so they do not necessarily decay at spatial infinity. Similarly, the asymptotic of φ ε in.3.) holds in C[, T ]; W s k, ). Proof. The proof proceeds along the way which we show in Section..3. Instead of the asymptotic expansion of φ ε itself, we consider the expansion of v ε = φ ε : k v ε = ε j v j + oε k ) in C[, T ]; H s ). j= This is due to the following two reason: Firstly, it is rather easier to analyze the system for a ε, v ε ) than the system for a ε, φ ε ) itself, and, secondly, once we obtain the above expansion of v ε then it is easy to construct each φ i from the corresponding v i. Since A ε is uniformly bounded and Aε ε= = A, we see that.3.) admits a solution even in the case ε =. We denote this solution by a, φ ), which solves t a + v )a + a v =, t v + v )v + f a ) =, a, x), v, x)) = A, Φ )..3.) 3