On nonlinear effects in multiphase WKB analysis for the nonlinear Schrödinger equation

We consider the Schrödinger equation with an external potential and a cubic nonlinearity, in the semiclassical limit. The initial data are sums of WKB states, with smooth phases and smooth, compactly supported initial amplitudes, with disjoint supports. We show that like in the linear case, a superposition principle holds on some time interval independent of the semiclassical parameter, in several régimes in term of the size of initial data with respect to the semiclassical parameter. When nonlinear effects are strong in terms of the semiclassical parameter, we invoke properties of compressible Euler equations. For weaker nonlinear effects, we show that there may be no nonlinear interferences on some time interval independent of the semiclassical parameter, and interferences for later time, thanks to explicit computations available for particular phases.

The potential V = V (x) is supposed real-valued, smooth, and at most quadratic: Typical examples are V = 0, V linear (V (x) = E • x), V harmonic (V (x) = ω 2 |x| 2 ), V ∈ S(R d ), or any sum of such potentials.As initial data, we consider the sum of WKB states: α j (x)e iϕj (x)/ε , with γ 0. The value of γ measures the size of the initial data, and thus the importance of nonlinear effects in the semiclassical limit ε → 0. The case γ = 0 is supercritical in terms of WKB analysis: the evolution of the phase describing the rapid oscillation is given by an eikonal equation which involves the leading order amplitude, and a standard application of the WKB asymptotic expansion leads to systems which are not closed, no matter how many correcting terms are considered (see e.g.[5,Chapter 1] or [11]).
This work was supported by Centre Henri Lebesgue, program ANR-11-LABX-0020-0.A CC-BY public copyright license has been applied by the author to the present document and will be applied to all subsequent versions up to the Author Accepted Manuscript arising from this submission.
The case N = 1, referred to as monokinetic case, is well understood for short time, as we recall below, in the sense that the asymptotic behavior of u ε as ε → 0 is described precisely, locally in time on some interval independent of ε.The large time behavior is, in general, unknown; the one-dimensional case, with V = 0, is an exception, since it is completely integrable, see e.g.[16,27].Consider the case γ = 0.When V ≡ 0, the leading order asymptotic description involves the compressible Euler equation (1.4) ∂ t ρ + div (ρv) = 0, This equation is quasilinear, while (1.2) is semilinear (the nonlinear term is viewed as a perturbation when solving the Cauchy problem).In Section 2, we recall how to justify, in this case, the existence of a WKB approximation of the form u ε (t, x) = a(t, x) + εa 1 (t, x) + . . .+ ε k a k (t, x) e iφ(t,x)/ε , for all k 0, for some T > 0 independent of ε.We choose to measure errors in L 2 ∩ L ∞ in the spatial norm, in order to avoid to introduce ε-dependent norms when derivatives are involved, due to rapid oscillations.This time T can be taken as the lifespan of the smooth solution to the Euler equation (1.4) with suitable initial data.When N 2, the new question arising is the nonlinear interaction of the WKB states.As the problem is supercritical, even a formal computation is a delicate issue: if we plug an approximate solution of the form Remark 1.2 (Infinitely many states).The case N = ∞ may also be addressed, under suitable assumptions on the growth in space of the phases φ j compared to the size of the support of α j , as j → ∞.More precisely, as will be clear from the proof of the main result, we can consider the case N = ∞ provided that we may find cutoff functions χ j so that or at least in a weaker form if φ 0 ∈ H s (R d ) for some s > 2+d/2.Another constraint, in this case, is that we have to find a common lower bound for the lifespan of all the approximate solutions (φ j , a j ) considered below, an aspect which is obvious when N is finite, since we consider the minimum of a finite set.
1.2.Main results.The nonlinear evolution of each initial WKB state will play a crucial role: Under our assumptions, for fixed initial data, we know that: • If d 3, the equation is energy-subcritical, and for fixed ε > 0, there exists a unique solution , and it is smooth.See e.g.[8].
, the equation is energy-critical: the above conclusion is known to remain when V = 0 ( [29]), when V is an isotropic quadratic potential ( [17]), or when V is harmonic at infinity ( [15]).• If d 5, the equation is energy-supercritical: only a local in time smooth solution is known to exist by classical theory.In the cases where the global existence of a smooth solution is not known, the local existence time might go to zero as ε → 0, so the existence of a smooth solution on a time interval independent of ε > 0 is already a nontrivial step.The description of the solutions u ε j as ε → 0 on some time interval [0, T j ] independent of ε was evoked above, and is recalled in Sections 2 (case V = 0) and 3 (V satisfying (1.1)).Our main result is the following nonlinear superposition principle:

, and initial data satisfying Assumption 1.1. There exists
where u ε j is the solution of (1.5).Let us discuss this result in the supercritical case γ = 0, as it is the case where Theorem 1.3 may be the most surprising.The result follows from a detailed WKB analysis, as well as a property of finite speed of propagation for the compressible Euler equation, discovered initially in [21].The key feature of our setting is the compact, disjoint supports of the initial amplitudes α j .In the case V = 0, as long as WKB analysis is valid for each u ε j in (1.5), u ε j remains supported in (essentially) supp α j up to O(ε ∞ ): all the amplitude terms of the WKB expansion (at leading order, as well as correctors at arbitrary order) remain compactly supported, and amplitudes associated with u ε j1 and u ε j2 , respectively, with j 1 = j 2 , do not interact.In the case V ≡ 0, u ε j remains supported in supp α j evolving according to the classical flow generated by V , up to O(ε ∞ ).In other words, we recover the same phenomenon, regarding the evolution of supports, as in the linear case (see e.g.[22,28]), even though the régime associated to (1.2) is strongly nonlinear (of course Theorem 1.3 is trivial in the linear case, as u ε ≡ u ε j ).In particular, the initial modes cannot interact at a "visible" order before WKB analysis for at least one of the u ε j 's ceases to be valid, that is, before the solution of the corresponding Euler equation (1.4) breaks down (see however Section 5 for a discussion on the influence of our proof strategy on this statement).Recent progress on this precise question, [24,25,3] (see also [23] for a relation with the nonlinear Schrödinger equation), suggests that the expected scenario is rather that of an implosion: the conclusion of Theorem 1.3 might remain valid even after WKB has ceased to be valid.Remark 1.4 (Wigner measures).Since the proof of Theorem 1.3 relies on WKB analysis, it also implies the characterization of Wigner measures.Recall that the Wigner transform of u ε is defined by The position and current densities can be recovered from w ε , by A measure µ is a Wigner measure associated to u ε (there is no uniqueness in general) if, up to extracting a subsequence, w ε converges to µ as ε → 0 (see e.g.[12,19]).In the context of Theorem 1.3, each wave function u ε j has a unique Wigner measure, and the sum of these Wigner measures is the Wigner measure of u ε .For instance, if V = γ = 0, where (ρ j , v j ) solves (1.4) with initial data (ρ j , v j ) |t=0 = (|α j | 2 , ∇(χ j ϕ j )), and )) is (any function) such that χ j ≡ 1 on the support of α j .See Section 5 for the dependence of this statement upon χ j .
The next result shows that in the weakly nonlinear case γ = 1, some explicit information is available, in the sense that indeed, nonlinear interferences are negligible on some time interval [0, T 0 ] with T 0 > 0 independent of ε, while nonlinear interferences occur later.
) with disjoint supports, such that the following holds.There exist T 1 > 0 and T 0 ∈ (0, T 1 ) independent of ε, such that the solution to and The proof of Proposition 1.5 relies on explicit computations available in this weakly nonlinear case, and the fact that for linear oscillations, no caustic appears in the case of a single WKB state: the nature of nonlinear interferences is shown in Section 6, and consists of nonlinear phase modulations.In an appendix, we give an alternative argument illustrating another type of nonlinear interferences at leading order, consisting of the creation of a new mode (when d 2): starting from three WKB states, a fourth one, associated with a new phase, may appear by resonant interaction.
1.3.Content.In Section 2, we recall the WKB construction introduced in [14] for the case γ = 0, and emphasize the finite speed of propagation which appears in our framework.In Section 3, we explain how to adapt the previous approach to the case where V satisfies (1.1), and address the case γ > 0. In Section 4, we complete the proof of Theorem 1.3.Section 5 clarifies the role of the cutoff functions used in the proof of Theorem 1.3.Propositions 1.5 is established in Section 6.In an appendix, we propose an alternative proof of Proposition 1.5, in the case d 2 with N = 3, showing that there are several sorts of nonlinear interferences in the weakly nonlinear case.

The monokinetic case without potential
In this section, we consider (1.2)-(1.3) in the monokinetic N = 1, and in the supercritical case γ = 0, with slightly different notations for future reference: (2.1) In view of the setting of this paper, we assume We first consider the case V ≡ 0, then introduce the main ideas that make it possible to incorporate a subquadratic potential V .
We recall the main steps to the construction introduced in [14] (see also [5,Section 4.2]).The idea introduced in [14] consists in writing the solution to (2.1) as with a ε complex-valued and φ ε real-valued, solving The key remark is that this leads to a symmetric hyperbolic system, perturbed be a skew-symmetric term.The hyperbolic system appears when considering the unknown Considering the gradient of the first equation in (2.3), the system can be written To be precise, the system is made symmetric thanks to the constant symmetrizer Once v ε is known, one recovers φ ε by integrating in time the first equation in (2.3), Assuming that a 0 , ∇φ 0 ∈ H s (R d ) for s large (we will always assume a 0 , φ 0 ∈ C ∞ 0 (R d ) in the forthcoming applications), the limit ε → 0 leads to an asymptotic expansion of the form The leading order term is obtained by simply setting ε = 0 in (2.4): (2.5) Working with the intermediary unknown v = ∇φ, we get a system of the form and we infer the following result from [21]: (2.5).Moreover, (φ, a) remains compactly supported for t ∈ [0, T * ], and The first part of the statement is a consequence of classical theory for symmetric hyperbolic systems (see e.g.[2,20]).The property stated that initial compactly supported condition lead to a zero speed of propagation is due to the structure of this hyperbolic system, and is well understood from the simplest model of the Burgers equation Suppose we have a smooth solution on some time interval [0, T * ].In particular, We have directly, for all (t, x) Gronwall lemma then shows that if u 0 (x 0 ) = 0, then u(t, x 0 ) = 0 for all t ∈ [0, T * ], hence the zero speed of propagation for smooth solutions.As the matrix A(U, ξ) is linear in U , the result follows in the setting of (2.5).Note that to prove this zero speed of propagation, we do not invoke the symmetry of A: it was used in order to get Sobolev estimates (which ensure that U ∈ L 1 ([0, T * ], W 1,∞ )), but only the fact that it is (at least) linear in U is used at this stage.We then have, for the same T * as in Proposition 2.1: There exists T * > 0 independent of ε ∈]0, 1] such that for all s 0, there exists C = C(s) such that To infer the pointwise description of u ε at leading order, we must in addition know φ ε up to o(ε), which is achieved by considering the linearization of (2.5).At the next step of the WKB expansion, we find that where the first corrector (φ (1) , a (1) ) solves the system: + ∇φ • ∇φ (1) + 2 Re aa (1) = 0, ∂ t a (1) + ∇φ • ∇a (1) + ∇φ (1) • ∇a |t=0 = 0, for some function F k which is a polynomial in its arguments, without constant term, and whose precise expression is unimportant here.The left hand side is always the linearization of the left hand side of (2.5) about (φ, a), and the right hand side depends on previous correctors.We infer (see [14,5]), by induction: Let T * > 0 given by Proposition 2.1.For all k 1, there exists a unique solution (φ 2 to the above system, and for all s 0, there exists C = C(k, s) such that In addition, if supp a 0 , supp φ 0 ⊂ K, then (φ (k) , a (k) ) remains compactly supported for t ∈ [0, T * ], and The support property is a consequence of the same argument as in the proof of Proposition 2.1.Using the embedding we also deduce from the above error estimate the bound, for k 1: with the convention (φ (0) , a (0 ) = (φ, a).The standard form of WKB expansions, , is then obtained by setting a = ae iφ (1) , a 1 = a (1) e iφ (1) + iaφ (2) e iφ (1) , etc. Remark 2.4 (Higher order nonlinearities).If instead of (2.1), one considers with σ 2 an integer, then the justification of WKB analysis requires a different approach.We refer to [1,9] for two different proofs, which show that the conclusions of the propositions stated in this section remain valid.
Remark 2.5 (Focusing nonlinearity).If instead of (2.1), one considers a cubic focusing nonlinearity, then the analogue of (2.5) is no longer hyperbolic, but elliptic.Working with analytic initial data (φ 0 , a 0 ) is then necessary in order to solve (2.5) ( [18,26]), and this is a framework where nonlinear WKB analysis is fully justified ( [11,30]).However, analyticity is incompatible with an initial compact support.On the other hand, in the weakly nonlinear case γ = 1 (and more generally if γ 1), it is possible to justify WKB analysis with a focusing nonlinearity and compactly supported initial data (see e.g.[4] or [5,Chapter 2]).

The monokinetic case with a potential
In this section, we first recall some elements of WKB analysis in the linear case.We then show how this case can be merged with the analysis presented in the previous section, when γ = 0. We sketch how the case of a weaker nonlinearity, 0 < γ < 1.To conclude, we briefly discuss the weakly nonlinear régime γ = 1, and more generally the situation γ 1.
3.1.Linear case.The eikonal equation associated to , that is, without initial rapid oscillation, reads: In this subsection, we assume that ϕ 0 is smooth and at most quadratic, in the same sense as in (1.1).This eikonal equation is solved by introducing the classical trajectories, solving As V is at most quadratic, from (1.1), the above system has a unique, global, smooth solution, and in addition uniformly in y ∈ R d , for any matricial norm on R d×d .Therefore, the Jacobi determinant J t (y) = det ∇ y x(t, y), remains non-zero and bounded on some time interval [0, T ] with T > 0. Since we also have, by uniqueness in ordinary differential equations, ∇φ eik (t, x(t, y)) = ξ(t, y), for any smooth solutions to (3.2), the global inversion theorem implies the following result (see also [5, Proposition 1.9]): Lemma 3.1.Let V satisfying (1.1), and ϕ 0 satisfying the same properties.There exists T > 0 and a unique solution
In the linear case, the leading order amplitude is given by the linear transport equation Following the classical trajectories, this transport equation becomes trivial, since A(t, y) := J t (y)a (t, x(t, y)) satisfies ∂ t A = 0.
3.2.Supercritical case: γ = 0. We consider the same framework as in the previous section, now with a potential: As noticed in [4], it is possible to adapt the above WKB analysis in the presence of an external potential satisfying (1.1) by simply mixing the standard approach followed in the linear case (see e.g.[28]) and Grenier's method.
3.2.1.Introducing the nonlinearity.As noticed in [4], the approach presented in the case V = 0 for the nonlinear case can be adapted by changing the representation (2.2) to u ε (t, x) = a ε (t, x)e iφ eik (t,x)/ε+iφ ε (t,x)/ε , where φ eik solves (3.2) with ϕ 0 ≡ 0, and requiring The new terms compared to (2.3) involve φ eik , and since φ eik is at most quadratic in space, it turns out that they can be estimated like (semilinear) perturbative terms (using commutator estimates for the transport part).The natural limit for (3.9) when ε → 0 is given by The following result is a consequence of [4]: 2 , and for all s 0, there exists C = C(s) such that

1), and T , φ eik given by Lemma 3.1. There exists
The correctors φ (j) , a (j) j 1 as obtained in the same fashion as in Section 2. The only difference is that the operator ∂ t is replaced by

Finite speed of propagation: following the classical trajectories.
In order to prove that if a 0 ∈ C ∞ 0 (R d ), the solution to (3.1) remains compactly supported in the support of a 0 transported by the classical flow (3.3), it is standard to introduce the following change of unknown function (e.g.[28,5]): A(t, y) := J t (y)a (t, x(t, y)) , where a solves the transport equation as given by WKB analysis.Indeed, using (3.4), we easily check that A is constant in time, ∂ t A = 0. Correctors (a (k) ) k 1 in the (linear) WKB analysis solve the equation with the convention a (0) = a.Setting we infer that supp A (k) (t, •) ⊂ supp a 0 , ∀t ∈ [0, T ], ∀k 0, where T is given by Lemma 3.1.Thus, for t ∈ [0, T ], up to O(ε ∞ ), u ε remains compactly supported, in the support of a 0 transported by the classical flow.
In the nonlinear case, we check that the same argument remains valid.Consider φ eik solution to (3.2), and (φ, a) solving (3.7).The natural adaptation of the above computation consists in showing that if φ 0 , a 0 ∈ C ∞ 0 (R d ), the new unknown (ψ, A), defined by (3.8) A(t, y) := J t (y)a (t, x(t, y)) , ψ(t, y) := φ (t, x(t, y)) , enjoys a zero speed of propagation.Note that in view of Proposition 3.2, we already know that φ, a ∈ C([0, T * ], H ∞ (R d )), so it suffices to check that (ψ, A) solves a system for which the argument presented on the toy model of Burgers equation in Section 2 remains valid.Introducing whose determinant is by definition J t (y), we find: We do not express the right hand side of the last equation in terms of (ψ, A): differentiating the first equation with respect to y, the bounds stated in Proposition 3.2 make it possible to infer an inequality of the form Integrating in time the equation solved by ψ, we conclude to the zero speed of propagation for (ψ, A).Arguing like in Section 2 for the correctors, we have: ) with supp φ 0 , supp a 0 ⊂ K.There for any t ∈ [0, T * ], where T * is given by Proposition 3.2, where ψ and A are related to φ and a through (3.8).The same is true for the correctors (ψ (k) , A (k) ) k 1 corresponding to the next terms (φ (k) , a (k) ) k 1 in the asymptotic expansion in (3.9).Remark 3.4 (Special potentials).In the case where V is linear in x or isotropic quadratic, explicit formulas allow to bypass the above arguments.
, we recover exactly (2.1).Otherwise, a (smooth) time dependent factor has appeared, which obviously does not change the conclusion of Propositions 2.1 and 2.3.The case of a potential with the opposite sign is obtained by changing ω to iω in the formulas.See e.g.[5,Section 11.2] and references therein regarding these changes of unknown functions.For such potentials, the classical trajectories given by (3.3) are computed explicitly, and we can check directly the conclusions of Proposition 3.3.

Weaker nonlinearity.
We now consider the case 0 < γ < 1.This case is still a supercritical case as far as WKB analysis is concerned, in the sense described in the introduction: a "natural" asymptotic expansion of the solution u ε still involves a system of equations which is not closed.As noticed in [4], this intermediary case can be handled like the case γ = 0, by replacing (3.9) with (3.9) The matrices A j and S now depend on ε, in an explicit way, and the asymptotic expansion of (φ ε , a ε ) involves more terms.Let N = [1/γ], where [r] is the largest integer not larger than r > 0: N new intermediary terms appear compared to the case γ = 0, where the estimate holds in L ∞ ([0, T ], H s ) for any s > 0. This can be seen by setting φε = ε −γ φ ε : the leading order term is given by    The leading order amplitude solves the same transport equation as in the linear case, and it is readily observed that the analogue of Proposition 3.3 remains valid, up to adapting the hierarchy of equations.
3.4.Weakly nonlinear and linearizable cases.We now assume γ 1.As in [4] (or [5, Chapter 2]), we present a strategy for any γ 1, and emphasize the fact that the value γ = 1 is specific.In this setting, the coupling between phase and amplitude changes dramatically: rapid oscillations are described by φ eik only, and the analysis consists in expanding the amplitude a ε = u ε e −iφ eik /ε in powers of ε: Like above, the case when γ > 1 is not an integer requires a special asymptotic expansion, and we do not discuss this case.When γ > 1, the leading order amplitude satisfies the same transport equation as in the linear case.When γ = 1, it satisfies Following the classical trajectories, that is resuming the change of unknown function (3.8), this equation reads In particular, ∂ t |A| 2 = 0, and the nonlinear effect in a consists of a phase selfmodulation.In particular, the support of A(t, •) is independent of t ∈ [0, T ].The same is true for all correctors in the asymptotic expansion, as can be checked easily.

Separation of states
We complete the proof of Theorem 1.3, by proving the nonlinear superposition.
We set Then a 0 , φ 0 ∈ C ∞ 0 (R d ), φ 0 is real-valued, and We can then resume the analysis from the monokinetic case as presented in Sections 2 and 3, with the same notations.Let φ eik be given by Lemma 3.1 (it does not depend on the initial data, but only on V ).
4.1.Supercritical case.When γ = 0, the WKB analysis for each u ε j , solution to (1.5), involves the following system: To simplify the discussion, suppose first that V = 0, hence φ eik = 0.Each solution to (4.1) remains smooth on some time interval [0, T j ] for some 0 < T j T , and, on this time interval, enjoys a zero speed of propagation.As a consequence of Proposition 2.1, we have since nonlinear terms containing two indices j 1 = j 2 involve two functions whose supports are disjoint.Also, for all k 1, the correctors satisfy we obtain Theorem 1.3 in the case V = 0.In the case where V is not trivial, we just have to resume the above arguments by replacing the functions (φ, a) (possibly with indices and/or superscripts) with (ψ, A), as defined by the change of unknown function (3.8), which involves only V (see (3.3)), and not the initial data.
4.2.Other cases.When γ > 0, we have seen that the leading order amplitude is the same as in the linear case, up to a phase modulation.Leading order oscillations are given by φ eik , where we now set ϕ 0 = φ 0 in (3.2).The features used in the supercritical case then remain, regarding the evolution of the support of the terms involved in WKB analysis.
For any such function χ, we have u ε |t=0 = a 0 e iχφ0/ε .However, the eikonal equation now depends on χ, as (3.2) becomes As recalled in Section 3.1 (in the case φ 0 = 0), the solution is constructed, locally in time, via the classical trajectories, or, equivalently, through characteristic curves.
As V is smooth, the slope of characteristic curves at time t = 0 is uniformly bounded on the support of a 0 .By finite speed of propagation, there exists T (χ) > 0 such that φ eik does not depend on χ for t ∈ [0, T (χ)].In practice, the introduction of χ may shorten the time interval of validity of WKB analysis, as we now illustrate.Let d = 1, V = 0, and φ 0 (x) = x 2 /2.The solution to the eikonal equation (without cutoff χ) is given explicitly by This is a case where there is no singularity for t 0 (but a caustic reduced to one point at t = −1).Indeed, the classical trajectories, solving ẋ(t, y) = ξ(t, y), x(0, y) = y ; ξ(t, y) = 0, ξ(0, y) = φ ′ 0 (y) = y, are given by x(t, y) = (1 + t)y, obviously inverted, for all t 0, as and the leading order amplitude in WKB analysis is given by For χ a (usual) cutoff function as above, χφ 0 has two humps: in the presence of χ, y → x(t, y) ceases to be invertible for all t 0 (φ ′ eik solves the Burgers equation), but for short time (independent of ε, but depending on χ), a(t)e iφ eik (t)/ε does not depend on χ.

Supercritical WKB analysis for the nonlinear Schrödinger equation.
In the case addressed in Section 2, the above eikonal equation is replaced by (2.5).By considering the gradient of the phase instead of the phase, the Burgers equation (in the case of WKB analysis for the linear Schrödinger equation without potential) is replaced by the symmetrization of the Euler equation.Like above, finite speed of propagation implies that the introduction of a cutoff function in the initial phase does not alter the solution to (2.5) on some time interval [0, T (χ)], for some T (χ) > 0 possibly depending on χ.This time is of course independent of ε, as ε is absent from (2.5).This is why in Remark 1.4, the Wigner measure does not depend on the χ j 's, even though its construction seems to depend on these cutoff functions: the time of validity that we can prove may, on the other hand, depend on the choice of these cutoff functions.
We conclude this discussion by an illustration similar to the one given in the previous subsection.Let a 0 ∈ C ∞ 0 (R d ), and assume that for s > d/2+1, a 0 H s (R d ) is sufficiently small.Suppose also that v 0 = ∇φ 0 satisfies: , and there exists δ > 0 such that for all x ∈ R d , dist(Sp(∇v 0 (x)), R − ) δ, where we denote by Sp(M ) the spectrum of a matrix M .Then it follows from the main result in [13] that (2.5) has a global (in the future) solution where v is the unique, global smooth solution to the (multidimensional) Burgers equation We may for instance consider φ 0 (x) = |x| 2 /2 (see the previous subsection), and then v(t, x) = x 1 + t .
On the other hand, if φ 0 is multiplied by a cutoff function χ, then the initial data in (2.5) belong to C ∞ 0 (R d ): it follows from [21] that the corresponding solution develops a singularity in finite time.Like in the previous subsection, the introduction of the cutoff χ reduces the lifespan of the solution involved in WKB analysis but, for short time, does not alter the asymptotic description of the solution u ε .

Weakly nonlinear case
In this section, we prove Proposition 1.5.Instead of (1.2)-(1.3),we consider the weakly nonlinear case, When d 2, the creation of new WKB terms is possible by resonant interactions, provided that N 3, as recalled in the appendix.The one-dimensional cubic case is special, as there are no nontrivial resonances, see [6].In order to present an argument including the cubic one-dimensional case, we propose a proof which does not use the creation of, e.g., a fourth term out of three.
Consider linear phases, The first part of Proposition 1.5 is simply a restatement of Theorem 1.3 in this case.To prove the appearance of nonlinear interferences, we will not consider cutoff functions, and rely on explicit computations.WKB analysis in the monokinetic case N = 1 leads to the hierarchy The eikonal equation is solved explicitly, As ∆φ = 0, the initial amplitude α is transported along the vector k with a phase self-modulation: In the case N = 2, no new WKB term is created, but interactions between the two modes lead to a modification of the phase modulation.As computed in [6, Section 3], we find In addition, we have for any T > 0 (independent of ε); see Corollary 5.13 and Theorem 6.5 in [6].
The leading order nonlinear interactions between the two modes correspond to the integrals in time in the exponentials.For small time though, the integrals are zero on the support of the transported amplitudes: in other words, there exists T 0 > 0 independent of ε such that for t ∈ [0, T 0 ], in agreement with the conclusion of Theorem 1.3 (up to the order of precision).The conclusion in Proposition 1.5 then follows from the property: This is possible as soon as the transport of the support of α 1 meets the support of α 2 , as transported in the above integral.Let α ∈ C ∞ 0 (R d ) supported in the ball centered at the origin, of radius 1, and set where (e 1 , . . ., e d ) is the canonical basis of R d .Setting k 2 − k 1 = λe 1 for λ > 0, we see that the above property is satisfied for some T 1 > 0. We also remark that T 1 → 0 as λ → ∞.

Appendix A. Weakly nonlinear case and creation of a new term
In this appendix, we prove that in the weakly nonlinear case, if d 2, then nonlinear interactions may lead to the creation of a new WKB term, which is a stronger phenomenon than that used in the proof of Proposition 1.5.Consider α j (x)e iϕj (x)/ε , with now N = 3, and d 2. The one-dimensional cubic case is special, as there are no nontrivial resonances, see [6].Again, we consider linear phases, Recall ( [10], see also [6,Lemma 2.2]) that the resonant set is defined by is characterized as follows: (k j , k ℓ , k m ) ∈ Res(n) when the endpoints of the vectors k j , k ℓ , k m , k n form four corners of a nondegenerate rectangle with k ℓ and k n opposing each other, or when this quadruplet corresponds to one of the following two degenerate cases: (k j = k n , k m = k ℓ ) or (k j = k ℓ , k m = k n ).Note that we always have (A.1) {(j, j, n), ((n, j, j), a j ≡ 0} ⊂ Res(n), where a j is the amplitude associated with the phase In order for the nonlinearity to create a term associated with a phase φ 4 , out of three phases associated with wave numbers k 1 , k 2 and k 3 , we must have This resonant condition is equivalent to the following conditions: and the endpoints of k 1 , k 2 and k 3 are not aligned (the case of alignment corresponds to the set on the left in (A.1)); this is possible with pairwise different k 1 , k 2 , k 3 and k 4 ∈ {k 1 , k 2 , k 3 } provided that d 2, see [6] (or [5, Section 2.6]).For instance if d = 2, we can choose, for λ > 0, k 1 = λ(1, 1), k 2 = λ(1, 0), k 3 = λ(0, 1), hence k 4 = (0, 0).
In higher dimension, we simply complete each vector by zero coordinates.Then a new term, associated with the phase φ 4 may be created by nonlinear resonance.Because of the geometric characterization of resonances, no other term can be created apart from this one, since we have completed a rectangle.The creation is effective only if the associated amplitude does not remain zero.The equation for the corresponding amplitude is If we assume that the mode 4 is not effectively created, that is a 4 ≡ 0, then the inclusion (A.1) is actually an equality, and |a k | 2 a j + i|a j | 2 a j , j = 1, 2, 3. hence a j (t, x) = α j (x − tk j )e −iSj (t,x) , j = 1, 2, 3, for some explicit real-valued phase, whose expression is irrelevant here (see [6,Section 3.1] for the formula).Given any T > 0, we may choose α 1 , α 2 , α 3 compactly supported, with disjoint supports, k 1 , k 2 , k 3 like above, so that a 2 ā1 a 3|t=T /2 ≡ 0.
This shows that the term a 4 is actually created, in the sense that a 4 does not remain trivial on [0, T ].The error estimate proved in [7] (see also [5,