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Singular solutions for the plasma at the resonance
Bruno Després, Lise-Marie Imbert-Gérard, Olivier Lafitte
To cite this version:
Bruno Després, Lise-Marie Imbert-Gérard, Olivier Lafitte. Singular solutions for the plasma at the resonance. Journal de l’Ecole Polytechnique, Ecole Polytechnique, 2017. �hal-01097364�
Singular solutions for the plasma at the resonance
∗Bruno Despr´es†, Lise-Marie Imbert-G´erard‡ and Olivier Lafitte§ December 19, 2014
Abstract
Little is known on the mathematical theory of hybrid and cyclotron solutions of the Maxwell equations with the cold plasma dielectric tensor. Such equations arise in magnetic plasmas such the ones needed for the modeling the an electromagnetic wave in Tokamaks. The behavior of solutions can be extremely different to those in vacuum. This work intends to contribute to the local theory by means of original representation formulas based on special functions and a certain eikonal equation, and with a careful treatment of the singularity for the hybrid resonance.
1 Introduction
The goal of this work is to study with an original mathematical method some singular solutions of the cold plasma model at different resonances, singular meaning here that at a pointx∗ a functionusatisfies|(x−x∗)u(x)| ≥c for some c >0. In a more general sense, the function ucan also be a Dirac mass or a principal value. The problem comes from the modeling of electromagnetic waves in strongly magnetized plasmas and from the so- called resonant heating phenomenon. We more specifically consider the one species cold plasma model which represents a collection of zero-temperature electrons immersed in a uniform static magnetic field. Resonances correspond to limits of this model, and will be studied in this work thanks to the introduction of the friction of electrons on the background ions.
In slab geometry in dimension two, (x, y) ∈ R2, and time harmonic regime, there is no resonance in the equation for the transverse magnetic polarization (also denoted as O-mode equations in plasma physics), while the equations for the transverse electric polarization (also denoted as X-mode) have two resonances. This work focuses on the latter equations, that can be reduced to
−∂y2E1 +∂xy2 E2 −ǫ11E1 −ǫ12E2 = 0,
∂xy2 E1 −∂x2E2 −ǫ21E1 −ǫ22E2 = 0, (1) where the electric field (E1, E2) is transverse to the bulk magnetic field. Denote byωthe pulsation of the wave.
A first order approximation as in [17, 18, 6, 11] through the cold plasma approximation yields the following dielectric tensor
ǫ=
1− ω
2 p
(ω2−ωc2) i ωcω
2 p
ω(ω2−ωc2)
−iω(ωω2cω−2pω2c) 1−(ω2ω−p2ω2
c)
, (2)
where the cyclotron frequency is defined byωc= e|Bm0|
e while the plasma frequency plasma frequency is defined byωp=q
e2Ne
ǫ0me. Assume that the magnetic field and the electronic density depend on the horizontal variable xonly: that isB0=B0(x) and Ne=Ne(x). It corresponds to a plasma facing a wall as depicted in figure 1.
This is typically what occurs at a wall in a Tokamak where an antenna sends an electromagnetic wave toward the plasma in order to probe or to heat the plasma. Heating of a magnetic fusion plasma with such devices
∗Lise-Marie Imbert-G´erard and Bruno Despr´es acknowledge the support of ANR under contract ANR-12-BS01-0006-01. More- over this work was carried out within the framework of the European Fusion Development Agreement and the French Research Federation for Fusion Studies. It is supported by the European Communities under the contract of Association between Euratom and CEA. The views and opinions expressed herein do not necessarily reflect those of the European Commission.
†Sorbonne Universit´es, UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris, France
‡Courant Institue of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY 10012-1185.
§Universit´e Paris 13, Sorbonne Paris Cit´e, LAGA (UMR 7539) 93430 - Villetaneuse, France, and DM2S/STMF, CEA Saclay, 91191 Gif sur Yvette cedex
x Antenna
x=−L
reflected wave incident wave
Figure 1: X-mode in slab geometry: the domain. In a real physical device an antenna is on the wall on the left and sends an incident electromagnetic wave through a medium which is assumed to be infinite for simplicity.
The incident wave generates a reflected wave which can be used to identify the properties of the plasma. The medium is filled with a plasma with dielectric tensor given by (3).
is important issue for the ITER project [12]. The physical coefficients at the wall of a Tokamak are described in figure 2. More realistic plasma parameters may be much more oscillating as shown in the numerical studies [8, 9].
x B
0
x=−L
Ne
Figure 2: The physical parameters for the X-mode equations in slab geometry. The electronic densityx7→Ne(x) is low at the boundary, and increases towards a plateau. The background magnetic field B0 is here taken as constant for simplicity.
Resonances correspond to a limit of the geometric optics, as the local wave number approaches infinity. They require a careful examination since they are related to singular behavior and absorption. The interested reader can refer to the physics textbooks [17, 18], or to [6] for a historical physical paper on the topic. A recent physical reference is [10] and therein. See also for singular solutions of wave equations for the theory of cloaking [20] and for metamaterials [5, 2]. As announced earlier, it is then natural to model the effect of collisions with a bath of static ions by an additional friction term, since the energy is absorbed by the ions. Describing the behavior of the solutions as the friction parameter ν >0 approaches zero will evidence the singularity of the solutions to the limit problem for ν = 0. The singular solutions will be built by means of new explicit representation formulas which allow a careful extraction of the singularities.
The singular behavior of the limit solutions is strongly related to the structure of the regularized dielectric tensor. The complex pulsationωe=ω+iνshifts the exact pulsation in the upper half-plane by the factorν >0.
So the regularized dielectric tensor reads
ǫν =
1− ωωe
2 p
ω(ωe2−ω2c) i ωcω
2 p
ω(ωe2−ω2c)
−i ωcω
2 p
ω(ωe2−ω2c) 1− ωωe
2 p
ω(eω2−ωc2)
, (3)
where ωp andωc respectively are the plasma and cyclotron frequencies, and the regularized dielectric tensor is translation invariant in they-direction. The equations are
−∂y2E1ν +∂2xyEν2 −ǫν11E1ν −ǫν12E2ν = 0,
∂xy2 E1ν −∂x2E2ν −ǫν21E1ν −ǫν22E2ν = 0. (4) For a recent mathematical treatment by means of singular integral equations refer to [11]. In fusion plasmas, the value of the friction parameter can be extremely small. For example relative values of ων ≈10−7are common in fusion plasmas. This extremely small value shows that the frictionless limit regimeν →0 is relevant for some fusion applications. In such regimes the formal first order approximation of the dielectric tensor reads
ǫν =ǫ0+iνD+O(ν2), (5)
with
ǫ0=
1− ω
2 p
ω2−ωc2 i ωcω
2 p
ω(ω2−ωc2)
−i ωcω
2 p
ω(ω2−ω2c) 1− ω
2 p
ω2−ω2c
andD=
λ1 −iλ2
iλ2 λ1
. (6)
The dissipation tensorDaccounts for the underlying physical dissipation. Its coefficients areλ1= ω
2 p(ω2+ω2c)
ω(ω2−ωc2)2 >0 and λ2= 2ωcω
2 p
(ω2−ω2c)2 ∈R. Since λ1−λ2= ω
2 p
ω(ω+ωc)2 >0, one has thatD=D∗ >0 is indeed a positive matrix.
As previously mentioned, and stressed in figure 2, the coefficients of these tensors are a priori non constant functions of the horizontal variable x.
The structure of the limit tensor (2) shows two different resonances in the limit model ν = 0, defined thanks to the dispersion relation, see [17]. The first one is related to thecyclotron resonanceand corresponds to a vanishing denominator: it writes
ωc(x) =ω
where ωc is a function of thex variable, a priori non constant. We will show that mathematical solutions of Maxwell’s equations (4) actually have no singular behavior: they are bounded in this regime, despite the fact that the dielectric coefficients are not. The other interesting regime corresponds to the vanishing of the diagonal part of ǫ0, that is to say
ωp2(x) +ω2c(x) =ω2.
This is thehybrid resonance. In this regime the dielectric tensor is non singular since the denominator is non zero, i.e. the coefficients of the dielectric tensor are bounded. However the solutions are known to be highly singular: they are not bounded near the resonance. As stressed in [11], a useful quantity that can be used to characterize the singular behavior of the solutions is the resonant heating of the plasma. The resonant heating [3, 11] is defined by
Q= lim
ν→0+
ℑ
Z
(ǫνEν·Eν)dx
(7) where the total electric field isEν = (E1ν, E2ν) and (A·B) =A·Bdenotes the hermitian product of the complex vectorsAandB. This formula has been justified in [11] where it is shown thatQ>0 in case of resonant heating using the simplificationǫν =ǫ+iνI. It means that the dissipation tensor introduced in theν expansion ofǫν (5) is diagonal and D=I. This simplification is mathematically justified since it corresponds to thelimiting absorption principle. In this work we will show that the resonant heating is well defined and takes the same value for a large class of dissipation tensors that includes physically based tensors.
The method of analysis usesWν =∂xE2ν−∂yE1νwhich is, after linearization around the bulk magnetic field, the parallel component of the magnetic field. The system (4) is then equivalent to the following first order system
∂yWν −ǫν11E1ν −ǫν12E2ν = 0,
−∂xWν −ǫν21E1ν −ǫν22E2ν = 0, Wν +∂yE1ν −∂xE2ν = 0.
Thanks to the fact that the coefficientsǫνij depend only onx, using the Fourier transform iny, denoting byiθ the Fourier coefficient, it yields
iθWν −ǫν11E1ν −ǫν12Eν2 = 0,
−dxdWν −ǫν21E1ν −ǫν22Eν2 = 0, Wν +iθE1ν −dxd Eν2 = 0.
(8) The Fourier coefficientθ can be considered as a frozen parameter from now on, and∂xwill be replaced by dxd for the derivative with respect tox. In this system, the derivative with respect toxappears only onWν and E2ν. The energy integral (7) can be expressed differently using the identity
Z b a
(ǫνEν·Eν)dx= Z b
a
iθWνE1ν− d dxWνE2ν
dx
= Z b
a
iθWνE1ν+Wν d dxE2ν
dx−Wν(b)E2ν(b) +Wν(a)E2ν(a) =− Z b
a |Wν|2dx−Wν(b)E2ν(b) +Wν(a)E2ν(a).
One obtains another expression ℑ
Z b a
(ǫνEν·Eν)dx=−ℑ Wν(b)E2ν(b)
+ℑ Wν(a)E2ν(a)
(9) which expresses a balance of energy.
It is convenient to consider a differential system with only two unknowns, namely (E2ν, Wν), while E1ν can be computed thanks to the first equation of (8) by
E1ν =iθWν−ǫν12E2ν
ǫν11 (10)
The reduced system then reads d dx
E2ν(x) Wν(x)
=Mν(x, θ)
E2ν(x) Wν(x)
, x∈R, (11)
where the matrix is defined as follows.
Definition 1. The matrix Mν(x, θ)is constructed from the dielectric tensor by
Mν(·, θ) =
−iθǫν12
ǫν11 1− θ2 ǫν11
−dν
ǫν11 −iθǫν21 ǫν11
(12)
where the coefficient dν is the determinant of the dielectric tensorǫν
dν =ǫν11ǫν22−ǫν12ǫν21. (13) The determinant of Mν(x, θ)will be denotedDν(·, θ) =det Mν(·, θ).
The determinant of the matrix can be simplified as follows Dν(·, θ) =−
iθǫν21 ǫν11
2
+ dν ǫν11
1− θ2
ǫν11
= dν
ǫν11 −θ2ǫν22 ǫν11 = dν
ǫν11 −θ2. (14) It is already foreseeable that the roots of ǫν11 will play a crucial part in some cases: the analysis of singular solutions relies on the analyticity of the coefficients of the dielectric tensor, at least locally aroundx=x∗, where x∗ is a hybrid resonance. This assumption is very useful to study the problem with a convenient shift in the complex plane. The main results of this paper stem from a precise local study of the solution of the reduced system when the entries of Mν have a simple pole atx∗, and the determinant of the matrix has a simple pole as well. The mathematical method of singularity extraction is completely new to our knowledge.
Our main results can be summarized as follows:
Lemma 2. In our geometry, the cyclotron singularity of the dielectric tensor (ω = ωc) is only an apparent singularity in the sense that the electromagnetic field is bounded and the limit heating is zeroQcyc= 0.
Theorems 1 and 2. The hybrid resonance (ω2 = ω2c +ωp2) yields a singular horizontal component of the electric field, i.e. |(x−x∗)E1ν=0| ≥c >0, and a generic positive resonant heatingQhyb>0. The limit of the horizontal component of the electric field can be expressed as the sum of a Dirac mass, of a Principal Value and of a bounded term, as in [11].
Corollary 1. The value of the hybrid resonant heating is independent of the local dissipation tensor, and moreover the local dissipation tensor can be non physical as well. The limit value for ν = 0+ of the electromagnetic unknowns is independent of the local dissipation tensor. It establishes the uniqueness of the limit solution with respect to the local dissipation tensor. This is a new result that was not possible with the techniques developed in [11] where the limit was established only forD=I.
Numerical section We provide numerical evidence of our claims with the help of a numerical method that can be used to treat other problems with same structure.
Extension to multi-species We finally provide some formulas which shows the method of singularity extrac- tion can be used for multi-species, such as ions and electrons. These formulae are easier to handle than the sum of dielectric tensors generally used both for the cyclotron resonances and the hybrid resonances.
The work is organized as follows. The cyclotron singularity of the dielectric tensor is studied in section 2, where the electromagnetic field is shown to stay bounded in the neighborhood of this singularity. The hybrid resonance is studied in detail in section 3 where the singularity is carefully extracted using convenient representation formulas with Bessel functions. We prove both the generic resonant heating and the fact that its value is independent of the local dissipation tensor. Numerical results are provided in section 4. Some alternative formulas are collected in the appendix.
2 The cyclotron resonance
The cyclotron singularity corresponds to a formal singularity in the denominator of (3). We consider the situation where there exists at least one xc in the domain such that ωc(xc) = ω. For the simplicity of the analysis, we will assume that dxdωc(xc)6= 0 so that xc is isolated. Clearly the dielectric tensor (3) is singular at xc at the limit ν = 0+, while it is not singular for the regularized system, i.e. ν 6= 0. In order to focus on the cyclotron resonance, we want it to be away from the hybrid resonance, that is ωp 6= 0. This is a generic case for a plasma.
Definition 2. A point xc is referred to as an isolated cyclotron singularity if there exists r > 0 such that ωc(xc) =ω and dxd ωc(xc)6= 0.
Note that such a singularity does not depend on ν, sinceωc does not depend on ν. Let us now consider the reduced system (11), where the system’s matrix Mν(·, θ) is defined with the physical dielectric tensor (3). We assume there is an isolated singularity xc. Then the system is well defined for all ν ∈R, and one can pass to the limit in the following sense.
Lemma 1. Assume that the magnetic field B0 and the densityNe are smooth. Assume an isolated cyclotron singularity at xc, and assume there is no hybrid resonance on [xc−r, xc+r]. Assume(Eν2, Wν)solves the well posed system (11)on[xc−r, xc+r] for all ν ∈R, with a given Cauchy data which admits a finite limit when ν goes to 0 written as (E2ν, Wν)(xc−r) = (a, b).
Then (E2ν, Wν)tends to a bounded limit (E2, W)asν tends to zero, and (E2, W) solves the limit system (11) on [xc−r, xc+r]with ν= 0 and with the same Cauchy data(E2, W)(xc−r) = (a, b)
Proof. Definition 1 with the physical tensor (3) yields after some careful calculations
Mν(·, θ) = 1 αν(·, θ)
θωcω2p (1−θ2)ω(˜ω2−ω2c)−ωω˜ p2
−ω(˜ω2−ωc2) + 2˜ωω2p−ω4p
ω −θωcωp2
(15)
whereαν(·, θ) =ω(˜ω2−ωc2)−ωω˜ p2. The determinant (14) of this matrix can be recast asDν(·, θ) = ǫdνν
11 −θ2=
ω(˜ω2−ωc2)−2˜ωω2p+ω
4p ω
αν(·,θ) −θ2. The limit value of α as ν approaches 0+ is α0(·, θ) = ω ω2−ω2c−ωp2
. Since the cyclotron resonance is not also an hybrid resonance, ωp>0, one has that
α0(xc, θ) =−ωω2p6= 0 sinceω=ωc(xc)6= 0. (16) In this situation the matrix Mν(·, θ) is locally bounded and smooth. So the solutions of the reduced system (11) are smooth locally around xc and one can pass to the limit in this system.
Note that there is still a particular behavior in the neighborhood of this cyclotron resonance, known as ’turning point behavior’ [7, 13], but which does not induce any singular regime, except in the limit ω→+∞, which is not considered here. The consequences of the representation (15) on physical quantities are characterized in the following result.
Lemma 2. Under the assumptions of Lemma 1, thenE1ν also tends to a bounded limitE1 and the value of the limit heating is zero
νlim→0+
Im
Z xc+r xc−r
(ǫνEν·Eν)dx
= 0. (17)
Proof. The horizontal component of the electric field can be evaluated using (10): after some computations the coefficients in (10) can be expressed as
iθ
ǫν11 =iθω ω˜2−ωc(·)2
αν(·, θ) and −ǫν12
ǫν11 = −iωc(·)ωp2 αν(·, θ) . Using (16), the limit ν= 0 value of these coefficients atxc is
iθ ǫ011
(xc) = 0 and −ǫ012
ǫ011
(xc) =i. (18)
So one can pass to the limit in (10) since the coefficients are bounded in the interval [xc−r, xc+r]. Therefore the horizontal part of the electric fieldE1ν admits a bounded and smooth limitE1.
Consider now the heating term. On the first hand we notice that the electric field can be split into two parts Eν=E2ν
i 1
+
E1ν−iE2ν 0
,
the first vector being an eigenvector of the dielectric tensor : ǫν i
1
= λ i
1
where the eigenvalue λ= 1− ω
2 p
ω(˜ω+ωc) is bounded so non singular atxc. The second vector isO(ν+|x−xc|) due to (10) and (18), which means that it counterbalances the singularity of the dielectric tensor. It shows that ǫνEν is bounded uniformly on [xc−r, xc+r] with respect to ν. SinceEν is also uniformly bounded, it yields that (ǫνEν,Eν) is bounded uniformly on [xc−r, xc+r] with respect toν.
On the other hand the tensorǫν tends almost everywhere (that is for x6=xc) to an hermitian tensor, so that ℑ(ǫνEν,Eν) tends almost everywhere to zero. The Lebesgue dominated convergence theorem states that
νlim→0+
ℑ
Z xc+r xc−r
(ǫνEν·Eν)dx
= 0.
The proof is ended.
It evidences the fact that there is no resonant cyclotron heating in our model. This result is compatible with the literature [17, 18, 6] where a resonant cyclotron heating is possible, but only in another configuration with wave number parallel to the bulk magnetic field. In our case it yields ∂x= 0 and so is excluded from our discussion.
3 The hybrid resonance
This section gathers the main theoretical contributions of this work. The required hypothesis on the general dielectric tensor
ǫν(x, y) =
ǫν11(x) ǫν12(x) ǫν21(x) ǫν22(x)
∈C2×2,∀ν∈R, (19) for the following work to hold will be emphasized in Assumptions 1, 2 and 3. Note that each one of these assumptions is satisfied by the physical tensor (3), but that all the following results hold for the wider range of tensors satisfying these three assumptions.
The hybrid resonance corresponds to a singularity in the limit problemν= 0, it more precisely corresponds to the vanishing ofǫ011, which occurs in the expression of thexcomponent of the electric fieldE01, see (10), as well as in the matrix of the (E20, W0) system, see (12) withν = 0.
Definition 3. A point xh is referred to as a local hybrid resonance for the limit system (4)-(19), i.e. with ν = 0, if the limit diagonal coefficient ofǫν11(x), is such that
ǫ011(xh) = 0 and∂xǫ011(xh)∈R∗ (20) and if the extra-diagonal part is locally non zero
ǫ012(xh)6= 0. (21)
We will moreover assume that
i∂νǫ011(xh)∈R∗ (22)
which is compatible with the physical context as visible in equation (6).
Going back to the physical dielectric tensor (3), one sees that ǫ011(xh) = 0 corresponds to ω2=ω2c(xh) +ωp2(xh).
This case is referred to as hybrid regime in the literature, see [17, 18, 6]. Unlike in the cyclotron resonance case, there exist singular solutions at the hybrid resonance. This is known since the seminal work of Budden [4], for an extremely particular case where ǫ011 = 1 +xx
h. A recent mathematical analysis, performed in [11]
with a singular integral equation, has evidenced the role of the limit resonant heating. Our main objective of the present work is to provide additional understanding of the resonant heating by means of a local extraction of the singularity.
Stemming directly from the definition, a first property of the 2×2 differential system (11) is the following:
Lemma 3. Assume xh is local hybrid resonance, then forν = 0the determinant (13) of the dielectric tensor is such
d0(xh) =−ǫ012(xh)2<0 (23) Hereafter follows the list of some assumptions on the general regularized tensor (19). This is required for the upcoming analysis.
Assumption 1 (Uniqueness and Analyticity of the coefficients). Assumexhis a local hybrid resonance. There existsΛ1>0 such that
• the dielectric tensorǫν is analytic in the ballx∈B(xh,Λ1)for allν ∈[−Λ1,Λ1],
• xh is the unique root ofǫ011 in B(xh,Λ1).
Note that, forFsatisfying this assumption of analyticity,F(z) =P
n≥0anzn, withP
|an|Rn<+∞forR <Λ1. One shall denote the derivative as usual by F′(z) = P
n≥0nanzn−1. One has the chain rule dxd (F(φ(x))) = F′(φ(x))dφdx(x) forφa function from an intervalI ofRto C.
Assumption 2 (Antisymmetry of the dielectric tensor). The dielectric tensorǫν is such that
ǫν12=−ǫν21,∀ν ∈[−Λ1,Λ1]. (24) As a result, the trace of the matrix Mν(x, θ)of the 2×2 system is identically zero on its domain.
The last assumption is satisfied by the physical tensor (2) and by its first order approximation (5).
Also note that as a consequence of this assumption, the determinant ofMν is less singular than expected: since all the entries of the matrix areO((ǫν11)−1), the determinant would be expected to beO((ǫν11)−2), but from (24) it actually is onlyO((ǫν11)−1). For the sake of clarity, consider the following notation:
Definition 4. Define for all ν∈[−Λ1,Λ1] aν =−iθǫν12
ǫν11 , bν = 1− θ2
ǫν11, cν =−dν
ǫν11, (25)
so that thanks to the symmetry property (24), the matrixMν introduced in Definition 1 reads Mν(x, θ) =
aν(x, θ) bν(x, θ) cν(x) −aν(x, θ)
. (26)
These quantities are well defined except possibly at the roots of ǫν11.
In the model caseǫν =ǫ0+iνDwith a diagonal dissipation tensorD=Istudied in [11], one hasǫν11=α(x) +iν and ǫν12 = iδ(x). In this case the coefficients are exactly aν(x, θ) = α(x)+iνθδ(x) , bν(x, θ) = 1− α(x)+iνθ2 and cν(x) =α(x)+iνδ2(x) −(α(x) +iν), and socν does not depend onθ.
Assumption 3. The regularized dielectric tensor coefficient ǫν11(z)is locally analytic with respect to both vari- ables z andν.
The techniques developed in this work are direct consequences of these properties. The study that follows is performed in two steps: the regularized problem, with ν 6= 0 is first analyzed, then the limit asν approaches zero is studied, to go back to the original problem. To this purpose it is crucial to develop uniform tools with respect to the regularization parameter ν, so that their properties still hold asν goes to zero.
3.1 A convenient second order equation
System (12) with matrix (26) can provide two different second order equations, by elimination of either of the two unknowns. In order to choose which of these two equations to study, consider the following consequence of Definition 3.
Lemma 4. Assume that xh is a local hybrid resonance. Then the dielectric tensor ǫν (3) is such that:
• There existsΛ2 satisfying 0<Λ2≤Λ1 andC >0 independent of the angle of incidenceθ such that
aν(x) cν(x)
≤C|θ|forx∈B(xh,Λ2)and for|ν| ≤Λ2. (27)
• The quantityabνν(x(xhh))
=ǫ
ν 12(xh)
θ
is unbounded forθ→0.
The fact that the upper extra diagonal coefficient bν does not dominate the diagonal part aν for vanishing θ results in a singularity in this regime which makes the analysis trickier than with for the equation on E2ν, therefore the formulation considered is the one on Wν:
d dx
1 cν
d dxWν
= a2ν cν
+bν− aν
cν
′!
Wν. (28)
Remark 1. At the resonance, the limit equation (28) has a ’regular singular point’ [7, 13]. Indeed, thanks to Definition 3 and Lemma 4, the c1
ν term is bounded while the ac2ν
ν,bν and
aν
cν
′
term all behave asO((ǫν)−1)at the resonance.
An adequate scaling of the unknown Wν then provides an equation with no first order term. The rescaled unknowny(x) =√ 1
−cν(x)Wν(x) indeed satisfies the equation d2y
dx2 = a2ν+bνcν−cν
aν
cν
′
+√
−cν
1
√−cν
′′!
y, x∈R. (29)
