HYBRID SINGULARITY FOR THE OBLIQUE INCIDENCE RESPONSE OF A COLD PLASMA

In this paper, we study the electromagnetic response of a cold plasma with dissipation ν > 0 in a general slab geometry (plasma density n0(x) and background magnetic field B0(x)~e3). We express the behavior of the electromagnetic field ( ~ E, ~ H) (at a given frequency ω) in the neighborhood of the hybrid resonance xh solution of ω = ωh(x). This generalizes to any incident wave vector (k1, k2, k3) the result obtained by Despres, Imbert-Gerard and Lafitte in the normal incidence case (the incoming electromagnetic field has its wave vector belonging to the orthogonal plane to ~e3, id est k3 = 0), hence treating a fully coupled system of ODEs of order 4. This is done by deducing from the complete system of ODEs an integro-differential system which differential part contains a regular singular point at ν → 0+. This new system is then solved in the neighborhood of this regular singular point using the variation of parameters method. In the oblique incidence case at the limit ν → 0+, we observe that (E2, E3, H2, H3) are integrable in the neighborhood of xh uniformly in ν → 0+, while the most singular term of the limit of ε0E1 when ν → 0+ is proportional to [P.V.( 1 x− xh )− iπsign(∂xωh(xh))δxh + a ln(x− xh) ].


Introduction
Following the seminal work of White and Chen [WC74], we study the electromagnetic field in a cold plasma under the influence of a strong magnetic field B 0 = B 0 (x) e z (such that ∇.B 0 = 0).This plasma is inhomogeneous, the density of electrons is n 0 (x), depending only also on x.The functions n 0 (x), B 0 (x) are assumed to be locally analytic.From the given functions B 0 (x), n 0 (x) and the constants e, m, ε 0 , (−e is the charge of the electron, m is its mass and ε 0 is the permittivity of the vacuum) one observes that two quantities of interest appear in this equation, which have both the dimension of a frequency: (1.1) ω c (x) = eB 0 (x) m , ω p (x) = e 2 n 0 (x) ε 0 m , These two frequencies are respectively called the cyclotron frequency and the plasma frequency.
This work was completed while the author was invited professor at IU Mathematics Department, Bloomington (Spring 2017).
The author wants to thank the anonymous referee for all his comments for improving the present paper.
From these two frequencies, one defines the upper hybrid frequency (1.2) ω h (x) = ω 2 p (x) + ω 2 c (x).The aim of the present paper is to study, in the framework of the cold plasma model, the response of the plasma under an electromagnetic wave of frequency ω, that is the electromagnetic field in the neighborhood of a point x h such that ω h (x h ) = ω, generalizing previous works as [DIGL17], [DIGW14].
Let us review the model used throughout the paper.The cold plasma model assumes that the ions and the electrons follow the classical law of motion (acceleration is equal to the magnetic force).Collisions are, nevertheless, taken into account through a dissipation parameter ν (expressed in s −1 and small with respect to the characteristic time of motion).Further, electroneutrality of the plasma is assumed.As the mass of the ions is much larger than the mass of electrons, one also neglects the velocity of ions in the plasma density current j.The motion of electrons is thus characterized by the linearized equation (hence the name of 'cold model') and the current density is given through (1.4) j = −en 0 (x) v.
The electromagnetic field ( E, B 0 + B) is given through the Maxwell equations with a source current density We assume that the electromagnetic field has a given frequency ω > 0: ( E, B)(x, y, z, t) = e −iωt ( E(x, y, z), B(x, y, z)), and we introduce ω ν = ω + iν.
Throughout the paper, we shall use E, B (and H) for the components of the electromagnetic field or for related fields.Usual studies of this problem rely on the construction of the effective dielectric tensor ε ν (see Section 2.1).The resulting PDE on E writes (1.6) ∇ ∧ ∇ ∧ E = ω 2 c 2 ε ν (x) E. As (n 0 , B 0 ) depend only on x, and as it is done in [WC74], [DIGL17], one assumes that the incident electromagnetic field outside the plasma is (1.7) ( E, B)(x, y, z) = e i(k2y+k3z) ( E(x), B(x)), where (k 2 , k 3 ) are the (y, z) components of the wave vector k of the electromagnetic field in the vacuum, such that k 2 2 + k 2 3 ≤ ε 0 µ 0 ω 2 .This assumption is in fact a consequence of the theorem of propagation of singularities for the Maxwell system of equations in R 3 × R t .In a previous paper, Despres, Imbert-Gérard and Lafitte [DIGL17] studied the so called ordinary and extraordinary modes of this plasma in the case k 3 = 0.This case is called the normal incidence case because k = (k 1 , k 2 , 0) is normal to B 0 .This work followed a first study of Despres, Imbert-Gérard and Weder [DIGW14] on slab geometry, where a particular form of n 0 (x) was assumed.
One of the main advantages of the normal incidence case is that the resulting system of ODEs consists in two decoupled systems of ODE of order 2. The system on (E 2 , B 3 ) contains the extraordinary mode, and the system on (E 3 , B 2 ) corresponds to the ordinary mode.One of the novelties of [DIGL17] was to bypass the computation of the effective dielectric tensor, hence avoiding a singularity for the ordinary mode (there is amplitude growth but not a mathematical singularity), still observing the singularity for the extraordinary mode.Bypassing this computation was done by expressing the current j and the first component of the electric field E 1 in terms of other components of the electromagnetic field.This trick applies here as well.Two assumptions are used in this paper, for ω given: (A0) There exists a unique Under assumptions (A0),(A1), the unique solution of D 0 (x) = 0 is x h and ∂ x D 0 (x h ) = 0.The point x h is the unique resonance.Moreover Despres, Imbert-Gérard and L. proved in [DIGL17] the local existence and uniqueness of X ν ∈ C, satisfying D ν (X ν ) = 0, and, for k 2 = 0, the existence of three analytic functions a ν 1 , b ν 1 , c ν 1 such that U = (E 2 , iωB 3 ) T satisfies: where the matrix is smooth in the neighborhood of X ν , with the relation (a ν 1 (X ν )) 2 + b ν 1 (X ν )c ν 1 (X ν ) = 0 (the matrix is of rank 1 at x = X ν ).Note that X ν = x h + iθν + O(ν 2 ) and θ is of the sign of ∂ x ω h (x h ).All solutions were then calculated in a neighborhood of x h , using the Bessel function of the first kind J 0 and the Bessel function of the second kind Y 0 , both being solutions of zJ + J + zJ = 0, given by T 0 being an even analytical function described in [DIGL17], ln z being given classically by ln These special functions are used also in the present paper.We treat in the present paper the case called Oblique Incidence, namely k 2 k 3 = 0.In this case, the system on ( E, H, j), H = µ −1 0 B, is transformed, using (1.7), into a system of ordinary differential equations on supplemented with relations calculating E 1 , E 3 , H 1 , H 3 (as well as j).
When considering the limit ν → 0 + , the system becomes a system of singular ODEs of order 4, which is not separable into ordinary and extraordinary modes, as it is mentioned in paragraph (c) of the Introduction of [WC74].It is stated in [WC74] that one needs to take into account Landau damping via the Vlasov equation, and that one needs to study the normal modes of the system.Furthermore, the authors mention that the problem of oblique incidence is not interesting because no new effects of wave amplification occur if k 3 = 0.However, the rigorous proof of this assertion is not present in [WC74] or in the references therein.We prove that the authors are correct by noticing that no new effects of wave amplification occur at other points and provide a rigorous mathematical proof of the amplification result in this more complicated set-up.In addition, we can, thanks to the present paper, • confirm that the point x h is a singularity of the system and wave amplification occurs at this point, • extend some of the results of [DIGL17] in the case (k 2 , k 3 ), k 2 k 3 = 0, • give a precise representation of the electromagnetic field in the neighborhood of x h .
For ν > 0 small, one has where P ν is a constant matrix of rank 1, R ν is analytic in the neighborhood of X ν , the projection of R ν (X ν ) on the image of P ν and on the image of the matrix (P ν ) T being non zero (which ensures that the system is coupled).In the limit ν → 0 + , the system has a singular point x h .Introduce b ν (x) = (x − X ν )M ν 12 (x) and a ν (x) = (x − X ν )M ν 11 (x) and obtain δ ν through (3.10).Let (1.11) Define, for λ = λ ν , of positive real part, given below by (3.3): (1.12) where It is important to use this function for representing the singular solutions because of the fact that one does not have the equality 2 ln(λ As the equation (1.10) is singular at x h for ν = 0, we have to modify the Cauchy data at x = x h for (1.9) in order to have an uniform in ν result.Instead of writing the Cauchy data for the components (E 2 , E 3 , H 2 , H 3 ), we consider the following condition where (A 0 , B 0 , C, D) is given in C 4 .Note that one chooses the notations (A 0 , B 0 ) for quantities which are not values of the fields ε 0 E 2 , ik 2 H 3 − ik 3 H 2 at any point, while C, D are the actual values at x h of the quantities (ε 0 (ik Note in addition that B 0 has nothing to do with the imposed magnetic field, and is an arbitrary constant.
We prove in the present paper the Theorem 1.1.There exists δ > 0 such that, for all 0 < |ν| ≤ δ, the unique solution This Theorem proves that the singular contribution when ν → 0 + (i.e. a non C 0 solution at ν = 0 and x → x h ) appears only through the action of M ν 2 .Remark 1.1.The matrix M ν 2 (x h ) contains terms of the form bounded uniformly in ν at x h (even worse E 2 , H 2 , H 3 are merely in L 1 ).We need to impose the modified Cauchy data (1.14) in order to track, uniformly in ν, continuous quantities, such as R(x, ν) This Remark leads to the following definition: In a similar way one calls It will be observed in Lemma 3.7 that the limit of ln(x − X ν ) when ν → 0 + is ln(x − x h ) + and the limit of ln(X ν − x) when ν → 0 + is ln(x h − x) − .We deduce the following Theorem 1.2.Under the same hypotheses as in Theorem 1.1, for each ν > 0 and for each (C, D) there exists ζ ν ∈ R given by (3.37), ζ ν = O(ν ln ν), (K ν , e ν ) ∈ R 2 given in (3.38) which limit is non zero when This second result allows us to use the absorption principle limit when . This is stated in Theorem 4.1, and this result is similar to the one obtained in [DIGL17].This singular behavior can also being captured through the value of the electromagnetic field at a point x 1 , such that have a finite limit when ν → 0 + .

One has also
Proposition 1.2.A similar result holds for Remark 1.2.The behavior, in the neighborhood of x h , of the electromagnetic field satisfying the Cauchy condition (1.15) is deduced from the given value of the field at a point x 1 at finite distance of x h .
The method that we develop in this paper for obtaining Theorems 1.1 and 1.2 amounts to studying, in the neighborhood of the point x h , a system of ODEs whose coefficients have a singularity at x h for ν = 0.This type of studies dates back to Budden [Bud61], p476 or Weyl [Wey70] where for both authors, the problem boils down to an ODE of order 2 (the Budden problem).We were able, in previous studies, to reduce rigorously a system of ODEs of order greater than 2 to a block-diagonal equivalent system of ODEs.For example, using the fact that the singularity at x h can be considered as a high frequency limit through a suitable change of variable, we obtained in [LWZ15] (for the study of the instability of a detonation profile) a linearized system of the form hU = ΘU that we studied in the high frequency regime h → 0. This was performed by block-diagonalizing Θ to deduce a block-diagonal equivalent system after further conjugation.In the study of the ablation growth rate in [Laf08], one of the key arguments was to prove that, in the neighborhood of a singularity at −∞ of the same type, one could boil down the problem to a representation using a solution of a singular ODE (namely the confluent hypergeometric equation).
In the present paper, we observe that the eigenvalues of the matrix M ν , depending on k 2 , k 3 , x, ω, ω ν , are ±k s (x), ±k r (x), where k s (x) has an infinite limit for (x, ν) → (x h , 0), while k r is regular in a complex neighborhood of (x h , 0).This renders the case studied here more complicated than the case studied in [DIGL17], because the eigenvector matrix is both complicated and singular.We propose here to treat successively the regular then the singular part of the system.The regular part of the system gives (ε 0 (ik Plugging this relation into the singular part of the system leads to an integro-differential equation on (ε 0 E 2 , ik 2 H 3 − ik 3 H 2 ).We use in this paper the variation of parameters method initiated in [DIGL17], namely constructing approximate solutions with the same singular behavior as the solutions obtained in [DIGL17] but we apply this method to an integrodifferential system, which is a powerful tool, as this method applies for example for all system of ODEs of the form (1.10).Note that the numerical analysis of the hybrid mode with a manufactured solution (see the talk of A. Nicolopoulos at WAFU, 2017 or at Waves, 2017 [Nic17]) is presented through a toy model that may be treated also with the methods of the present paper.The cold plasma model in the oblique incidence case appears already in [WC74], and the authors assert that the TM mode has not an accessible resonance.They prove this assertion only in the limit B 0 large, reverting to an ODE where they neglect terms small with respect to B 0 even if they may be singular at x h (see equations ( 59) and (65) of [WC74]).Note that the case B 0 large cannot be adressed in the present paper because it would lead to a point xh close to a cyclotron frequency.In order for this paper to be self-contained, we derive this system of equations for ( E, H, j) in Section 2 and deduce from it a system of ODEs of the form (1.10).We then choose an approximate fundamental solution for studying this integrodifferential system and obtain in Section 3 the representation of all solutions of the system of ODEs of order 4.

Cold plasma model
We consider here the real physical problem (namely coupling the equation for the current j with Maxwell equations in the cold plasma approximation).
2.1.Physical equations.In this section, we assume that the frequency is known, equal to ω.Indeed, all quantities shall depend on (x, ω).The equation on the electric current j reads:

In previous papers ([DIGW14], [DIGL17]
) one studied points of interest in the plasma, namely the cyclotron resonances (points x c such that ω c (x c ) = ω) and the hybrid resonances (points x h such that ω 2 c (x h ) + ω 2 p (x h ) = ω 2 ).We proved that, in the limit ν → 0 + , the electromagnetic field is not singular at x c (hence it is not a resonance but an artificial singularity created by the dielectric tensor which links the electric current to the electric field (see forthcoming studies [LM17])).The equation for the electric current is coupled with Maxwell's equations One considers electromagnetic waves of the form (1.7), and ( E(x), H(x), j(x)) is a solution of a system of first order differential equations in x.For convenience, we still keep the notation ∂ x for the derivative with respect to x and we omit the dependency on x in ω p and in ω c .
2.2.Physics revisited: cut-off points, resonances and turning points.This work gives more precise ideas on the notions of cut-off and resonances in the case of B 0 and n 0 depending only on x.Indeed, cut-off frequencies and resonances are, by definition in the Physics literature, obtained for B 0 , n 0 constants, not taking into account the spatial dependency.Let us observe for the moment that the cut-off points described in the Physics literature do not correspond to the cutoff points of the system of ODEs obtained after Fourier transform in (y, z), while Physics resonances correspond to resonances also for this system of ODEs, as we see below.
Most of the textbooks in Plasma Physics use the dielectric tensor approach, in which the current j is expressed in terms of E through the relation j = σ E (σ being called the conductivity tensor).See for example Stix [Sti62].One thus deduces ∇∧ H = (σ −iωε 0 Id) E. Let the dielectric tensor ε being equal to Id+ i ε0ω σ and introduce .
Denoting by the system on E writes, when E is expressed in a basis of the form ( a, e 3 ∧ a, e 3 ), with a. e 3 = 0 Physics textbooks begin by assuming that ω p and ω c are independent on x.System (2.3) is a constant coefficient system, hence one can perform the Fourier transform (replacing ∇ by k).The relation on k for which this system has a non trivial solution yields: This is called the dispersion relation.
Physics textbooks give the following expression: ), from which one deduces the definitions of cut-off points and resonance points (see [IG17] for example).
When n 0 and B 0 depend only on x, this dispersion relation is a relation between x, k 1 , k 2 , k 3 , ω.
Definition 2.1.A cut-off point x 0 for the plasma characterized by (n 0 (x), B 0 (x), ω) is a solution of F (n 0 (x 0 ), B 0 (x 0 ), 0, 0, 0, ω) = 0.A resonance for the plasma characterized by (n 0 (x), B 0 (x), ω) is a point x 0 such that there exists a sequence Note that we used a dispersion relation obtained in the case of constant coefficients to deduce definitions and properties in the case n 0 , B 0 depending on x.One then performs the Fourier transform in y, z of the system on (E, B, j), to deduce the system of ODEs (stated below in (2.13)).We observe that the eigenvalues , ω be given and consider the limit ν = 0. Assume x 0 is a point such that there exist (x n 0 , k n 1 ), solution of the dispersion relation, with It is a resonance for the problem.Assume x 0 is a point such that (x 0 , 0) is a solution of the dispersion relation.It is not a cut-off for the problem except when (k 2 , k 3 ) = (0, 0).Points (x 0 , 0) solution of the dispersion relation are turning points for the system of ODEs.

Derivation of the system of ordinary differential equations.
Let us rewrite all the equations which are needed, in the suitable order, after using the relation j = j(x)e ik2y+ik3z as well: (2.4) Observe that only four of the nine equations exhibit a derivative in x, (H 2 , E 2 , H 3 , E 3 ).We express j 1 , j 2 , E 1 through the last three equations of the system (2.4), and replace the resulting quantities in the first four equations, leading to a system of ODEs on E 2 , E 3 , H 2 , H 3 .After a straightforward but tedious calculation, one obtains, using Notation (1.8) (2.5) The singularity at X ν for the electric current which was identified as the key tool for the study of the problem is common to the framework of this paper and to the framework of [DIGL17].No other singularity occurs here in the matrix, hence no other resonance is present. .
Introduce the values equal to which is equal to 0 when x = X ν .The assertions of Lemma 2.1 are proven.The system of equations (2.4) implies the Lemma 2.2.Let T, R being given respectively by There exists three matrices G 11 (x, ν), G 12 (x, ν), G 22 (x, ν), depending on x, ν and the constants k 2 , k 3 , ω, given by (2.12) below, bounded in V such that the system of ODEs deduced from (2.4) is The matrix M ν is thus Proof.Using (2.7) in the first four equation of (2.4), one obtains This system is equivalent to expressing H 3 , E 3 with, respectively, T, H 2 and R, E 2 .For simplicity, denote by (2.12) This ends the proof of Lemma 2.2 hence the expression of the matrix M ν .Introduce (2.14) that is Lemma 2.3.The point x h is the unique resonance of the oblique incidence cold plasma model for ν = 0 according to Definition 2.2.
Proof of Lemma 2.3.Assume ν = 0 and x = x h .It is equivalent to say that (x, k 1 , k 2 , k 3 ) is solution of the dispersion relation and to say that Indeed, a solution (x, k 1 , k 2 , k 3 ) of the dispersion relation is a value of (k 1 , k 2 , k 3 ) for which the system (2.4) with ν = 0 and where the coefficients ω c and ω p are frozen at x, ∂ x being replaced by ik 1 has a non trivial solution.
The modal system (where ∂ x is replaced by ik 1 ) is p iω ε 0 E 2 .As x = x h , one deduces j 1 , j 2 , E 1 from the last three equations, and, consequently (2.16) is equivalent to (2.18) p E 3 , j 3 = 0 and H 1 = 0, hence a trivial solution.When k 3 = 0, the non trivial solutions of (2.16) are deduced from non trivial solutions of Resonances (as defined by Definition 2.2) are then points x 0 such that there exists a sequence (k n 1 , x 0 n ) such that ik n 1 is an eigenvalue of where x 0 n → x 0 and |k n 1 | → +∞.On the other side, values of (k 1 , x) such that P (n 0 (x), B 0 (x), k 1 , k 2 , k 3 ) = 0 are given by a quadratic equation on k 2 1 which coefficients depend on (x, k 2 , k 3 , ω), the coefficient of (k 2 1 ) 2 being D 0 (x).This expression extends to the case ν = 0.As a consequence, there exists two functions, smooth in V, non zero at X ν , denoted respectively by r ν and m ν , such that the solutions of Dν (x) (singular at X ν ).As eigenvalues of (2.11) are for x = x h and ν = 0 are solutions of P = 0, the eigenvalues of M 0 (x) are thus, for x = x h , equal to ± r 0 (x), ± m0(x) D0(x) .Hence, for any sequence of points x 0 n converging to x h , the sequence m0(x 0 n ) D0(x 0 n ) goes to infinity, hence x h is a resonance of the problem.No other point x 0 is associated to an infinite limit of an eigenvalue, hence Lemma 2.3 is proven.
Remark 2.1.Thanks to (2.6), the functions ik 2 (x − X ν )A ν (x), ik 2 (x − X ν )B ν (x) have a ν , b ν as limits when x → X ν , hence are bounded in a complex neighborhood of X ν (excluding X ν ) and can be extended at X ν .System (2.13) is of the form 1 (1.9)where P ν is the constant coefficient matrix of rank 1 As the eigenvalue-eigenvector decomposition does not lead to a simple behavior of the solution near X ν , we develop a new method, treating successively the regular part and the singular part.This is done in Section 3.
1 The procedure described in [LWZ15] does not apply to this system in the neighborhood of the point Xν , because there is no splitting of the eigenvalues in the neighborhood of this point (the unique eigenvalue of P ν is zero).One could have used block-diagonalization techniques, separating the singular and regular eigenvalues in the neighborhood of x h but, after inspection, the eigenvectors are extremely complicated.

Resolution of System (2.8)
One observes that, in Lemma 2.2, the ordinary differential equations on (ε 0 E 2 , T ) have coefficients A ν , B ν , C ν which have a singularity at X ν , while the ordinary differential equations on (ε 0 R, H 2 ) have regular (uniformy bounded) coefficients.Hence one calls in the sequel singular unknowns the quantities (ε 0 E 2 , T ) and regular unknowns the quantities (ε 0 R, H 2 ).
3.1.Reduction to a system of two integro-differential equations.Previous work of the author( [Laf08]) use this transformation into an integro-differential system.We express (ε 0 R, H 2 ) in terms of (ε 0 E 2 , T ), and we use the result for obtaining an integrodifferential system on (ε 0 E 2 , T ).Denote by U x h the fundamental solution of Its expression is obtained in Lemma 5.1.System (2.13) implies (3.1) The operator U x h is explicit in the case of a toy model presented in the literature (for example [Nic17], following closely the ideas of the slab geometry model of [DIGW14]), on which numerics are possible.Let us describe this model.One chooses, in equation (1.6), the following effective dielectric tensor: Note that this model is suitable only for small values of x.
One thus considers (1.6) with ω 2 µ 0 = 1, ε 0 = 1 and the above dielectric tensor.Introducing B given by i −1 ∇ ∧ E, the Maxwell equations write: (3.2) Two relations do not involve derivatives here, hence we reduce to a system of 4 equations with 4 unknowns, that we choose to be . It is a constant coefficient linear system on (B 1 , B 2 ) with source terms expressed in terms of (T, E 2 ).In this simple case, U x h is explicit and one expresses B 1 , B 2 in terms of T, E 2 and of B 1 (0), B 2 (0).(Lemma 3.1, of which the proof is left to the reader).This result leads to an integrodifferential system similar to (3.1), it is then solved using the general method.
The solution of (3.3) is equal to 3.2.An approximate fundamental matrice for (3.1).In this Section, we construct an approximate fundamental matrix M ν 2 (x), from which one deduces a Volterra equation obtained for are given in (2.14).The fundamental matrix M ν 2 (x) that shall be used in the proof of the main result below is given by (1.12).We present its construction and show that it is associated with a C 0 kernel in the Volterra equation.Let λ be any complex number and let γ ν be given by (3.12).Lemma 3.2.Let C, S be two constants.The functions ũ ṽ = CBe 1 (x) + SBe 2 (x) are solutions of the system ũ ṽ + γ ν (x) 0 0 1 0 ũ ṽ .
Proof of Lemma 3.2.As the second order singular system writes We replace into the equation for v, one obtains To avoid complicatedness, assume b ν (X ν ) > 0 (this can be achieved by considering −v instead of v in the system).If one writes u = b ν (x)U , the equation on U is where Equation (3.6) has a regular singular point at X ν hence an approximate solution can be expressed by means of the Bessel functions.The aim of what follows is to construct the approximate solutions of (3.6) and to perform the variation of parameters method.For this purpose, consider ũ and U 0 given by We have the identity Use then the relation (3.5) to define the function ṽ: (3.9) ṽ = 1 If one introduces the quantity (3.10) which rewrites where Be 1 , Be 2 are the fundamental solutions standing for the Bessel solution of the first of second kind, introduced in (1.11), where λ is to be chosen below.
The function r has a finite limit when x → X ν (by taking the limit in the complex plane) if and only if λ = λ ν such that γ ν (X ν ) = 0, that is One then deduces that r is smooth.This ends the proof of Lemma 3.3.
From now on, we will omit the subscript ν in λ ν except when considering the limit ν → 0. Using (3.11) and (2.14), one observes that the ODE satisfied by Be 1 is (3.16) Observe that Be 2 satisfies also (3.16).One deduces that Remark 3.1.A simple fundamental solution associated with the reduced system This matrix is used in the computation of manufactured solutions of the cold plasma equations (see [DIGW14], [Nic17]), and it is another approximate solution of (3.4).
Let Θ * ν be the matrix It is bounded in the neighborhood of X ν , uniformly in ν.System (3.4) rewrites From the identity M ν 1 is an approximate solution for (3.4).The additional feature of the present paper is that we replace Θ * ν by (x−X ν )r(x) 0 0 1 0 which, in addition to being bounded, is 0 at the singular point.

Contraction argument. Define the integral operator
A key ingredient of our study is the following contraction result: Proposition 3.1.Let |ν| ≤ Λ, Λ small enough.There exists δ > 0 such that T ν is a contraction on (C 1 ([x h − δ, x h + δ]) 2 , with a constant independent on ν.
Proof of Proposition 3.1.We observe that the entries of (the most singular terms which appear here are (x−X ν )(ln(x−X ν )) 2 ).The contraction lemma is thus a classical result for the first part of the operator (involving r).The C 1 behavior of the solutions comes from the fact that, uniformly in ν ≥ 0, f →

One has
Lemma 3.4.There exists δ > 0, such that for all p, p 1 > 1 there exists a constant Using the Holder inequality, it is thus bounded by The second term of the estimate is the estimate of the contribution of the regular term.It deals with integrals of the form Using the Holder inequality for (p, q) such that 1 p + 1 q = 1, one has Applying again the Holder inequality with (p 1 , q 1 ) such that 1 p1 + 1 q1 = 1 and integrating, we obtain the estimate, thanks to p for x h < x and a similar equality for x < x h : Combining the two inequalities, for all p, p 1 > 1 there exists a constant C * such that Note that the the entries of matrix used for constructing T ν , id est the continuity of (x − X ν ) ln(y − X ν ), allows us to consider C 1 solutions of (I − T ν ) −1 (f 0 ).It is an improvement of the Volterra operator that one would obtain with the manufactured solution when f 0 is a function belonging to C 1 locally in the neighborhood of x h , the coupling term being automatically in C 1 thanks to the C 0 behavior of M ν 2 .As the point x h is a singular point for the system when ν = 0, even if the system has analytic coefficients for x ∈ [x h − δ, x h + δ] for all ν > 0, one constructs in this paper another Volterra operator T ν * , which corresponds to selecting the values of (ε 0 E 2 , T ) at x * = x h in the variation of parameters method.This is done in the next subsection.
3.4.Variation of parameters methods.The aim of this Section is to prove the following Proposition.Let us fix a point x * , |x * − x h | ≥ δ 4 .Using the contraction T ν * defined by (3.19) (similar to T ν ), the inverse of (Id − T ν * ) being n≥0 (T ν * ) n , one can find all the components of the electromagnetic field for ν > 0 in terms of (ε 0 R, H 2 ) evalated at x h and (ε 0 E 2 , T ) evaluated at x * .

One deduces that (M
) is bounded for 0 < ν < δ and has a finite limit when ν → 0 + , even if (M ν 2 (x h )) −1 is singular when ν → 0 + .This leads to the unique solution of the system of ODEs of Lemma 2.2, From now on, we replace Λ and δ by min(Λ, δ) and one denotes it by δ.
(1) The singular unknowns in the system of Lemma 2.2 are given by (2) The regular unknowns in the system of Lemma 2.2 are then given by Proof.One uses the variation of parameters on the approximate fundamental solutions.Introduce The first step is to obtain (3.21) The system on (A, B) becomes: Using the operator T ν * defined above, one obtains The operator T ν * is a contraction, hence this system has a unique solution in The expression of ε 0 R H 2 is obtained by replacing (3.22) in (5.2).This ends the proof of Proposition 3.2.
We then deduce the unique solution with given data at x * for (ε 0 E 2 , T ) and given data at x h for (ε 0 R, H 2 ).This is expressed in the following fundamental Lemma: and denote by (ε 0 E 2 , T, ε 0 R, H 2 ) be the unique solution constructed through Proposition 3.2.

i) The value
) exist for all ν = 0 and has a finite limit when ν → 0 + .ii) For all (A 0 , B 0 ) ∈ C 2 , there exists is (A 0 , B 0 ) (uniform continuity with respect to initial conditions for the problem on (A, B) T for ν ≥ 0).
Proof.We have the equality Use Lemma 3.7, item i), below.The limit, when ν → 0 + , in L 1 of ln(y − X ν ) being ln(y − x h ) + and the limit when ν → 0 + in L 1 of ln(X ν − y) being ln(x h − x) − one may compute the limit of the matrix M ν 2 when ν → 0 + .In this limit, three types of terms appear: the function ln(x − X ν ), the coefficients b ν , a ν , δ ν , and and the functions J 0 , T 0 , R 0 , Z 0 .As these four functions are analytic in z 2 , the limit when ν → 0 is expressed in terms of z 2 = λ 2 0 (x − x h ).The limit of (b ν , a ν , δ ν ) is smooth and yields (b 0 , a 0 , δ 0 ).Call then M 0,+ 2 the limit of M ν 2 when ν → 0 + in L 1 ([x h − δ, x h + δ]).One deduces the limit of the operator T ν * when ν → 0 + .It can be denoted by T 0,+ * .It is a continuous operator from C 1 ([x h − δ, x h + δ]) to itself.We consider then the limit ν → 0 + in the equality at the beginning of this proof to deduce the limit of (A ν , B ν ) in terms of A * , B * , C, D, called (A 0 , B 0 ).Remark 3.2.Let ln(x−x h ) − = ln(x − x h ) + and notice that this is lim ν→0− ln(x− X ν ).The same limit is performed when ν → 0 − and one replaces ln(x − x h ) + by ln(x − x h ) − as well as ln(x h − x) − by ln(x h − x) + .
The second item comes from a careful study of this relation, solving the system on (A * , B * ), (A ν , B ν ) being given.Moreover, as the limit when ν → 0 + of this relation is one obtains (A 0 , B 0 ).As there exists C 0 independent of ν such that |T ν (f )(x h )| and Proof of Theorem 1.1.Lemma 3.5, item ii) allows us to use, uniformly in ν > 0, this argument to construct the unique solution using the operator T ν on C 1 ([x h − δ, x h + δ]) defined by (3.18) to rewrite Proposition 3.2 as well as Lemma 3.5.to obtain the expression of the solution of (1.10): Consider (A 0 , B 0 , C, D) ∈ C 4 , and assume (ε 0 R, H 2 )(x h ) = (C, D) and, for ν > 0, (M ν 2 ) −1 (x h ) One obtains, through Proposition 3.2 and through the second item of Lemma 3.5, which gives The regular unknowns are then given by (3.23) Once this solution is well defined for all ν > 0, revert to expression (3.21).The system on (A, B) T yields (by integrating from x h to x instead of integrating from x * to x and by using (A, B) T (x h ) = (A 0 , B 0 )): One then deduces the equality This allows us to introduce the matrices These are C 1 2 × 2 matrices on the (complex) ball B(x h , δ).One has Denoting by R the matrix , this ends the proof ot Theorem 1.1.
Proposition 1.1 follows.Let  be given independent of ν, and let We observe that the limit of M ν 2 (x 1 ) and of R(x 1 , ν) exist when ν → 0 + , hence the result.
3.5.Proof of Theorem 1.2.The Bessel functions J 0 and Y 0 have to be studied carefully.One knows that J 0 and T 0 are even analytical functions, hence J 0 and T 0 are odd hence there exists Z 0 , R 0 , H 0 , even analytical functions such that J 0 (z) = zZ 0 (z) and , where γ is the Euler constant.One has the equality, for In all what follows, remember that all the functions J 0 , T 0 , R 0 , Z 0 are expressed as analytic even sequences p≥0 a p z 2p and are evaluated at z 2 = λ 2 ν (x − X ν ).To simplify notations, we will denote these quantities suppressing the argument thereafter.We use the function Y * 0 (λ ν ,

√
x − X ν ) and modify the analytical functions accordingly.Our aim is to compute the leading order term of (ε 0 E 2 , T, εR, H 2 ) and, finally, of E 1 The second preliminary calculation concerns the term (M ν 2 (y)) −1 G 12 (y).Define (3.27) One has One has (3.29) . The inequality on ν a comes from the Cauchy-Schwartz inequality applied to Remark that Lemma 3.6 allows us to rewrite (3.29) as (3.31) where the vector Ãν and the matrix Bν are analytic in x (and no longer analytic in ν).In means in particular that all the terms containing ln(x − X ν ) have been collected in the matrix (x−X ν ) ln(x−X ν ) Ãν (x) 0 .One observes that, however, there exists an analytic function For simplicity, introduce One finally obtains We have a more precise estimate.Using Ãν , Bν , define There exists a vector r ν , analytic in x, such that Indeed, The expression of s ν and of ε ν 3 lead to Equality (3.34).The estimate Observe then that hence for all α < 1, one has Calculation of the super singular term.One is then left with the estimation of the 'super singular term', namely E 1 because one needs to divide by D ν (x).For this purpose, we plug (3.33) in (3.26) and we notice that ).

One used that e
, where v ν 1 is given by (3.25) and that K ν is the limit, when x → X ν , of (D 2 ).One checks that ζ ν = O(ν ln ν) thanks to (3.32) and Xν x h K ν (y)dy = O(ν).There exist an analytic function R ν 0 (x) such that R ν 0 (X ν ) = 0 and a C 0 function R ν 1 (x) (in which one collects all the terms of order at least O This equality is a consequence of J and of the existence of Rν 00 , continuous, such that Collecting all the terms containing ln(x − X ν ), and observing that the leading order term is associated with v ν 1 (x), given by (3.25), with , the value of its coefficient at X ν being also B 0 −ζ ν , one denotes by R ν 0 what is left in the coefficient of ln(x − X ν ).One has finally ν (x)− ν (Xν ) x−Xν bounded and continuous, hence is added to the term R ν 1 (x).Theorem 1.2 is proven.In the next paragraph, we study the limit of the terms appearing in all the quantities.Evaluation of the singular integrals.We have the two following Lemmas Lemma 3.7.i) The limit in L 1 ([x h − δ, x h + δ]) of ln(x − X ν ) is the function ln(x − x h ), x > x h , ln(x h − x) + iπsign(i∂ ν X ν | ν=0 ), x < x h .This limit is denoted by ln(x − x h ) + , x = x h (as in [DIGL17]).ii) One has, in D , lim ν→0+ 1 x − X ν = P.V.( ) − isign(i∂ ν X ν | ν=0 )δ x h .
iii) The limit in D of Observe in addition that T ν (f )(x) − T ν (f )(X ν ) = O(|X ν − x h | 1 p ) for all p > 1. Proof of Lemma 3.7.Item i) is a consequence of the fact that Hence, for x < x h , x − X ν = (x h − x)[−1 + i ν x h −x 1 0 (i∂ ν X ντ dτ )], hence the result.Item ii) is a classical result of distribution theory, using and using the change of variable x − X ν = | X ν |t in the integral < 1 x−Xν , ψ >, which leads, with ψ(x+ X ν ) = ψ p (x)+xψ imp (x), to the expression dt, with ψ p (0) → ψ(x h ).Let us prove the point iii).The derivative of (y − X ν ) ln(y − X ν ) with respect to y is ln(y − X ν ) + 1, hence one has the identity ln(x − X ν ) = (x h − X ν ) ln(x h − X ν ) x − X ν + x − x h x − X ν + x x h ln(y − X ν )dy x − X ν .
The distribution 1 x−Xν converges to P.V.( 1 x−x h ) + isδ x h (s given above), hence, as the sequence of numbers (x h − X ν ) ln(x h − X ν ) converges to 0, the first term converges to 0. The function x−x h x−Xν converges to 1 in the distribution sense (product of (x − x h ) and of 1 x−Xν , which limit is P.V.( 1 x−x h ) − iπsign(∂ x ω h (x h ))δ x h ).The point iv) comes from the identity Assume that, in the vacuum (which is a region with no surrounding magnetic field and no plasma hence ω c = ω p = 0, assumed to be x > C, C large enough), the electromagnetic wave is a plane wave with k 2 = 0 (oblique incidence, not parallel to the direction of the background magnetic field).Deduce from the Cauchy problem on (E 2 , E 3 , H 2 , H 3 ) the electromagnetic field and assume it is independent on ν (this is not true and one should perform the proof with a kernel, smooth uniformly in ν ≥ 0 when the wave reaches the boundary of the plasma but we skip this step, taking into account the linearity of the equations).Introduce which limit at ν → 0 + is equal to Observe moreover that K ν and e ν have a limit when ν → 0. We rely on the following Lemma 5.1.Denote by U x * the fundamental solution (which is a matrix) of the system U = G 22 (x)U with the Cauchy data U x * (x * ) = Id.Denote by U 1 , V 1 , U 2 , V 2 the entries of U x * , such that (1) Let u 0 be given by W 0 1 + W 0 2 x x * a 12 (y)e − y x * a22(z)dz dy.The components U i , V i are given by V(u 0 )e x x * a11(y)dy = W 0 1 U 1 (x) + W 0 2 U 2 (x), K(a 21 (W 0 1 U 1 + W 0 2 U 2 )) + W 0 2 e x x * a22(y) dy = W 0 1 V 1 (x) + W 0 2 V 2 (x).