Solitons in ferromagnets and the KdV Hierarchy

Solitons in ferromagnets and the KdV Hierarchy

H. Leblond

Laboratoire P.O.M.A., U.M.R-C.N.R.S. 6136 Universit´e d’Angers

2B^d Lavoisier 49045 ANGERS Cedex 01, France (Received 9 November 2000)

The higher order terms in the perturbative expansion that describes KdV solitons propagation in ferro- magnetic materials are considered. They satisfy inhomogeneous linearized KdV equations, explicitly written down. The parity and homogeneity properties of the expansion show that half of these equations admit a zero solution. Long time propagation is investigated, through the consideration of the unbounded or secular solutions and a multi-time expansion. It is governed by all equations of the KdV Hierarchy. Major result of the paper is that the multi-scale expansion can be achieved up to any order with all its terms bounded, what is a necessary condition for its convergence.

Key words: KdV Hierarchy, Higher order KdV, KdV solitons, ferromagnets PACS numbers: 03.40.K (41.20.J, 75.50.G)

1 Introduction

Investigations of the higher order equations in multi-scale expansions are important, first from the point of view of the pure soliton theory, but also for applications of this theory to various domains in Physics. Perturbation theories for solitons have been developed [1], and have found applications, e.g. in the frame of the physics of optical soli- tons in fibers, related to optical telecommunications [2]. The higher order equations are an alterna- tive way to deal with the corrections computed by these theories. From the purely theoretical point of view, it has been shown that the solvability of the higher order equations can in some sense be re- lated to the problem of the complete integrability of the starting model [3],[4]. Furthermore, a physical interpretation of the equations of the Korteweg-de Vries (KdV) Hierarchy has been found by Kraenkel, Manna and Pereira, in the frame of water waves the- ory [5],[6],[7].

In the present article, this kind of expansion is ap- plied to the problem of electromagnetic waves prop- agation in a ferro- or ferrimagnetic medium with a

negligible conductivity. Although it doesn’t take into account the properly ferrimagnetic properties of this materials, the model discussed here describes in a quite realistic way the propagation of electro- magnetic waves in ferrites [8],[9]. A wave propa- gation mode that gives rise to solitons in such a material has been discovered by the present author [10]. These solitons are described by the Korteweg- de Vries (KdV) equation, which is known to describe the propagation of long waves in shallow water [11].

Recall that the KdV equation is the first one that has been solved by the inverse scattering transform (IST) method [12],[13]. In fact, the partial differ- ential system yielded by the Maxwell and the Lan- dau equations is not linear at all. The weakly non- linear approximation, that leads to the asymptotic equations of nonlinear Schr¨odinger or Korteweg-de Vries type, isa priorinot justified here. This differs strongly from the case of nonlinear optics. But, in the same way as the scientists of the seventies built experimental situations where a linear theory can be used, the weakly nonlinear situation of a rela- tively small signal added to some constant field is considered here. A signal comparable in its ampli- 67

tude to this constant field is only rejected by tech- nical grounds: there is no physical reason for not considering it. These considerations make evident the interest for studying the higher order terms in the weakly nonlinear approximation for this physi- cal situation. This is still true even if the model is a rough approximation from another point of view.

The approximation described by the KdV equa- tion is valid up to some given propagation time only.

After this time, the higher order terms get the same order of magnitude as the main one, and therefore can no more be neglected. These terms are called secular. An approximation on larger time intervals is obtained by incorporating these corrections into the main term. This appears as a velocity renor- malization in [14], and through the consideration of the higher order time variables in the multiple time formalism of Kraenkel, Manna and Pereira [6]. The latter formalism, that involves the KdV hierarchy, is used in the present paper. The main term is shown to obey all equations of the KdV hierarchy. It is but clear that it cannot give a precise approximation of the real pulse during an infinite time: this would mean that the starting system was completely in- tegrable, what is certainly not the case, even if it has never been proved. As in [15] and [16], the deviation from integrability should appear through a modification of the soliton shape and amplitude, and through the arising of radiation, especially dur- ing wave interaction. In the perturbative formalism, these effects are taken into account by the higher order terms. Conversely, the integrability of the starting equation appears through their vanishing [3]. When considering only the main time scale, at which the KdV equation yields a correct approxi- mation, to take the higher order terms into account gives a finite but arbitrary accuracy. If propagation on a longer time is considered, the way in which the higher order terms depend on the higher order time variables must been clarified. This has only been done in a recent paper by the present author [17]: they obey the linearized KdV Hierarchy, and this allows to prove that no unbounded term does appear in the expansion. The paper gives the first example of application of this general theory.

It is organized as follows: in section 2 is described

the model and summarized the algebraic part of the multi-scale expansion. The equations of the per- turbative scheme are written down at an arbitrary order. We introduce some algebraic approach that is rather unusual in a physical frame, and close to that used for abstract mathematical studies, but it remains completely explicit. Section 3 is a study of the homogeneity and parity properties of the ex- pansion, that are of major importance for the gen- eral theory to apply. We define adequate algebraic tools, in order to study and justify these physical properties. In particular, it is found that the equa- tions of odd order have a zero r.h.s., despite the corresponding terms in the expansion do not vanish completely. This is a essential assumption in the general theory [17], which remained to be proved at least in some particular situation. Note that in [3],[5],[6] these terms did not appear at all. In sec- tion 4, the problem of secularities is presented and solved. The approximation obtained at each order is clarified. It is stated that the precision obtained depends on both the numbers of terms considered in the expansion, and on the order of magnitude of the propagation time. Then the evolution of the higher order terms with regard to the higher order time variables is considered. Section 5 yields a conclu- sion. The mathematical proofs and tools, a large part of which have been built specially, together with some more technical results, are presented in appendices: appendix A describes the derivation of the evolution equations up to any order in the multi- scale expansion. In appendix B, the linear part of the r.h.s. of these equations are computed explic- itly, and in appendix C is given the proof of some statement given in section 3.

2 An adequate perturbative formalism

2.1 The Maxwell-Landau equations We consider a classical model describing wave prop- agation in an infinite and isotropic ferromagnetic di- electric, immersed in a constant and uniform mag- netic field. It consists in the Maxwell equations, with the simplest constitutive relation, linear and

isotropic, for the electric part, and the torque equa- tion, also called Landau equation, that describes the evolution of the magnetization densityM~ in the magnetic fieldH:~

∂_tM~ =−δµ₀M~ ∧H~ (1) (δ is the gyromagnetic ratio, and, in all this pa- per, we use the notation∂_tfor the partial derivation operator _∂t^∂ ). Eq. (1), despite it neglects the in- homogeneous exchange interaction, anisotropy and damping, yields a good approximation for the de- scription of wave propagation and interactions in ferro- or ferrimagnetic media [18],[9]. The electric field E~ can be eliminated from the Maxwell equa- tions to yield the following wave equation:

−∇(~ ∇ ·~ H) + ∆~ H~ = 1

c²∂_t²(H~ +M~) (2) where ∆ is the Laplacian operator, andc= 1/√

µ₀ε˜ is the speed of light based on the dielectric constant

˜

εof the medium. ˜εis scalar and real, and defined so that the electric inductionD~ and the electric fieldE~ satisfy the relation D~ = ˜ε ~E. In fact, the use of the second-order equation (2) is not suited to the math- ematical techniques that we need to write the evo- lution equations up to any order in the multiscale expansion: the first order form (i.e., the Maxwell equations themselves) is more convenient.

After rescaling E,~ H,~ M,~ t into _c^δ2E,~ ^δµ_c⁰H,~

δµ0

c M~,ct, we get the following fundamental system:

∂_tE~ = ∇ ∧~ H~

∂_tH~ = −∇ ∧~ E~ +M~ ∧H~

∂_tM~ = −M~ ∧H~

(3)

System (3) can be written in the following matrix form:

∂_tu=A~·∇u~ +B(u, u) (4) where the functionu of the variables~xandtis val- ued inIR⁹,

u=



 E~ H~ M~



 (5)

A~ = (A_x, A_y, A_z), where A_x,A_y, A_z, are the three 9×9 antisymmetrical matrices given by:

A_s=





0 R_s 0

−R_s 0 0

0 0 0



 (s=x, y, z) (6)

with:

R_x=





0 0 0

0 0 −1

0 1 0



, R_y =





0 0 1

0 0 0

−1 0 0





R_z =





0 −1 0

1 0 0

0 0 0





(7)

Thus,A~ is defined in such a way that:

A~·∇~ =





0 ∇∧~ 0

−∇∧~ 0 0

0 0 0



 (8)

B is a bilinear symmetrical mapping fromIR⁹×IR⁹ into IR⁹, defined by:

B µ

 E~ H~ M~



,



 E~⁰ H~⁰ M~⁰





¶

=

1 2





0

M~ ∧H~⁰+M~⁰∧H~

−(M~ ∧H~⁰+M~⁰∧H)~



 (9)

2.2 Multiple time formalism

The propagation mode governed by KdV was ob- tained in ref. [10] by using the following multi-scale expansion: the fields M~ and H~ were expanded in a power series of a small parameterε:

M~ = M~₀+ε ~M₁+ε²M~₂+· · ·

H~ = H~₀+ε ~H₁+ε²H~₂+· · · (10) whereH~₀is the constant field created in the medium by the external applied field, and M~₀ the corre- sponding magnetization density. The other terms are functions of the slow variables:

ξ=ε(x−V t)

τ =ε³t (11)

It was found thatM~₁andH~₁must be zero, and that M~₂ and H~₂ are proportional to some real function ϕ, solution of the KdV equation:

∂_τϕ+q ϕ∂_ξϕ+r∂_ξ³ϕ= 0 (12) Explicit expressions for the constants q and r, and for the velocity V were also found. Note that, as in usual KdV-type expansions, there is no carrier wave, fast oscillations or Fourier expansion here.

Confusion with envelope solitons, as those described by the nonlinear Schr¨odinger (NLS) equation, in op- tical fibers, ferrites, water waves, or elsewhere, must be avoided. Here, in the same way, we expandu in a power series of ε:

u=u₀+εu₁+ε²u₂+· · · (13) We will see below that, as is described in ref.

[6],[14], secularities (linear growth of the solutions with time) appear in the higher order terms. When considering propagation times with an order of mag- nitude larger than 1/ε³, these secularities must be removed, and this is achieved, according to the mul- tiple time formalism of Kraenkel et al., and to the general theory of [17], by imposing to the terms of higher order some particular dependency with re- gard to higher order time variables (the equations of the KdV Hierarchy). Let us introduce now these variables τ₂,τ₃,..., which are defined by:

τ_j =ε^2j+1t (j≥1) (14) (Thus, τ = τ₁). The slow three-dimensional space variable writes:

~ξ= (ξ, η, ζ) =ε(~x−V t)~ (15) where V~ is a speed vector to be determined. Re- porting this expansion in eq. (4), and collecting the terms at each power in ε, we get the following equation for every nonnegative integerp :

−V~ ·∇~_~ξu_p−1+^X

j≥1

∂_τju_p−2j−1

=A~·∇~_~ξu_p−1+ ^X

k+j=p

B(u_k, u_j) (16) (We make the convention thatu_p is zero ifpis nega- tive.) We will see that equation (16) can be reduced

to an explicit recurrence formula giving 8 of the 9 degrees of freedom ofu_p, and a solvability condition, that yields, at each order, an evolution equation for this 9^th degree of freedom. The evolution equation obtained for a rankp in eq. (16) governs the evolu- tion of the term of orderε^p−3.

2.3 The evolution equation at an arbitrary order

Before we solve eq. (16), we define the boundary conditions as follows: The first termu₀=



 E~₀ H~₀ M~₀





corresponds to the steady state induced by the con- stant and uniform applied magnetic field. The mag- netization density is assumed to have its saturation value m = ||m||, with an arbitrary direction, that~ we choose in the xy-plane without loss of general- ity: m~ =



 m_x

m_t 0



 . The corresponding magnetic field is parallel to the magnetization, given byα ~m, where α is some positive real number, fixed. The applied electric field is zero. Let us call ˜m the 9- components vector ˜m =



 0 α ~m

~ m



 , and assume that u_{0 ξ→−∞}−→ m˜ , and that all other u_j and all the derivatives tend to zero asξ −→ −∞. Further- more, we will assume that all the u_j are bounded, and restrict us to one spatial dimension, that is:

V~ =



 V

0 0



 ∇~_ξ~=





∂_ξ 0 0



 (17)

It can seem rather unphysical to neglect any trans- verse variation of the wave profile, while we intend to study the longitudinal evolution up to any or- der in the perturbation theory. But it must be no- ticed that these two approximations are in fact in- dependent one from each other. It is often possible to choose experimental conditions very close to the one-dimensional case. This will justify that we ne- glect the transverse variations. On the other hand, as soon as the values taken by the order parameterε

become appreciable, the higher orders in perturba- tion theory become non negligible, whatever is the global precision expected.

Eq. (16) is reduced to a linear 3×3 algebraic system:

L ~H_n=W~_n−1 (18) in which the operatorLis defined by eq. (A10), and the r.h.s. W~_n−1 expresses in terms of the previous orders u₁, ...u_n−1 as in eq. (A11). Eq. (18) is equivalent to eq. (16), if we take into account the formulas:

E~_n= 1 V

X

j≥1

Z _ξ

−∞∂_τjE~_n−2j− 1

VR_xH~_n (19) M~_n=^X

j≥1

Z _ξ

−∞∂_τjm~_I(u_n−2j)−ΓH~_n (20) that give the rest of u_n in terms of H~_n and of the previous orders. The details of the computation and the definition of the notations are given in appendix A. It is seen that system (18) has a rank 2; thus at each order, H~_n (or u_n ) expresses in an unique way in terms of the previous orders, apart from a scalar functionϕ_nto be determined (eqs. (A18) ff.).

The searched evolution equations are the solvability conditions for eq. (18), which read:

~

m·W~_n= 0

for each n. We compute first the term involving ϕ_n : to obtain a nontrivial solution, we must im- pose some condition, that determines the velocity V. Then ϕ_n vanishes from the equation, and we look at the term involving ϕ_n−1 . Taking n = 2, we see that ϕ₁ is necessarily zero. Owing to the vanishing of some coefficients, this implies that the function ϕ_n−1 completely vanishes from the equa- tionm~ ·W~_n= 0 . Thus this equation yields an evo- lution equation forϕ_n−2. For the main termn= 4, it is the Korteweg-de Vries (KdV) equation:

∂_τ1ϕ₂+qϕ₂∂_ξϕ₂+r∂_ξ³ϕ₂= 0 (21) q andr are some scalar coefficients, explicitly given by eqs. (A43-A44). For the higher order terms, n=l+2≥5, it is the same equation, but linearized

around the main order solutionϕ₂ , and inhomoge- neous. Its r.h.s. member Ξ_l expresses in terms of the solutions ϕ₂, ϕ₃,... up to ϕ_l−1 of the previous orders, their derivatives and primitives. This equa- tion reads:

∂_τ1ϕ_l+q∂_ξ(ϕ₂ϕ_l) +r∂_ξ³ϕ_l

= Ξ_l(ϕ₂, ϕ₃, . . . ϕ_l−1) (22) 2.4 About the formal solution

We can now write a formal solution:

u^(p)_ε = Xp l=2

ε^lu_l(ξ, τ₁) (23) where u_l is given by the recurrence formula (A19), and theϕ_n functions are solutions of the equations (21) and (22). u^(p)_ε is obviously expected to give some approximation of the exact solution. In which sense this must be precise. Convergence theorems have recently been given for the multi-scale expan- sions leading to the nonlinear Schr¨odinger equation, and to the Davey-Stewartson system, in a general frame including the model considered in the present paper [19]. Leaving by side many technical points, we notice that these theorems state: for ε small enough, there exists some T, such that for times t ≤ T /ε (the quoted paper considers 3 time vari- ables, t/ε, t, and εt), the difference between the main term of the expansion and the exact solution is uniformly bounded by a O(ε). Thus the con- vergence is proved only when the slowest variable belongs to some bounded interval. We do not in- tend to give any convergence proof in the present paper, and even we will not make any attempt to specify the norms, but we feel useful to clarify what kind of convergence can be eventually conjectured here, in order to clarify the meaning of the expan- sion. Here it can be expected that the approximate solutionu^(p)_ε differs from the exact solution for some O^¡ε^p+1¢, on a time interval bounded by someT /ε³. Difficulties arise when solving eqs. (21) and (22), that a priori forbid to propagate the solution on longer times. They are discussed in section 4. This discussion uses several algebraical and symmetry properties of the expansion, that give the matter of the next section.

3 Homogeneity properties of the r.h.s. members Ξ

3.1 Homogeneity

When writing the equations of the perturbative scheme (16), we selected the coefficients of some given power of ε, thus these coefficients must have some homogeneity properties. These are essential in the general frame of the multi-time formalism.

Indeed, the introduction of the KdV Hierarchy is based on the fact that it is the only system that both is compatible and possesses the same homo- geneity properties as the terms of the perturbative expansion. Let us first define this homogeneity in a rigorous mathematical way. ϕ_k appears in the ex- pression ofu_k, which is the coefficient of orderε^kin the expansion of the fields. ∂_ξis defined as∂_x =ε∂_ξ, and, while ∂_t = −εV ∂_ξ+ε³∂_τ1 +ε⁵∂_τ5 +. . ., each

∂_τj is the coefficient of some factor of orderε^2j+1. Every termAin the expression of the r.h.s. mem- ber Ξ_n(ϕ₁, ϕ₂, . . . ϕ_n) of the linearized KdV equa- tion (22) is a polynomial expression in the func- tions ϕ₁, ϕ₂,..., and the operators ∂_ξ, ^³R_−∞^ξ ∂_τ1^´,

³R_ξ

−∞∂_τ2^´,... This is easily checked using the recur- rence formulas (A19)(A11). The important feature is that each time derivation is coupled with a spatial primitivation, and reciprocally. We callβ_k=β_k(A) the total exponent ofϕ_kinA(for eachk),θ=θ(A) the one of ∂_ξ, and α_j = α_j(A) the total exponent of ^³R_−∞^ξ ∂_τj^´ (for eachj). Then Awould appear as the coefficient of:

ε^β1(ε²)^β2(ε³)^β3. . .×(ε^k)^βk×ε^θ

×^Y

j≥1

µ1

ε×ε^2j+1

¶_αj

=ε^d(A) (24) where the exponentd(A) is equal to:

d(A) =^X

j≥1

2jα_j(A) +θ(A) +^X

k≥1

kβ_k(A) (25) u_n is a term of order εⁿ, thus is homogeneous in relation to the “d-degree” d, with:

d(u_n) =n (26)

In the same way, the r.h.s. W~_n of system (18) ap- pears there with index n=p−1, it is thus homo- geneous withd-degree:

d(W~_n) =p=n+ 1 (27) Ξ_n is computed from W~_n+2, thus is homogeneous withd-degree:

d(Ξ_n) =n+ 3 (28) The use of an explicit mathematical definition for this homogeneity allows to prove formulas (26) to (28) in a more rigorous way. They are indeed proved by mathematical induction using recurrence formu- las (A19) and (A11). This proof does not present any special difficulty and is left to the reader.

3.2 The terms of odd order

Currently, in KdV-type expansions, the odd-order terms are taken to be zero in the following sense: in the formula analogous to (10), only even order terms are written. As an example, in ref. [14], where the quantity that we call ε is called ε¹², there are only integer exponents in the expansion of the “fields”.

Here, we can see that the termsu_nwith odd values of integernare not zero. But the parity of the case studied by Kodama and Taniuti is yet partly con- served in the following sense: if no nonzero initial condition is imposed at the odd orders, the r.h.s. Ξ_n of the higher order equations and the corresponding solutionsϕ_nare zero at all odd orders. This feature is not trivial, and in the general theory it must be taken as an hypothesis. In the case of ferromagnet it is possible to prove it, using homogeneity properties of the expansion, in the following way.

Let us call Π thexy plane, and ∆ thezaxis. The proof rests on the following property: most of the operators that intervene in the algebraic computa- tion permute Π and ∆, and appear with a certain kind of parity. The “degree” associated with this parity, which we will call “δ-degree” has a rather exotic definition, because it depends simultaneously on the degree relative to theϕ_jand on the belonging of the term to Π or ∆. This definition is imposed by the algebraic properties of the problem. First we

define a Π-degree:

if~v∈IR³: δ_Π(~v) =

( 0 if ~v∈Π

1 if ~v∈∆ (29) (δ_Π(~v) is not defined in the other cases.) For the 9-components vectors, we take an analogous definition, but with the following particularity:

δ_Π(





~e

~h

~ m



) = 0 if ~h ∈ Π and m~ ∈ Π, but ~e ∈ ∆

(and in the same way, δ_Π(





~e

~h

~ m



) = 1 if ~h ∈ ∆,

~

m ∈ ∆, and ~e ∈ Π) For a polynomial expression A, in ϕ₁, ϕ₂, ϕ₃..., belonging to IR³ or to IR⁹, the δ-degree will be defined by:

δ(A) =^X

k≥1

kβ_k(A) +δ_Π(A) (30) (β_k is defined as above.) We show by mathematical induction in appendix C that:

δ(u_k)≡k[2] (31) i.e. that these two quantities have the same parity.

Let us assume that theϕ_n, with an odd value ofn, are not all equal to zero. We call n₀ the smallest odd value of n for which ϕ_n0 6= 0. The equation that governs the evolution of ϕ_n0 is m~ ·W~_n0+2 = 0 . Consider a monomialA~ofW~_n0+2 which doesn’t contain ϕ_n0. Because of (C60), and because n₀ is odd, δ(A) =~ δ(W~_n0+2) is odd. On the other hand, onlyϕ_kwith even values ofkappear inA~(the other are zero). Thus, owing to the definition of the δ- degree, is found that:

δ(A)~ ≡δ_Π(A)~ ≡1 [2]

Thus A~ belongs to ∆ and m~ ·A~ = 0 This yields finally:

Ξ_n0 = 0 (32)

ϕ_n0 is thus solution of an homogeneous linear equa- tion; if the initial condition is zero,ϕ_n0 is zero. Thus all the ϕ_n, and all the Ξ_n, with odd values ofn are zero.

3.3 Parity properties

Let us present some other interesting symmetry property of the perturbative expansion, related to the symmetry of the basic equations (1) and (3), that corresponds to a change of the exterior field in its direction, for the opposite one. Mathemati- cally, the considered transformation consists in re- placing the vector parameter m~ by−m~ . It is clear that nothing is modified if the propagation direction changes in the same time. The transformm~ 7→ −m~ is thus equivalent to the transform V 7→ −V. Be- side this, it is easily seen that eqs. (1) and (3) are not modified by the following transform σ:

~

m 7−→ m~^σ =−~m u=



 E~ H~ M~



7−→ u^σ =





−E~

−H~

−M~





à x t

! 7−→

à x^σ t^σ

!

= Ã −x

−t

!

(33)

We have, for each j:

u_j =u_j(ε(x−V t), ε³t, ε⁵t, . . .) (34) The transformσ(33) conserves this relation only ifε changes also intoε^σ =−ε. Thenξis unchanged and τ_j is transformed into −τ_j, and ϕ_k into (−1)^kϕ_k . The action of the transform σ on the perturbative expansion can thus be defined algebraically in the following way:

m_x 7−→ −m^σ _x m_t 7−→ −m^σ _t

∂_ξ 7−→^σ +∂_ξ for allj ≥1 : ^R_−∞^ξ ∂_τj 7−→^σ +^R_−∞^ξ ∂_τj for all k≥1 : ϕ_k 7−→^σ (−1)^kϕ_k

(35)

From recurrence formulas (A19) and (A11), we show by induction that:

u^σ_k = (−1)^k+1u_k

W~_n^σ = (−1)ⁿ⁺¹W~_n (36)

Thus, for all n≥1:

q^σ =q r^σ =r

and Ξ^σ_n= (−1)ⁿΞ_n (37) As seen above, Ξ¹_n= 0 for all odd value of n, thus, from (37), we see that the transformσdoesn’t mod- ify anything in the whole perturbative expansion.

4 Boundedness of the expansion terms

4.1 The problem of secularities

It is well known that the KdV equation (21) is com- pletely integrable, i.e. that the Cauchy problem for it can be solved by use of the Inverse Scatter- ing Transform (IST) method [12],[13],[20]. The IST method gives also explicit formulas for the resolu- tion of the linearized KdV equation (22) [14],[21].

Despite this latter equation is linear, the use of the IST method for solving it is necessary, while the main term ϕ₂, solution of the KdV equation, that intervenes in it as an essential parameter, can be only expressed in terms of its inverse transform. Un- fortunately, the solution computed this way is not always bounded (I mean uniformly bounded inξ as τ −→ +∞): there exists solutions linearly growing with the time τ, called secular. This phenomena occurs when a term in the r.h.s. “resonates”, that is, from the mathematical point of view, is the solu- tion of the homogeneous equation [14]. The mathe- matical characterization of these secular-producing terms is studied in [21].

When propagating during timest≤T /ε³only, in the present frame, such a secular term wτ₁ remains bounded (ifw is so):

|wτ₁|=^¯¯¯wε³t^¯¯¯≤ |w|ε³T

ε³ =|w|T (38) Therefore, the removal of the secular terms is not needed to give an expansion of the solution at any order in ε, valid on ^h0,_ε^T3

i

only. It is but nec- essary when longer time propagation is intended.

Let us consider the term of order ε⁴: it writes

ε⁴u₄ = ε^4³u^b₄+wτ₁^´ , where u^b₄ is the nonsecu- lar part andwτ₁ the secular term. When t∼T /ε⁵, it becomesε⁴u₄ 'ε⁴u^b₄ +ε²wT, of order ε². Thus u⁽²⁾_ε = ε²ϕ₂(ξ, τ₁) ˜u₁ yields no more a correct ap- proximation, even at leading order. To prolongate its validity range to such a time interval requires the removal of the secularities.

It has been stated in the documentation [6],[14], that the secular producing term are the linear ones.

Recent work [21] show that they are rather the conserved densities of the KdV equation. It con- firms but that the method of Kraenkel, Manna, and Pereira [5],[6], for removing the secular terms, is correct. This method consists in introducing addi- tional evolution equations in such a way that the linear terms in the r.h.s. cancel. These equations describe the evolution of the main term relative to the higher order time variablesτ₂,τ₃.... , and belong to the so-called KdV Hierarchy. This is described in a general frame in [17], we recall below the main results and precise the application of the theory to the present special case.

4.2 The second order equation and the KdV Hierarchy

What happens at second order

Using the above homogeneity properties, it has been shown that, if the corresponding initial conditions are zero, all the Ξ_n and all theϕ_n with odd values of the integernare zero. Then the second equation of the perturbative expansion is the equation (22) forϕ₄. The approximate solution corresponding to this step of the perturbative expansion reads:

u⁽⁴⁾_ε =ε²ϕ₂u˜₁+ε³u⁰₃(ϕ₂)

+ε^4³ϕ₄u˜₁+u⁰₄(ϕ₂)^´ (39) We use the notations defined in appendix A, eq.

(A19) and following. It can be conjectured that

¯¯

¯u⁽⁴⁾_ε −u^¯¯¯

L^∞ ∈O^¡ε^5¢fort≤T /ε³, for someT, and εsmall enough. ϕ₄ is secular, thus this approxima- tion is not valid for larger t. The secularities must thus be removed.

According to [21], the secular solutions of the lin- earized KdV equation are removed when the linear

terms vanish from the r.h.s. members of these equa- tions. This involves the computation of their lin- ear part. The r.h.s. Ξ₄(ϕ₂) of the second equation of the perturbative expansion is given by formula (B53); The indices (α_j)_j≥1, k in this sum take the following values only: ((α_j)_j≥1, k) = ((0, . . .),5);

((1,0, . . .),3); ((2,0, . . .),1); ((0,1,0, . . .),1) . Fur- thermore, Ξ((0,1,0, . . .),1) =−1 (eq. (B55)), and, asϕ₂ satisfies the KdV equation (21), we have:

Z _ξ

−∞∂_τ1ϕ₂=−r∂_ξ²ϕ₂+O₂ Thus the expression of Ξ₄(ϕ₂) reduces to:

Ξ₄(ϕ₂) =−∂_τ2ϕ₂−r₂∂_ξ⁵ϕ₂+O₂ (40) whereO₂ designates here an expression inϕ₂ with- out linear terms, and the coefficientr₂ is given by:

−r₂= Ξ((2,0, . . .),1)(−r)²

+ Ξ((1,0, . . .),3)(−r) + Ξ((0, . . .),5) (41) The removal of the secular-producing terms is achieved, according to the multi-time formalism of Kraenkel, Manna, and Pereira [6], and to the gen- eral theory of [17], by imposing the following evo- lution equation forϕ₂ , relative to the second-order time variableτ₂:

−1

16r₂∂_τ2ϕ₂ =∂_ξL²ϕ₂ (42) with:

L= 1

4∂_ξ²− q

6rϕ₂+ q 12r

Z _ξ

−∞dξ(∂_ξϕ₂). (43) It is easily checked that Ξ₄(ϕ₂) doesn’t contain any linear term more.

In fact, for t ≤ T /ε³, this procedure does not modify u⁽⁴⁾_ε except for some O^¡ε^4¢. On the other hand,u⁽²⁾_ε =ε²ϕ₂(ξ, τ₁, τ₂)˜u₁is expected to yield an approximation in the sense: ^¯¯¯u⁽²⁾_ε −u^¯¯¯

L^∞ ∈ O^¡ε^3¢ fort ≤T /ε⁵. Notice that the τ₂ evolution of ϕ₄ is a priorinot known, what implies that the approxi- mate value u⁽⁴⁾_ε of u to within a O^¡ε^5¢, is no more valid whentreaches the order of magnitude of 1/ε⁵.

Generalization by means of the KdV Hierarchy

The linear, thus the secular-producing, terms inϕ₂ are removed by the same way at each order. We impose for eachp≥2:

−1

(−4)^pr_p∂_τpϕ₂=∂_ξL^pϕ₂ (44) withLas above. The scaling coefficientr_pis defined by r₁=r and the following recurrence formula:

r_p+1 = X

(αj)1≤j≤p−1, k≥0

³P_p−1

j=12jαj

´

+k=2p+3

Ξ((α_j)_{1≤j≤p−1}, k)

p−1Y

j=1

(−r_j)^αj (45)

We use here the same scheme as in the first case p= 2. We have:

L^p= µ1

4∂_x²

¶_p

+O₁ (46)

thus eq. (44) yields:

∂_τpϕ₂ =−r_p∂_x^2p+1ϕ₂+O₂. (47) We assume that this identity is satisfied for all p ≤ p₀, and consider the r.h.s. member Ξ_n(ϕ₂, ϕ₄, . . . , ϕ_n−2) where ∂_τp0ϕ₂ appears for the first time: n = 2p₀ . Making use of eqs. (47) and (45) in the expression (B53) of Ξ_n shows that the linear terms in ϕ₂ vanish from it. This proves that the equation (44) effectively governs the τ_p0 evolu- tion of ϕ₂. By mathematical induction this state- ment is valid for any p. The equations (44) are the so-called KdV Hierarchy; it is the only com- patible set of partial differential equations that has the same homogeneity properties as the sequence of the equations of the perturbative frame. Thus the main term obeys universal equations. All the dependency with regard to the particular physical situation is contained in the scaling coefficients r_p of the higher order time variables τ_p. The study of these coefficients is left for further publication.

This concerns the main term only, the higher order terms depend highly on the initial model through

the r.h.s. members Ξ_p, and give account for most features particular to the physical situation, espe- cially the lack of integrability.

The approximate solution u⁽²⁾_ε = ε²ϕ₂(ξ, τ₁, τ₂, . . . , τ_p) defined this way is expected to differ from the exact solution u for some O^¡ε^3¢, when t goes up to T /ε^2p+1. Despite this paper does not intend nor pretend to prove the convergence, it has to be noticed that this conjecture is reasonable when all higher order terms are bounded with regard to each of the involved time variables. This point yields the matter of next subsection.

4.3 The third order and the linearized KdV Hierarchy

Which approximations can be built from the third order equations

The third equation of the perturbative expansion is eq. (22), written for l = 6. It involves the time derivatives ∂_τ1ϕ₂, ∂_τ2ϕ₂, ∂_τ3ϕ₂, ∂_τ1ϕ₄, ∂_τ2ϕ₄, and ∂_τ1ϕ₆ . Solving the corresponding step of the perturbative scheme consists thus in building 3 functions: ϕ₂(ξ, τ₁, τ₂, τ₃) of 3 time variables, ϕ₄(ξ, τ₁, τ₂) of 2 time variables, and ϕ₆(ξ, τ₁) of only one time variable. These functions will be built in such a way that they are bounded with regard to each of their time variables. Then the approximate solution u⁽²⁾_ε = ε²ϕ₂(ξ, τ₁, τ₂, τ₃)˜u₁ is expected to be close to the exact solution u up to times t with the same order of magnitude as 1/ε⁷. We give first an heuristic reasoning in order to show that this convergence conjecture is reasonable. The bound- edness will be discussed thereafter. The basic idea of the reasoning is that the order of magnitude of the rest is the same as the one of the first non- written term. We assume that the following order is solved. The functions ϕ₄ and ϕ₆ grow at most linearly with regard to the following time variable, the first non-written one, andϕ₈evolves in the same way with regard toτ₁. This statement gives account for the existence of secularities. Then, fort∼T /ε⁷, ε⁴ϕ₄ ∈ O^¡ε⁴τ₃^¢ = O^¡ε^4¢, and, in the same way, ε⁶ϕ₆ and ε⁸ϕ₈ ∈ O^¡ε^4¢. Thus the first terms of the expansion that are left by side are O^¡ε^4¢, and

the main term is an approximation within this ac- curacy (precisely, u = ε²ϕ₂u˜₂ +ε³u⁰₃ +O^¡ε^4¢).

An analogous reasoning shows that, for t ≤ T /ε⁵, u=u⁽⁵⁾_ε +O^¡ε^6¢, with:

u⁽⁵⁾_ε = ε²ϕ₂(ξ, τ₁, τ₂, τ₃)˜u₁+ε³u⁰₃

+ε^4³u⁰₄+ϕ₄(ξ, τ₁, τ₂)˜u₁^´+ε⁵u⁰₅. (48) It shows also that, fort≤T /ε³, u=u⁽⁷⁾_ε +O^¡ε^8¢. All these statements, although they are not proved, involve the boundedness of the higher order terms, not only with regard to the main time variable τ₁, but also with regard to the higher order time vari- ables, namely theτ₂-dependency ofϕ₄, at this point of the expansion.

The linearized KdV hierarchy

The discussion of the boundedness of ϕ₆ involves the computation of the linear terms in the r.h.s.

member Ξ₆(ϕ₂, ϕ₄) of the linearized KdV equation under consideration. Ξ₆(ϕ₂, ϕ₄) is given by formula (B53). We make use of the first three equations of the Hierarchy for ϕ₂ (equation (47) written for p= 1, 2, 3. For p= 1 it is the KdV equation (21) itself), and Ξ₆ reads:

Ξ₆(ϕ₂, ϕ₄) =−∂_τ2ϕ₄ +Ξ((2,0, . . .),1)

ÃZ _ξ

−∞∂_τ1

!₂

∂_ξϕ₄ +Ξ((1,0, . . .),3)

ÃZ _ξ

−∞∂_τ1

!

∂_ξ³ϕ₄ +Ξ((0, . . .),5)∂_ξ⁵ϕ₄+O₂

(49) (O₂ represents here a function ofϕ₂ andϕ₄ without linear terms). Theτ₁-evolution ofϕ₄ is known, and described by the linearized KdV equation (22), with the r.h.s. member Ξ₄(ϕ₂). Thus we have:

Z _ξ

−∞∂_τ1ϕ₄=−r∂_ξ²ϕ₄+O₂. (50) Reporting eq. (50) into (49), we get:

Ξ₆(ϕ₂, ϕ₄) =∂_τ2ϕ₄+r₂∂⁵_ξϕ₄+O₂ (51)

(HereO₂ represents some expression inϕ₂,ϕ₄, their derivatives and primitives, without linear terms.) The terms involvingϕ₄ in the r.h.s. of eq. (51) are a priorisecular producing. It is shown that they in fact vanish as follows.

Every functionϕ_n can be divided into two parts ϕ_n = ϕ⁽¹⁾_n +ϕ⁽²⁾_n . The first term ϕ⁽¹⁾_n is a solu- tion of the homogeneous linearized KdV equation with the given initial data, andϕ⁽²⁾n is a solution of the inhomogeneous equation, with the r.h.s. mem- ber Ξ_n(ϕ₂, . . . ϕ_n−2), and zero initial data. For each p, the τ_p dependency of both ϕ⁽¹⁾n and ϕ⁽²⁾n is not free. For ϕ⁽²⁾₄ , it is completely determined by the spectral transform, through the explicit resolution formula of the linearized KdV equation [14], while ϕ⁽¹⁾_n satisfies all the equations of the linearized KdV Hierarchy [17]:

∂_τpϕ⁽¹⁾_n + (−4)^pr_p∂_ξD_pϕ⁽¹⁾_n = 0 (52) In eq. (52),D_p is defined by:

D_pϕ⁽¹⁾_n = µ

d₁L^p−1+Ld₁L^p−2+· · ·

· · ·+L^p−1d₁

¶ q

6rϕ₂+L^pϕ⁽¹⁾_n (53)

and:

d₁ = 6r q

dL

dϕ₂(ϕ⁽¹⁾_n ) =−ϕ⁽¹⁾_n +1 2

Z _ξ

dξ(∂_ξϕ⁽¹⁾_n ) (54) Further, it has been shown in [21] that ϕ⁽²⁾₄ is not secular producing, whileϕ⁽¹⁾n is so. It can be admit- ted that all linear terms involvingϕ⁽¹⁾_n also are sec- ular producing, while those involvingϕ⁽²⁾_n are not.

Boundedness at third order

Eq. (52), written for n = 4 and p = 2, together with eqs. (53), (54), and (46), yields:

∂_τ2ϕ⁽¹⁾₄ =−r₂∂_ξ⁵ϕ⁽¹⁾₄ +O₂ (55) and all linear terms vanish from Ξ₆(ϕ₂, ϕ₄), except those involvingϕ⁽²⁾₄ . Because the latter are not sec- ular producing, it is seen that all secular produc- ing terms are removed, and thus ϕ₆ is not secular.

The same feature holds at every higher order. Ac- cording to [14], all nonsecular solutions of the lin- earized KdV equation (22) are bounded, thus ϕ₆ is bounded. This shows that the complete expansion can be solved with all terms bounded with regard toτ₁.

The boundedness of ϕ₄ with regard τ₂ rests on completely different grounds. The solutions ϕ₄ of the linearized KdV equation can be expanded into a sum and integral of the squared Jost functions Ψ_k [14],[21]. Obviously, its dependency with regard to the higher order time variable τ₂ appears in this expansion through the dependency of the Ψ_k and through those of the coefficients. These coefficients are, apart from factors depending on the Cauchy data, scalar products of the squared Jost function Φ_k and the initial Cauchy data ϕ₄|_t=0, or scalar products of the Φ_k with the r.h.s. member Ξ₄(ϕ₂).

Thus all terms in this expansion are functions of ϕ₂, algebraically or through the inverse scattering transform. Beside ϕ₂, they can depend only on the initial Cauchy data ϕ₄|_t=0, that is constant with regard to any time variable. Thus the dependency of ϕ₄ with regard to τ₂ comes from the one of ϕ₂ only. Thus if the regularity properties of the Hi- rota τ-function ensure that ϕ₂ is bounded with re- gard to τ₂, ϕ₄ will also be bounded. Notice that these considerations do not yield a rigorous mathe- matical proof for the boundedness of ϕ₄, for which precise regularity hypothesis, adequate norm defini- tions, and estimations would have to be given. We have, not properly speaking proved, but show that it is reasonable to admit that ϕ₄ is bounded with regard toτ₂. The same reasoning holds, in the same heuristic way, to show that the higher order terms are all bounded regard to the higher order time vari- ables. This ensures the validity of the above men- tioned long-time approximation.

5 Conclusion

We have studied the higher order terms in the per- turbative expansion that describes KdV solitons in ferromagnetic materials. Using various mathemati- cal techniques, we have been able to write down the