Self-similar solutions of the one-dimensional Landau-Lifshitz-Gilbert equation

We consider the one-dimensional Landau-Lifshitz-Gilbert (LLG) equation, a model describing the dynamics for the spin in ferromagnetic materials. Our main aim is the analytical study of the bi-parametric family of self-similar solutions of this model. In the presence of damping, our construction provides a family of global solutions of the LLG equation which are associated to a discontinuous initial data of infinite (total) energy, and which are smooth and have finite energy for all positive times. Special emphasis will be given to the behaviour of this family of solutions with respect to the Gilbert damping parameter. We would like to emphasize that our analysis also includes the study of self-similar solutions of the Schr\"odinger map and the heat flow for harmonic maps into the 2-sphere as special cases. In particular, the results presented here recover some of the previously known results in the setting of the 1d-Schr\"odinger map equation.


Introduction and statement of results
In this work we consider the one-dimensional Landau-Lifshitz-Gilbert equation (LLG) where m = (m 1 , m 2 , m 3 ) : R × (0, ∞) −→ S 2 is the spin vector, β ≥ 0, α ≥ 0, × denotes the usual cross-product in R 3 , and S 2 is the unit sphere in R 3 .
Here we have not included the effects of anisotropy or an external magnetic field.The first term on the right in (LLG) represents the exchange interaction, while the second one corresponds to the Gilbert damping term and may be considered as a dissipative term in the equation of motion.The parameters β ≥ 0 and α ≥ 0 are the so-called exchange constant and Gilbert damping coefficient, and take into account the exchange of energy in the system and the effect of damping on the spin chain respectively.Note that, by considering the time-scaling m(s, t) → m(s, (α 2 + β 2 ) 1/2 t), in what follows we will assume w.l.o.g. that α, β ∈ [0, 1] and α 2 + β 2 = 1. (1.1) The Landau-Lifshitz-Gilbert equation was first derived on phenomenological grounds by L. Landau and E. Lifshitz to describe the dynamics for the magnetization or spin m(s, t) in ferromagnetic materials [24,11].The nonlinear evolution equation (LLG) is related to several physical and mathematical problems and it has been seen to be a physically relevant model for several magnetic materials [19,20].In the setting of the LLG equation, of particular importance is to consider the effect of dissipation on the spin [27,7,6].
The Landau-Lifshitz family of equations includes as special cases the well-known heat-flow for harmonic maps and the Schrödinger map equation onto the 2-sphere.Precisely, when β = 0 ).The opposite limiting case of the LLG equation (that is α = 0, i.e. no dissipation/damping and therefore β = 1) corresponds to the Schrödinger map equation onto the sphere Both special cases have been objects of intense research and we refer the interested reader to [21,14,25,13] for surveys.
Of special relevance is the connection of the LLG equation with certain non-linear Schrödinger equations.This connection is established as follows: Let us suppose that m is the tangent vector of a curve in R 3 , that is m = X s , for some curve X(s, t) ∈ R 3 parametrized by the arc-length.It can be shown [7] that if m evolves under (LLG) and we define the so-called filament function u associated to X(s, t) by u(s, t) = c(s, t)e i s 0 τ(σ,t) dσ , (1.2) in terms of the curvature c and torsion τ associated to the curve, then u solves the following non-local non-linear Schrödinger equation with damping where A(t) ∈ R is a time-dependent function defined in terms of the curvature and torsion and their derivatives at the point s = 0.The transformation (1.2) was first considered in the undamped case by Hasimoto in [18].Notice that if α = 0, equation (1.3) can be transformed into the well-known completely integrable cubic Schrödinger equation.
The main purpose of this paper is the analytical study of self-similar solutions of the LLG equation of the form for some profile m : R → S 2 , with emphasis on the behaviour of these solutions with respect to the Gilbert damping parameter α ∈ [0, 1].
For α = 0, self-similar solutions have generated considerable interest [22,21,4,15,9].We are not aware of any other study of such solutions for α > 0 in the one dimensional case (see [10] for a study of self-similar solutions of the harmonic map flow in higher dimensions).However, Lipniacki [26] has studied self-similar solutions for a related model with nonconstant arc-length.On the other hand, little is known analytically about the effect of damping on the evolution of a one-dimensional spin chain.In particular, Lakshmanan and Daniel obtained an explicit solitary wave solution in [7,6] and demonstrated the damping of the solution in the presence of dissipation in the system.In this setting, we would like to understand how the dynamics of self-similar solutions to this model is affected by the introduction of damping in the equations governing the motion of these curves.
(iii) Moreover, there exists a constant C(c 0 , α, p) such that for all t > 0 for all p ∈ (1, ∞).In addition, if α > 0, (1.10) also holds for p = 1.Here, χ E denotes the characteristic function of a set E.
The graphics in Figure 1 depict the profile m c 0 ,α (s) for fixed c 0 = 0.8 and the values of α = 0.01, α = 0.2, and α = 0.4.In particular it can be observed how the convergence of m c 0 ,α to A ± c 0 ,α is accelerated by the diffusion α.Notice that the initial condition m c 0 ,α (s, 0) = A + c 0 ,α χ (0,∞) (s) + A − c 0 ,α χ (−∞,0) (s), (1.11) has a jump singularity at the point s = 0 whenever the vectors A + c 0 ,α and A − c 0 ,α satisfy In this situation (and we will be able to prove analytically this is the case at least for certain ranges of the parameters α and c 0 , see Proposition 1.5 below), Theorem 1.1 provides a bi-parametric family of global smooth solutions of (LLG) associated to a discontinuous singular initial data (jump-singularity).
As has been already mentioned, in the absence of damping (α = 0), singular self-similar solutions of the Schrödinger map equation were previously obtained in [15], [22] and [4].In this framework, Theorem 1.1 establishes the persistence of a jump singularity for self-similar solutions in the presence of dissipation.Some further remarks on the results stated in Theorem 1.1 are in order.Firstly, from the self-similar nature of the solutions m c 0 ,α (s, t) and the Serret-Frenet equations (1.6), it follows that the curvature and torsion associated to these solutions are of the self-similar form and given by c c 0 ,α (s, t) = c 0 √ t e − αs 2 4t and τ c 0 ,α (s, t) = βs 2 √ t . (1.12) As a consequence, the total energy E(t) of the spin m c 0 ,α (s, t) found in Theorem 1.1 is expressed as It is evident from (1.13) that the total energy of the spin chain at the initial time t = 0 is infinite, while the total energy of the spin becomes finite for all positive times, showing the dissipation of energy in the system in the presence of damping.
Secondly, it is also important to remark that in the setting of Schrödinger equations, for fixed α ∈ [0, 1] and c 0 > 0, the solution m c 0 ,α (s, t) of (LLG) established in Theorem 1.1 is associated through the Hasimoto transformation (1.2) to the filament function which solves and is such that at initial time t = 0 Here δ 0 denotes the delta distribution at the point s = 0 and √ z denotes the square root of a complex number z such that Im( √ z) > 0.
Notice that the solution u c 0 ,α (s, t) is very rough at initial time, and in particular u c 0 ,α (s, 0) does not belong to the Sobolev class H s for any s ≥ 0. Therefore, the standard arguments (that is a Picard iteration scheme based on Strichartz estimates and Sobolev-Bourgain spaces) cannot be applied at least not in a straightforward way to study the local well-posedness of the initial value problem for the Schrödinger equations (1.15).The existence of solutions of the Scrödinger equations (1.15) associated to an initial data proportional to a Dirac delta opens the question of developing a well-posedness theory for Schrödinger equations of the type considered here to include initial data of infinite energy.This question was addressed by A. Vargas and L. Vega in [29] and A. Grünrock in [12] in the case α = 0 and when A(t) = 0 (see also [2] for a related problem), but we are not aware of any results in this setting when α > 0 (see [14] for related well-posedness results in the case α > 0 for initial data in Sobolev spaces of positive index).Notice that when α > 0 , the solution (1.14) has infinite energy at the initial time, however the energy becomes finite for any t > 0.Moreover, as a consequence of the exponential decay in the space variable when α > 0, u c 0 ,α (t) ∈ H m (R), for all t > 0 and m ∈ N. Hence these solutions do not fit into the usual functional framework for solutions of the Schrödinger equations (1.15).
As already mentioned, one of the main goals of this paper is to study both the qualitative and quantitative effect of the damping parameter α and the parameter c 0 on the dynamical behaviour of the family { m c 0 ,α } c 0 ,α of self-similar solutions of (LLG) found in Theorem 1.1.Precisely, in an attempt to fully understand the regularization of the solution at positive times close to the initial time t = 0, and to understand how the presence of damping affects the dynamical behaviour of these self-similar solutions, we aim to give answers to the following questions: Q1: Can we obtain a more precise behaviour of the solutions m c 0 ,α (s, t) at positive times t close to zero?
Q2: Can we understand the limiting vectors A ± c 0 ,α in terms of the parameters c 0 and α?
In order to address our first question, we observe that, due to the self-similar nature of these solutions (see (1.5)), the behaviour of the family of solutions m c 0 ,α (s, t) at positive times close to the initial time t = 0 is directly related to the study of the asymptotics of the associated profile m c 0 ,α (s) for large values of s.In addition, the symmetries of m c 0 ,α (s) (see Theorem 1.2 below) allow to reduce ourselves to obtain the behaviour of the profile m c 0 ,α (s) for large positive values of the space variable.The precise asymptotics of the profile is given in the following theorem.
(ii) (Asymptotics).There exist an unit vector A + c 0 ,α ∈ S 2 and B + c 0 ,α ∈ R 3 such that the following asymptotics hold for all s ≥ s 0 = 4 8 + c 2 0 : ) Here, sin( φ) and cos( φ) are understood acting on each of the components of φ = (φ 1 , φ 2 , φ 3 ), with for some constants a 1 , a 2 , a 3 ∈ [0, 2π), and the vector B + c 0 ,α is given in terms of A + c 0 ,α = (A + j,c 0 ,α ) 3 j=1 by As we will see in Section 2, the convergence and rate of convergence of the solutions m c 0 ,α (s, t) of the LLG equation established in parts (ii) and (iii) of Theorem 1.1 will be obtained as a consequence of the more refined asymptotic analysis of the associated profile given in Theorem 1.
(c) We also remark that using the Serret-Frenet formulae and the asymptotics in Theorem 1.2-(ii), it is straightforward to obtain the asymptotics for the derivatives of m c 0 ,α (s, t).
(d) When α = 0 and for fixed j ∈ {1, 2, 3}, we can write φ j in (1.19) as and we recover the logarithmic contribution in the oscillation previously found in [15].Moreover, in this case the asymptotics in part (ii) represents an improvement of the one established in Theorem 1 in [15].
When α > 0, φ j behaves like and there is no logarithmic correction in the oscillations in the presence of damping.
Consequently, the phase function φ defined in (1.19) captures the different nature of the oscillatory character of the solutions in both the absence and the presence of damping in the system of equations.
(e) When α = 1, there exists an explicit formula for m c 0 ,1 , n c 0 ,1 and b c 0 ,1 , and in particular we have explicit expressions for the vectors A ± c 0 ,1 in terms of the parameter c 0 > 0 in the asymptotics given in part (ii).See Appendix.
(h) Finally, the amplitude of the leading order term controlling the wave-like behaviour of the solution m c 0 ,α (s) around A ± c 0 ,α for values of s sufficiently large is of the order c 0 e −αs 2 /4 /s, from which one observes how the convergence of the solution to its limiting values A ± c 0 ,α is accelerated in the presence of damping in the system.See Figure 1.
We conclude the introduction by stating the results answering the second of our questions.Precisely, Theorems 1.3 and 1.4 below establish the dependence of the vectors A ± c 0 ,α in Theorem 1.1 with respect to the parameters α and c 0 .Theorem 1.3 provides the behaviour of the limiting vector A + c 0 ,α for a fixed value of α ∈ (0, 1) and "small" values of c 0 > 0, while Theorem 1.4 states the behaviour of A + c 0 ,α for fixed c 0 > 0 and α close to the limiting values α = 0 and α = 1.Recall that A − c 0 ,α is expressed in terms of the coordinates of A + c 0 ,α as (see part (ii) of Theorem 1.1).
Theorem 1.4.Let c 0 > 0, α ∈ [0, 1] and A + c 0 ,α be the unit vector given in Theorem 1.2.Then A + c 0 ,α is a continuous function of α in [0, 1], and the following inequalities hold true: Here, C(c 0 ) is a positive constant depending on c 0 but otherwise independent of α.
As a by-product of Theorems 1.3 and 1.4, we obtain the following proposition which asserts that the solutions m c 0 ,α (s, t) of the LLG equation found in Theorem 1.1 are indeed associated to a discontinuous initial data at least for certain ranges of α and c 0 .Proposition 1.5.With the same notation as in Theorems 1.1 and 1.2, the following statements hold: (i) For fixed α ∈ (0, 1) there exists c * 0 > 0 depending on α such that for all c 0 ∈ (0, c * 0 ).
We would like to point out that some of our results and their proofs combine and extend several ideas previously introduced in [15] and [16].The approach we use in the proof of the main results in this paper is based on the integration of the Serret-Frenet system of equations via a Riccati equation, which in turn can be reduced to the study of a second order ordinary differential equation given by when the curvature and torsion are given by (1.7).
Unlike in the undamped case, in the presence of damping no explicit solutions are known for equation (1.29) and the term containing the exponential in the equation (1.29) makes it difficult to use Fourier analysis methods to study analytically the behaviour of the solutions to this equation.The fundamental step in the analysis of the behaviour of the solutions of (1.29) consists in introducing new auxiliary variables z, h and y defined by in terms of solutions f of (1.29), and studying the system of equations satisfied by these key quantities.As we will see later on, these variables are the "natural" ones in our problem, in the sense that the components of the tangent, normal and binormal vectors can be written in terms of these quantities.It is important to emphasize that, in order to obtain error bounds in the asymptotic analysis independent of the damping parameter α (and hence recover the asymptotics when α = 0 and α = 1 as particular cases), it will be fundamental to exploit the cancellations due to the oscillatory character of z, y and h.
The outline of this paper is the following.Section 2 is devoted to the construction of the family of self-similar solutions { m c 0 ,α } c 0 ,α of the LLG equation.In Section 3 we reduce the study of the properties of this family of self-similar solutions to that of the properties of the solutions of the complex second order complex ODE (1.29).This analysis is of independent interest.Section 4 contains the proofs of the main results of this paper as a consequence of those established in Section 3. In Section 5 we give provide some numerical results for A + c 0 ,α , as a function of α ∈ [0, 1] and c 0 > 0, which give some inside for the scattering problem and justify Remark 1.6.Finally, we have included the study of the self-similar solutions of the LLG equation in the case α = 1 in Appendix.
Acknowledgements. S. Gutiérrez and A. de Laire were supported by the British project "Singular vortex dynamics and nonlinear Schrödinger equations" (EP/J01155X/1) funded by EPSRC.S. Gutiérrez was also supported by the Spanish projects MTM2011-24054 and IT641-13.
Both authors would like to thank L. Vega for many enlightening conversations and for his continuous support.

Self-similar solutions of the LLG equation
First we derive what we will refer to as the geometric representation of the LLG equation.To this end, let us assume that m(s, t) = X s (s, t) for some curve X(s, t) in R 3 parametrized with respect to the arc-length with curvature c(s, t) and torsion τ(s, t).Then, using the Serret-Frenet system of equations (1.6), we have and thus we can rewrite (LLG) as in terms of intrinsic quantities c, τ and the Serret-Frenet trihedron { m, n, b}.
We are interested in self-similar solutions of (LLG) of the form for some profile m : R −→ S 2 .First, notice that due to the self-similar nature of m(s, t) in (2.2), from the Serret-Frenet equations (1.6) it follows that the unitary normal and binormal vectors and the associated curvature and torsion are self-similar and given by Assume that m(s, t) is a solution of the LLG equation, or equivalently of its geometric version (2.1).Then, from (2.2) As a consequence, Thus, we obtain for some positive constant c 0 (recall that we are assuming w.l.o.g. that α 2 + β 2 = 1).Therefore, in view of (2.4), the curvature and torsion associated to a self-similar solution of (LLG) of the form (2.2) are given respectively by Notice that given (c, τ) as above, for fixed time t > 0 one can solve the Serret-Frenet system of equations to obtain the solution up to a rigid motion in the space which in general may depend on t.As a consequence, and in order to determine the dynamics of the spin chain, we need to find the time evolution of the trihedron { m(s, t), n(s, t), b(s, t)} at some fixed point s * ∈ R.
The rest of the paper is devoted to establish analytical properties of the solutions { m c 0 ,α (s, t)} c 0 ,α defined by (2.9) for fixed α ∈ [0, 1] and c 0 > 0. As already mentioned, due to the self-similar nature of these solutions, it suffices to study the properties of the associated profile m c 0 ,α (•) or, equivalently, of the solution { m c 0 ,α , n c 0 ,α , b c 0 ,α } of the Serret-Frenet system (1.6) with curvature and torsion given by (2.6) and initial conditions (2.8).As we will continue to see, the analysis of the profile solution { m c 0 ,α , n c 0 ,α , b c 0 ,α } can be reduced to the study of the properties of the solutions of a certain second order complex differential equation.
3 Integration of the Serret-Frenet system

Reduction to the study of a second order ODE
Classical changes of variables from the differential geometry of curves allow us to reduce the nine equations in the Serret-Frenet system into three complex-valued second order equations (see [8,28,23]).Theses changes of variables are related to stereographic projection and this approach was also used in [15].However, their choice of stereographic projection has a singularity at the origin, which leads to an indetermination of the initial conditions of some of the new variables.For this reason, we consider in the following lemma a stereographic projection that is compatible with the initial conditions (2.8).Although the proof of the lemma below is a slight modification of that in [23, Subsections 2.12 and 7.3], we have included its proof here both for the sake of completeness and to clarify to the unfamiliar reader how the integration of the Frenet equations can be reduced to the study of a second order differential equation.
j=1 be a solution of the Serret-Frenet equations (1.6) with positive curvature c and torsion τ .Then, for each j ∈ {1, 2, 3} the function solves the equation with initial conditions Moreover, the coordinates of m, n and b are given in terms of f j and f ′ j by The above relations are valid at least as long as m j > −1 and |f j | > 0.
Proof.For simplicity, we omit the index j.The proof relies on several transformations that are rather standard in the study of curves.First we define the complex function On the other hand, the Serret-Frenet equations imply that Using again the Serret-Frenet equations, we also obtain Let us consider now the auxiliary function Differentiating and using (3.4), (3.5) and (3.6) Noticing that we can recast the relation and recalling the definition of ϕ in (3.6), we have ϕN = 1 − m, so that Finally, define the stereographic projection of (m, n, b) by Observe that from the definitions of N and ϕ, respectively in (3.3) and (3.6), we can rewrite η as η = ϕe −i s 0 τ (σ) dσ , and from (3.7) it follows that η solves the Riccati equation (recall that ψ = ce i s 0 τ (σ) dσ ).Finally, setting we get A straightforward calculation shows that the inverse transformation of the stereographic projection is so that we obtain (3.2) using (3.11) and the above identities.
Going back to our problem, Lemma 3.1 reduces the analysis of the solution { m, n, b} of the Serret-Frenet system (1.6) with curvature and torsion given by (2.6) and initial conditions (2.8) to the study of the second order differential equation with three initial conditions: For (m 1 , n 1 , b 1 ) = (1, 0, 0) the associated initial condition for f 1 is and for (m 3 , n 3 , b 3 ) = (0, 0, 1) is It is important to notice that, by multiplying (3.12) by f ′ and taking the real part, it is easy to see that d ds Thus, with E 0 a constant defined by the value of E(s) at some point s 0 ∈ R. The conservation of the energy E(s) allows us to simplify the expressions of m j , n j and b j for j ∈ {1, 2, 3} in the formulae (3.2) in terms of the solution f j to (3.12) associated to the initial conditions (3.13)-(3.15).
Indeed, on the one hand notice that the energies associated to the initial conditions (3.13)-(3.15)are respectively On the other hand, from (3.16), it follows that Therefore, from (3.17), the above identity and formulae (3.2) in Lemma 3.1, we conclude that The above identities give the expressions of the tangent, normal and binormal vectors in terms of the solutions {f j } 3 j=1 of the second order differential equation (3.12) associated to the initial conditions (3.13)- (3.15).By Lemma 3.1, the formulae (3.18) and (3.19) are valid as long as m j > −1, which is equivalent to the condition |f j | = 0.As shown in Appendix, for α = 1 there is s > 0 such that m j (s) = −1 and then (3.18) and (3.19) are (a priori) valid just in a bounded interval.However, the trihedron { m, n, b} is defined globally and f j can also be extended globally as the solution of the linear equation (3.12).Then, it is simple to verify that the functions given by the l.h.s. of formulae (3.18) and (3.19) satisfy the Serret-Frenet system and hence, by the uniqueness of the solution, the formulae (3.18) and (3.19) are valid for all s ∈ R.

The second-order equation. Asymptotics
In this section we study the properties of the complex-valued equation for fixed c 0 > 0, α ∈ [0, 1), β > 0 such that α 2 + β 2 = 1.We begin noticing that in the case α = 0, the solution can be written explicitly in terms of parabolic cylinder functions or confluent hypergeometric functions (see [1]).Another analytical approach using Fourier analysis techniques has been taken in [15], leading to the asymptotics as s → ∞, where the constants C 1 , C 2 and O(1/s 2 ) depend on the initial conditions and c 0 .
For α = 1, equation (3.20) can be also solved explicitly and the solution is given by In the case α ∈ (0, 1), one cannot compute the solutions of (3.20) in terms of known functions and we will follow a more analytical analysis.In contrast with the situation when α = 0, it is far from evident to use Fourier analysis to study (3.20) when α > 0.
For the rest of this section we will assume that α ∈ [0, 1).In addition, we will also assume that s > 0 and we will develop the asymptotic analysis necessary to establish part (ii) of Theorem 1.2.At this point, it is important to recall the expressions given in (3.18)- (3.19) for the coordinates of the tangent, normal and binormal vectors associated to our family of solutions of the LLG equation in terms f .Bearing this in mind, we observe that the study of the asymptotic behaviour of these vectors are dictated by the asymptotic behaviour of the variables associated to the solution f of (3.20).
As explained in the remark (a) after Theorem 1.2, we need to work with remainder terms that are independent of α.To this aim, we proceed in two steps: first we found uniform estimates for α ∈ [0, 1/2] in Propositions 3.2 and 3.3, then we treat the case α ∈ [1/2, 1) in Lemma 3.6.In Subsection 3.3 we provide some continuity results that allows us to take α → 1 − and give the full statement in Corollary 3.14.Finally, notice that these asymptotics lead to the asymptotics for the original equation (3.20) (see Remark 3.9).We begin our analysis by establishing the following: and f be a solution of (3.20).Define z, y and h as z = |f | 2 and y + ih = f f ′ .Then (i) There exists E 0 ≥ 0 such that the identity holds true for all s ∈ R. In particular, f , f ′ , z, y and h are bounded functions.Moreover, for all exists.
(iii) Let γ := 2E 0 − c 2 0 z ∞ /2 and s 0 = 4 8 + c 2 0 .For all s ≥ s 0 , we have where Proof.Part (i) is just the conservation of energy proved in (3.16).Next, using the conservation law in part (i), we obtain that the variables {z, y, h} solve the first-order real system To show (ii), plugging (3.27) into (3.29) and integrating from 0 to some s > 0 we obtain (3.32) We continue to prove (iii).Integrating (3.31) between s > 0 and +∞ and using integration by parts, we obtain From (3.30) and (3.33), we get In order to compute the integrals in (3.34), using (3.27) and (3.28), we write Then, integrating by parts and using the bound for y in (3.24), we conclude that Finally, using (3.27) and the boundedness of z and y, an integration by parts argument shows that Bearing in mind that α 2 + β 2 = 1, from (3.37) and (3.38), we obtain the following identity for all s > 0. In order to prove (iii), we first write Therefore, we can recast (3.39) as (3.25) with we obtain Hence, choosing s 0 = 4 8 + c 2 0 , so that 4 t 2 8 + c 2 0 e −αt 2 /2 ≤ 1/2, from (3.24) and (3.25) we conclude that there exists a constant , for all α ∈ [0, 1) and t ≥ s 0 , which implies that and the proof of (iii) is completed.
Formula (3.25) in Proposition 3.2 gives z in terms of y and h.Therefore, we can reduce our analysis to that of the variables y and h or, in other words, to that of the system (3.27)-(3.29).In fact, a first attempt could be to define w = y + ih, so that from (3.28) and (3.29), we have that w solves From (3.43) in Proposition 3.2 and (3.45), we see that the limit w * = lim s→∞ w(s)e (α+iβ)s 2 /4 exists (at least when α = 0), and integrating (3.45) from some s > 0 to ∞ we find that In order to obtain an asymptotic expansion, we need to estimate ∞ s e (−α+iβ)σ 2 /4 (z − z ∞ ), for s large.This can be achieved using (3.43), and the asymptotic expansion However this estimate diverges as α → 0. The problem is that the bound used in obtaining (3.46) does not take into account the cancellations due to the oscillations.Therefore, and in order to obtain the asymptotic behaviour of z, y and h valid for all α ∈ [0, 1), we need a more refined analysis.In the next proposition we study the system (3.27)-(3.29),where we consider the cancellations due the oscillations (see Lemma 3.5 below).The following result provides estimates that are valid for s ≥ s 1 , for some s 1 independent of α, if α is small.Proposition 3.3.With the same notation and terminology as in Proposition 3.2, let . Then for all s ≥ s 1 , y(s) = be −αs Proof.First, notice that plugging the expression for z(s) − z ∞ in (3.25) into (3.28), the system (3.28)-(3.29)for the variables y and h rewrites equivalently as where and R 0 is given by (3.40).
Introducing the new variables, we recast (3.50)-(3.51)as with where R 1 is the function defined in (3.52).In this way, we can regard (3.54) as a non-autonomous system.It is straightforward to check that the matrix At this point we remark that the condition t ≥ t 1 , with t 1 := s 2 1 /4 and s 1 ≥ 2c 0 ( 1 β −1) 1/2 , implies that so that we get with F = (F, 0).From the definition of w and taking into account that u and v are real functions, we have that w 1 = w2 and therefore the study of (3.59) reduces to the analysis of the equation: with with Since we recast G as Now, from the definition of K and ∆, we have Also, since s 1 = max{4 8 + c 2 0 , 2c 0 (1/β − 1) 1/2 }, for all t ≥ t 1 = s 2 1 /4, we have in particular that t ≥ 8 + c 2 0 and t ≥ c 2 0 (1/β − 1), hence and Since e we conclude that Here we have used the inequality which follows by integrating by parts.
In order to handle the terms involving G 2 and G 3 , we need to take advantage of the oscillatory character of the involved integrals, which is exploited in Lemma 3.5.From (3.57), (3.65) and (3.66), straightforward calculations show that the function defined by f = γ/(2t 1/2 ∆ 1/2 ) satisfies the hypothesis in part (ii) of Lemma 3.5 with a = 1/2 and L = C(E 0 , c 0 )/β.Thus invoking this lemma with f = γ/(2t 1/2 ∆ 1/2 ) and noticing that where we have used (3.57) and (3.64), we conclude that with For G 3 , we first write explicitly (recall the definition of R 1 in (3.52)) For the second term, using (3.57), (3.65) and (3.63), it is easy to see that the function f defined by f = (c 2 0 γ)/(4t 3/2 ∆ 1/2 ) satisfies as a consequence, invoking part (i) of Lemma 3.5, we obtain and Indeed, recall that λ + = αK 2 + iβ∆ 1/2 so that First, we notice that where both integrals are finite in view of (3.69).Moreover, by combining with the fact that |1 − e −x | ≤ x, for x ≥ 0, we can write , for all t ≥ c 2 0 /4. (3.77) The above argument shows that with H 1 (t) satisfying (3.77).
For the second term of the eigenvalue, using the definition of ∆ in (3.55), we write Proceeding as before and using that |1 − e ix | ≤ |x|, for x ∈ R, and that we conclude that with The claim follows from the above identity, the bounds for H 1 and H 2 , and the fact that C α,c 0 . From (3.74), the claim and writing for some real constants a and b such that b ≥ 0 and a ∈ [0, 2π), it follows that

.81)
The above bound for R w 1 (t) easily follows from the bounds for R 3 (t) and H(t) in (3.74) and (3.75) respectively, and the fact that Going back to the definition of w in (3.58), we have (u, v) = P (w 1 , w 2 ), that is with The asymptotics for y and h given in (3.47) and (3.48) are a direct consequence of (3.53) and the above identities and bounds.
Finally, we compute the value of b.In fact, from (3.47) and (3.48) On the other hand, since y + ih = f f ′ and using the conservation of energy (3.16) so that, taking the limit as s → ∞ and recalling that z = |f | 2 , (3.49) follows.
Remark 3.4.From the definitions of b in (3.49), and be ia in (3.80) (in terms of C α,c 0 , w 1 (t 1 ) and w ∞ in (3.80)), it is simple to verify that b and be ia depend continuously on α ∈ [0, 1), provided that z ∞ is a continuous function of α.In Subsection 3.3 we will prove that z ∞ depends continuously on α, for α ∈ [0, 1], and establish the continuous dependence of the constants b and be ia with respect to the parameter α in Lemma 3.13 above.
In the proof of Proposition 3.3, we have used the following key lemma that establishes the control of certain integrals by exploiting their oscillatory character.Lemma 3.5.With the same notation as in the proof of Proposition 3.2.
for some constants L, a > 0.Then, for all t ≥ t 1 and l ≥ 1 Here C(l, a, c 0 ) is a positive constant depending only on l, a and c 0 .

Previous lines show that
as desired.
We remark that if α ∈ [0, 1/2], the asymptotics in Proposition 3.3 are uniform in α.Indeed, Therefore in this situation we can omit the dependence on s 1 in the function φ(s 1 ; s), because the asymptotics are valid with We continue to show that the factor 1/β 2 in the big-O in formulae (3.47) and (3.48) are due to the method used and this factor can be avoided if α is far from zero.More precisely, we have the following: integrating the above identity between s and infinity, Now, integrating by parts and using (3.41) (recall that 1 ≤ 2α), we see that Next, notice that from (3.43) in Proposition 3.2, we also obtain The above argument shows that for all s ≥ s 0 The asymptotics for y and h in the statement of the lemma easily follow from (3.94) bearing in mind that w = y + ih and recalling that the function φ behaves like (1.21) when α > 0.
In the following corollary we summarize the asymptotics for z, y and h obtained in this section.Precisely, as a consequence of Proposition 3.2-(iii), Proposition 3.3 and Lemma 3.6, we have the following: 1).With the same notation as before, for all s ≥ s 0 = 4 8 + c 2 0 , y(s) = be −αs Here, the bounds controlling the error terms depend on c 0 and the energy E 0 , and are independent of α ∈ [0, 1).
Remark 3.8.In the case when s < 0, the same arguments to the ones leading to the asymptotics in the above corollary will lead to an analogous asymptotic behaviour for the variables z, h and y for s < 0. As mentioned at the beginning of Subsection 3.2, here we have reduced ourselves to the case of s > 0 when establishing the asymptotic behaviour of the latter quantities due to the parity of the solution we will be applying these results to.
Remark 3.9.The asymptotics in Corollary 3.7 lead to the asymptotics for the solutions f of the equation ∞ is strictly positive.Indeed, this implies that there exists s * ≥ s 0 such that f (s) = 0 for all s ≥ s * .Then writing f in its polar form f = ρ exp(iθ), we have ρ 2 θ ′ = Im( f f ′ ).Hence, using (3.22), we obtain ρ = z 1/2 and θ ′ = h/z.Therefore, for all s ≥ s * , dσ. (3.98) Hence, using the asymptotics for z and h in Corollary 3.7, we can obtain the asymptotics for f .In the case that α ∈ (0, 1], we can also show that the phase converges.Indeed, the asymptotics in Corollary 3.7 yield that the integral in (3.98) converges as s → ∞ for α > 0, and we conclude that there exists a constant θ ∞ ∈ R such that The asymptotics for f is obtained by plugging the asymptotics in Corollary 3.7 into the above expression.

The second-order equation. Dependence on the parameters
The aim of this subsection is to study the dependence of the f , z, y and h on the parameters c 0 > 0 and α ∈ [0, 1].This will allow us to pass to the limit α → 1 − in the asymptotics in Corollary 3.7 and will give us the elements for the proofs of Theorems 1.3 and 1.4.

Dependence on α
We will denote by f (s, α) the solution of (3.20) with some initial conditions f (0, α), f ′ (0, α) that are independent of α.Indeed, we are interested in initial conditions that depend only on c 0 (see (3.13)-(3.15)).Moreover, in view of (3.17), we assume that the energy E 0 in (3.16) is a function of c 0 .In order to simplify the notation, we denote with a subindex α the derivative with respect to α and by ′ the derivative with respect to s. Analogously to Subsection 3.2, we define Observe that in Proposition 3.2-(ii), we proved the existence of z ∞ (α), for α ∈ [0, 1).For α ∈ (0, 1], the estimates in (3.24) hold true and hence z(s, α) is a bounded function whose derivative decays exponentially.Therefore, it admits a limit at infinity for all α ∈ [0, 1] and z ∞ (1) is well-defined.
The next lemma provides estimates for z α , h α and y α .
The estimate for z ∞ near zero is more involved and it is based in an improvement of the estimate for the derivative of z ∞ .
Lemma 3.12.The function z ∞ is continuous in [0, 1].Moreover, there exists a constant C(c 0 ) > 0, depending on c 0 but not on α such that for all α ∈ (0, Proof.As in the proof of Lemma 3.11, we recall that the functions y(s, α), h(s, α) and z(s, α) are smooth in any compact subset of R × [0, 1).From now on we will use the identity (3.39) fixing s = 1.We can verify that the two integral terms in (3.39) are continuous functions at α = 0, which proves that z ∞ is continuous in 0. In view of Lemma 3.11, we conclude that z ∞ is continuous in [0, 1].

Now we claim that
In fact, once (3.118) is proved, we can compute which implies (3.117).
It remains to prove the claim.Differentiating (3.39) (recall that s = 1) with respect to α, and using that y(1, •), h(1, •) and z(1, •) are continuous differentiable in [0, 1/2], we deduce that there exists a constant C(c 0 ) > 0 such that with and By (3.24) and (3.100), z is uniformly bounded and z α grows at most as a cubic polynomial, so that the first and the last integral in the r.h.s. of (3.120) are bounded independently of α ∈ [0, 1/2].In addition, (3.100) also implies that which shows that the remaining integral in (3.120) is bounded.
Thus, the above argument shows that The same arguments also yield that the first two integrals in the r.h.s. of (3.121) are bounded by C(c 0 )α −1/2 .Using once more that |z α | ≤ C(c 0 )s 2 α −1/2 , we obtain the following bounds for the remaining two integrals in (3.121) In conclusion, we have proved that which combined with (3.119) and (3.123), completes the proof of claim.
We end this section showing that the previous continuity results allow us to "pass to the limit" α → 1 − in Corollary 3.
and this relation is valid for any α ∈ (0, 1].Let α ∈ (0, 1).In view of (3.92), letting s → ∞, we have where C(α, c 0 ) is the constant in (1.21).Notice that the r.h.s. of (3.125) is well-defined for any α ∈ (0, 1] and by the arguments given in the proof of Lemma 3.11 and the dominated convergence theorem, the r.h.s. is also continuous for any α ∈ (0, 1].Therefore, the limit L in (3.124) exists and is given by the r.h.s. of (3.125) evaluated in α = 1 and divided by ie iC(1,c 0 ) .Moreover, , so that by the compactness of the the unit circle in C, there exists θ ∈ [0, 2π) such that e iθ = L/b(1) and we can extend a by defining a(1) = θ.
The following result summarizes an improvement of Corollary 3.7 to include the case α = 1 and the continuous dependence of the constants appearing in the asymptotics on α.Precisely, we have the following:

Dependence on c 0
In this subsection, we study the dependence of z ∞ as a function of c 0 , for a fixed value of α.
To this aim, we need to take into account the initial conditions given in (3.13)- (3.15).More generally, let us assume that f is a solution of (3.20) with initial conditions f (0) and f ′ (0) that depend smoothly on c 0 , for any c 0 > 0, and that E 0 > 0 is the associated energy defined in (3.16).To keep our notation simple, we omit the parameter c 0 in the functions f and z ∞ .Under these assumptions, we have Proposition 3.15.Let α ∈ [0, 1] and c 0 > 0. Then z ∞ is a continuous function of c 0 ∈ (0, ∞).Moreover if α ∈ (0, 1], the following estimate hold (3.126) Proof.Since we are assuming that the initial conditions f (0) and f ′ (0) depend smoothly on c 0 , by classical results from the ODE theory, the functions f , y, h and z are smooth with respect to s and c 0 .From (3.39) with s = 1, we have that z ∞ can be written in terms of continuous functions of c 0 (the continuity of the integral terms follows from the dominated convergence theorem), so that z ∞ depends continuously on c 0 .

Proof of the main results
In Section 3 we have performed a careful analysis of the equation (3.12), taking also into consideration the initial conditions (3.13)- (3.15).Therefore, the proofs of our main theorem consist mainly in coming back to the original variables using the identities (3.18) and (3.19).For the sake of completeness, we provide the details in the following proofs.This proves part (i) of Theorem 1.2.
Second, in Section 3 we have seen that one can write the components of the Frenet trihedron { m, n, b} as with f j solution of the second order ODE (3.12) with initial conditions (3.13)-(3.15)respectively, and associated initial energies (see (3.17)) Notice that the identities (4.1)-(4.2) rewrite equivalently as in terms of the quantities {z j , y j , h j } defined by Denote by z j,∞ , a j , b j , γ j and φ j the constants and function appearing in the asymptotics of {y j , h j , z j } proved in Section 3 in Corollary 3.14.
Taking the limit as s → +∞ in (4.1)-(4.2),and since | m(s)| = 1, we obtain that there exists The asymptotics stated in part (ii) of Theorem 1.2 easily follows from formulae (4.1)-(4.2) and the asymptotics for {z j , y j , h j } established in Corollary 3.14.Indeed, it suffices to observe that from the formulae for b j and γ j in terms of the initial energies E 0,j and z j,∞ given in Corollary 3.14, (4.3) and (4.5) we obtain ) Next, from the parity of the components of the profile m(•) and the asymptotics established in parts (i) and (ii) in Theorem 1.2, it is immediate to prove the pointwise convergence (1.9).In addition, A − = (A + 1 , −A + 2 , −A + 3 ) in terms of the components of the vector A + = (A + j ) 3 j=1 .Now, using the symmetries of m(•), the change of variables η = s/ √ t gives us Therefore, it only remains to prove that the last integral is finite.To this end, let s 0 = 4 8 + c 2 0 .On the one hand, notice that since m and A + are unitary vectors, Proof of Theorem 1.3.The proof is a consequence of Proposition 3.15.In fact, recall the relations (4.5) and (3.17), that is Thus the continuity of A + c 0 ,α with respect to c 0 , follows from the continuity of z ∞ in Proposition 3.15.

Some numerical results
As has been already pointed out, only in the cases α = 0 and α = 1 we have an explicit formula for A + c 0 ,α (see (4.13)-(4.16)).Theorems 1.3 and 1.4 give information about the behaviour of A + c 0 ,α for small values of c 0 for a fixed valued of α, and for values of α near to 0 or 1 for a fixed valued of c 0 .The aim of this section is to give some numerical results that allow us to understand the map For a fixed value of α, we will discuss first the injectivity and surjectivity (in some appropriate sense) of the map c 0 → A ± c 0 ,α and second the behaviour of A + c 0 ,α as c 0 → ∞.
In Figure 2 we plot the function θ c 0 ,α associated to the family of solutions m c 0 ,α (s, t) established in Theorem 1.1 for α = 0, α = 0.4 and α = 1, as a function of c 0 > 0. The curves θ c 0 ,0 and θ c 0 ,1 are exact since we have explicit formulae for A + 1,c 0 ,α when α = 0 and α = 1 (see (4.13) and (4.16)).We deduce that in the case α = 0, there is a bijective relation between c 0 > 0 and the angles in (0, π).In the case α = 1, there are infinite values of c 0 > 0 that allow to reach any angle in [0, π].If α ∈ (0, 1), numerical simulations show that there exists θ * α ∈ (0, π) such that the angles in (θ * α , π) are reached by a unique value of c 0 , but for angles in [0, θ * α ] there are at least two values of c 0 > 0 that produce them (See θ c 0 ,0.4 in Figure 2).These numerical results suggest that, due to the invariance of (LLG) under rotations 2 , for a fixed α ∈ [0, 1) one can solve the following inverse problem: Given any distinct vectors A + , A − ∈ S 2 there exists c 0 > 0 such that the associated solution m c 0 ,α (s, t) given by Theorem 1.1 (possibly multiplied by a rotation matrix) provides a solution of (LLG) with initial condition (5.2) Note that in the case α = 1 the restriction A + = A − can be dropped.
The next natural question is the injectivity of the application c 0 −→ θ c 0 ,α , for fixed α.Precisely, can we generate the same angle using different values of c 0 ?In the case α = 0, the 2 In fact, using that it is easy to verify that if m(s, t) is a solution of (LLG) with initial condition m 0 , then mR := R m is a solution of (LLG) with initial condition m 0 R := R m 0 , for any R ∈ SO(3).
plot of θ c 0 ,0 in Figure 2 shows that the value of c 0 is unique, in fact one has following formula sin (θ c 0 ,0 /2) = A 1,c 0 ,0 = e − c 2 0 2 π (see [15]).In the case α = 1, we have sin ( As before, if α ∈ (0, 1) we do not have an analytic answer and we have to rely on numerical simulations.However, it is difficult to test the uniqueness of c 0 numerically.Using the command FindRoot in Mathematica, we have found such values.For instance, for α = 0.4, we obtain that c 0 ≈ 2.1749 and c 0 ≈ 6.6263 give the same value of A + c 0 ,0.4 .The respective profiles m c 0 ,0.4 (•) are shown in Figure 3.This multiplicity of solutions suggests that the Cauchy problem for (LLG) with initial condition (5.2) is ill-posed, at least for certain values of c 0 .This interesting problem will be studied in a forthcoming paper.The rest of this section is devoted to give some numerical results on the behaviour of the limiting vector A + c 0 ,α .In particular, the results below aim to complement those established in Theorem 1.3 on the behaviour of A + c 0 ,α for small values of c 0 , when α is fixed.We start recalling what it is known in the extremes cases α = 0 and α = 1.Precisely, if α = 0, the explicit formulae (4.13)-(4.15)for A + c 0 ,0 allow us to prove that and also that {A + 3,c 0 ,0 : c 0 ∈ (0, ∞)} = (0, 1).When α = 1 the picture is completely different.In fact A + 3,c 0 ,1 = 0 for all c 0 > 0, and the limit vectors remain in the equator plane S 1 × {0}.The natural question is what happens with A + c 0 ,α when α ∈ (0, 1) as a function of c 0 .Although we do not provide a rigorous answer to this question, in Figure 4 we show some numerical results.Precisely, Figure 4 depicts the curves A + c 0 ,0.01 , A + c 0 ,0.4 and A + c 0 ,0.8 as functions of c 0 , for c 0 ∈ [0, 1000].We see that the behaviour of A + c 0 ,α changes when α increases in the sense that the first and second coordinates start oscillating more and more as α goes to 1.In all the cases the third component remains monotonically increasing with c 0 , but the value of A + 3,1000,α seems to be decreasing with α.At this point it is not clear what the limit value of A + 3,c 0 ,α as c 0 → ∞ is.For this reason, we perform a more detailed analysis of A + 3,c 0 ,α and we show the curves A + 3,1,α , A + 3,10,α , A + 3,1000,α (for fixed α ∈ [0, 1]) in Figure 5. From these results we conjecture that {A + 3,c 0 ,• } c 0 >0 is a pointwise nondecreasing sequence of functions that converges to 1 for any α < 1 as c 0 → ∞.This would imply that, for α ∈ (0, 1) fixed, A 1,c 0 ,α → 0 as c 0 → ∞, and since A 1,c 0 ,α → 1 as c 0 → 0 (see (1.24)), we could conclude by continuity (see Theorem 1.3) that for any angle θ ∈ (0, π) there exists c 0 > 0 such that θ is the angle between A + c 0 ,α and − A + c 0 ,α (see (5.1)).This provides an alternative way to justify the surjectivity of the map c 0 → A + c 0 ,α (in the sense explained above).The curves in Figure 5 also allow us to discuss further the results in Theorem 1.4.In fact, when α is close to 1 the slope of the functions become unbounded and, roughly speaking, the behaviour of A + 3,c 0 ,α is in agreement with the result in Theorem 1.4, that is Numerically, the analysis is more difficult when α ∼ 0, because the number of computations needed to have an accurate profile of A + 3,c 0 ,α increases drastically as α → 0 + .In any case, Figure 5 suggests that A + 3,c 0 ,α converges to A + 3,c 0 ,0 faster than √ α| ln(α)|.We think that this rate of convergence can be improved to α| ln(α)|.In fact, in the proof of Lemma 3.10 we only used energy estimates.Probably, taking into account the oscillations in equation (3.102) (as did in Proposition 3.3), it would be possible to establish the necessary estimates to prove the following conjecture: | A + c 0 ,α − A + c 0 ,0 | ≤ C(c 0 )α| ln(α)|, for α ∈ (0, 1/2].

Appendix
In this appendix we show how to compute explicitly the solution m c 0 ,α (s, t) of the LLG equation in the case α = 1.As a consequence, we will obtain an explicit formula for the limiting vector A + c 0 ,1 and the other constants appearing in the asymptotics of the associated profile established in Theorem 1.2 in terms of the parameter c 0 in the case when α = 1.
We start by recalling that if α = 1 then β = 0. We need to find the solution { m, n, b} of the Serret-Frenet system (1.We see that when α = 1, and thus β = 0, (6.2) is a separable equation that we write as: Also, using (1.8) and (6.1) we get the initial conditions η 1 (0) = 0 and η 2 (0) = 1.In particular, if c 0 is small (6.3) is the global solution of the Riccati equation, but it blows-up in finite time if c 0 is large.As long as η j is well-defined, by Lemma 3.1, A priori, the formulae in (6.4) are valid only as long as η is well-defined, but a simple verification show that these are the global solutions of (1.6), with n 1 (s) = − sin (c 0 Erf(s)) and n 2 (s) = cos (c 0 Erf(s)) .
In conclusion, we have proved the following: for all s ∈ R. In particular, the limiting vectors A + c 0 ,1 and A − c 0 ,1 in Theorem 1.2 are given in terms of c 0 as follows: A ± c 0 ,1 = (cos(c 0 √ π), ± sin(c 0 √ π), 0).The conclusion follows from the definitions of A + c 0 ,1 , B + c 0 ,1 and a.
Remark 6.3.Notice that a is not a continuous function of c 0 , but the vectors (B + j sin(a j )) 3 j=1 and (B + j cos(a j )) 3 j=1 are.

2 .
With regard to the asymptotics of the profile established in part (ii) of Theorem 1.2, it is important to mention the following: (a) The errors in the asymptotics in Theorem 1.2-(ii) depend only on c 0 .In other words, the bounds for the errors terms are independent of α ∈ [0, 1].More precisely, we use the notation O(f (s)) to denote a function for which exists a constant C(c 0 ) > 0 depending on c 0 , but independent on α, such that

11 )
and equation (3.1) follows from (3.9).The initial conditions are an immediate consequence of the definition of η and f in (3.8) and (3.10).

( 3 .
40)Let us take s 0 ≥ 1 to be fixed in what follows.For t ≥ s 0 , we denote • t the norm of L ∞ ([t, ∞)).From the definition of R 0 in (3.40) and the elementary inequalities .66) From Proposition 3.2, u and v are bounded in terms of the energy.Thus, from the definition of G 1 and the estimates (3.56), (3.57) and (3.65), we obtain .82) This last inequality is a consequence of (3.53), (3.57), (3.62), (3.63) and the bounds for y and h established in (3.24) in Proposition 3.2.
.71) Using(3.44) and (3.57), we see that |G 3,1 (t)| ≤ C(E 0 , c 0 )e −4αt /t 2 , so that we can treat this term as we did for G 1 to obtain As has been already mentioned (see Section 2), part (i) of Theorem 1.1 follows from the fact that the triplet { m, n, b} is a regular-(C ∞ (R; S 2 )) 3 solution of (1.6)-(2.6)-(2.8)and satisfies the equation − s