Soliton Dynamics in Planar Ferromagnets and

Anti-Ferromagnets

Fang Hua Lin  Jalal Shatah

August 19, 2003

1 Introduction

Topological solitons have been the subject of much research for the past several decades. They are static solutions to geometric evolution equations with certain topological properties. Topological solitons arise in a variety of physical problems such as domain walls and magnetic bubbles in ferromagnetic continuum ([?], [?], [?], [?]), vortices in superfluids and superconductors ([?]), and defects in liquid crystals [?]. They also arise in the study of Bose-Einstein condensates, skyrmions, monopoles and instantions which are solutions of models of high-energy physics ([?], [?], [?], [?]).

One important area of research is to study the dynamical behavior when many solitons are present in the system. In the work of Lin and Xin [?] they successfully employed various conservation laws, in particular the conservation of topological vorticity, to establish rigorously the Kirchoff low for vortex dynamics described by Gross-Pitaevskii equations. The idea of employing conservation laws, such as the conservation of the topological vorticity, to study dynamical behavior of vortices may not be new to many physicists working on the subject. However the purpose of [?] and this work is to present rigorous mathematical proofs. These rigorous proofs have lead to several subtle mathematical issues which pose a challenge to explain at the level of physics. In particular, it is not known to the authors if there are any numerical simulations or physical experiments to study the effects of “sound waves” or “radiations” from the infinity on the dynamics of these topological solitons. In the present article, we shall concentrate on two particular examples: Topological solitons in planar ferromagnet described by the Landau-Lifschitz equations, and soliton in two dimensional anti-ferromagnets. One of the reasons for choosing these two specific examples (although our method applies to several other models as well) is that there is sparse experimental study of topological solitons in ferromagnetic films with an easy-plane anisotropy for which the relevant topological solitons are theoretically predicted to be half-bubbles or vortices. In fact, direct experimental evidence for the existence of pure anti-ferromagnetic solitons is absent although theoretical arguments suggests that such solitons should exist for the essentially the same reason as in the ordinary ferromagnets, however their dynamics should be significantly different as we will show below. Note that the rigorous static theory for the planar ferromagnetic vortices has recently established in [?]. The paper is organized as follows. In Section 2, we described the main results, in particular, the dynamical law for solitons in planar ferromagnetics and anti-ferromagnets. Various conservation laws, as well as their physical relevance, are also stated. In Section 3, we sketch the proofs of these results. The research of the second author is partially supported by a NSF-grant. The research of the first author is supported by a NSF-grant and a Chang-Jiang Professor Fund at the Zhejiang University.

2 Main Results

Ferromagnets The Landau-Lifschitz equations describing the planar ferromagnets with easy plane anisotropy are given by

(2.1)  \frac{∂}{∂t} m = m × f,  f = Δm − m_{3} e^{⇀}_{3} ,

where m : ℝ^{2} × ℝ → 𝕊^{2} is the spin density vector with |m|^{2} = m_{1}^{2} + m_{2}^{2} + m_{3}^{2} = 1, and e^{⇀}_{3} = (0, 0, 1) is the unit vector along the symmetry axis. Physical units have been normalized in (??) and, as a consequence, the magnon speed in the long-wavelength limit is normalized to unity (c ≡ 1). Equation (??) can also be rewritten as

(2.2)  m × \frac{∂m}{∂t} = −Δm + m_{3} e^{⇀}_{3} − (|∇m|^{2} + m_{3}^{2}) m .

These equations are infinite dimensional Hamiltonian system with the Hamiltonian E given by

\begin{matrix}e(m) = \frac{1}{2} (|∇m|^{2} + m_{3}^{2}) \\ E(m) = ∫_{ℝ^{2}} e(m)dx. \end{matrix} ; (2.3)

Besides the energy the flow keeps the symplectic form invariant. From the Lagrangian formulation of the equations one can easily compute the symplectic form to be ω(x, t) ∈ ℝ,

(2.4)  ω(x, t) = m · \frac{∂m}{∂x_{1}} ∧ \frac{∂m}{∂x_{2}} .

which is also the topological vorticity. Introducing the stress tensor

(2.5)  σ_{ij} = \frac{|∇m|^{2} + m_{3}^{2}}{2} δ_{ij} − \frac{∂m}{∂x_{i}} · \frac{∂m}{∂x_{j}} ,

1 ≤ i, j ≤ 2, then the conservation law for the topological vorticity is

(2.6)  \frac{∂}{∂t} ω = −ϵ_{ij} \frac{∂}{∂x_{i}} \frac{∂}{∂x_{k}} σ_{jk} .

Here (ϵ_{ij}) is the 2-D anti-symmetric tensor, and here the usual summation convention is employed in (??). Note that (??) can be rewritten as

(??′)  \frac{∂ω}{∂t} = − curl (div σ) .

Equations (??) or (??′) are derived formally under the assumption that the solution m(x, t) of (??) is classical (say C^{2}). From these equations one can formally deduce that the total topological vorticity ∫_{ℝ^{2}} ω(x, t) dx is a constant of time (assuming the vector m(x, t) is essentially a constant vector at |x| = ∞). In addition to this conserved quantity we can derive from (??) the conservation of the Linear momentum defined by p_{i} = −ϵ_{ij} I_{j},

I_{j} = ∫_{ℝ^{2}} x_{j} ω(x, t) dx,

and the angular momentum a

a = \frac{1}{2} ∫_{ℝ^{2}} |x|^{2} ω(x, t) dx.

In addition to these conserved quantities, the easy plane anisotropy conserves the total (azimuthal) spin:

M = ∫_{ℝ^{2}} m_{3}(x, t) dx.

In the study of the dynamics of topological solitons (magnetic vortices) of equation (??) formal arguments indicate that the behavior of these vortices are very similar to the ordinary vortices observed in classical fluids and superfluids (see [?], [?]). That is when these magnetic half-bubbles are far apart from each other ,and they move slowly in time ,in other words,it is in a regime of “particles + fields, see [?], these magnetic bubbles dynamics are very much like the dynamics of vortices in classical fluids whose essential features we now recall. A single vortex or anti-vortex is always spontaneously pinned and hence can move only together with the background fluid. On the other hand, in the presence of other vortices the nontrivial relative motion of vortices to the fluid is possible, and, in fact, they display characteristics of the 2-D Hall motion of interacting electric charges in a uniform magnetic field. In particular, two like vortices orbit around each other while a vortex antivortex pair undergoes Kelvin motion along parallel trajectories that are perpendicular to the line connecting the vortex and the antivortex. In [?], we shall prove rigorously this latter fact is indeed true globally in time for the planar ferromagnetism. To study the dynamics of topological vorticies we consider the Landau-Lifschitz equation (??) with a given initial data m(x, 0) = m_{0}(x). Typical data m_{0}(x) that we consider would have n degree 1 vortices and n degree −1 vortices (so that the total topological vorticity is zero). These vortices are moving at the speed ≃ ϵ^{2} ≪ 1 (note the speed of light is normalized to be 1 in equation (??)), and mutual distance between any two is of order \frac{1}{ϵ} (we shall see this is the right scale to have nontrivial vortex dynamics). After a proper scaling in both time and space, equation (??) reduces to the following

(2.7)  \{ \begin{matrix}m^{ϵ} × \frac{∂m^{ϵ}}{∂t} & = −Δm^{ϵ} + \frac{m_{3}}{ϵ^{2}} e^{⇀}_{3} − (|∇m^{ϵ}|^{2} + \frac{m_{e}^{2}}{ϵ^{2}} ) m^{ϵ} \\ m^{ϵ} |_{t=0} & = m_{0}^{ϵ}(x). \end{matrix}

Note that in the equation (??), vortices move at speed one, the distance between vortices is also of size one, and the “core” size of the magnetic bubbles are ≃ ϵ. Our main result is the following:

Theorem 1. Assume that the initial data has total 2n-vortices such that it has n degree + 1 vortices and n degree − 1 vortices, the total initial energy E(m_{0}^{ϵ}) = 2πn log \frac{1}{ϵ} + O(1), and m_{0}^{ϵ}(x) ≃ constant for |x| ≃ +∞. Then there is a solution of (??) with the following properties:

(i)  \frac{|∇m^{ϵ}|^{2}(x, t) + |m_{3}^{ϵ}(x, t)|^{2}/ϵ^{2}}{2π log \frac{1}{ϵ}} ⇀ ∑_{j=1}^{2n} δ_{a_{j}(t)}

as Radon measures when ϵ → 0^{+}, for every t ∈ [0, T]. Moreover, a_{j}(t)’s are Lifschitz continuous in t ∈ [0.T], for some T > 0. (ii) If, in addition, the initial energy is given by

E(m_{0}^{ϵ}) = 2πn log \frac{1}{ϵ} + W (a(0)) + o_{ϵ}(1),

then

(∗)  \frac{d}{dt} a_{j}(t) = n_{j} J ∇_{a_{j}} W (a(t)), for t ∈ [0, T].

Here n_{j} = degree of vortices at a_{j} = ±1,

a(t) = (a_{1}(t), a_{2}(t), … a_{2n}(t)) and J = (\begin{matrix}0 & 1 \\ −1 & 0 \end{matrix}) .

Where W is the so-called renormalized energy (see [?]) which is a function of a_{1}, a_{2}, … a_{2n} ∈ ℝ^{2}.

We remark that the Kirchoff dynamic law for vortices (∗) is established under the restrictive assumptions on the energy of the initial data m_{0}^{ϵ}. The reason for these assumption is to avoid the difficulty of dealing with the so-called “sound wave” effects due to the excess of energy in the data and, the the difficulty with the possible radiation effect from the infinity due to nonzero total topological vorticity. For more precise explanation see remarks in Section 3.

Antiferromagnets From the hydrodynamics approach via V. G. Baryakhar, M. V. Chetkin, B. A. Ivanov and S. N Gadeskii (Dynamics of Topological Magnetic Solitons—Experiment and Theory, Berlin, Springer), and the preliminary analysis in [?], the equations for planar antiferromagnets are given by

(2.8)  \{ \begin{matrix}n_{tt} − Δn + n_{3} e^{⇀}_{3} + e^{⇀}_{3} ∧ n_{t} = f,  (x, t) ∈ ℝ^{2} × ℝ_{+} , \\ n × f = 0,  n ∈ 𝕊^{2}. \end{matrix}

Here f = (|∇n|^{2} − |n_{t}|^{2} + n_{3}^{2} + n_{2} \frac{∂}{∂t} n_{1} − n_{1} \frac{∂}{∂t} n_{2})n. One should point out the term e^{⇀}_{3} ∧ n_{t}

plays a key role in the vortex dynamics of (??) even though it is a first order term in a second order system. This term is due to a constant applied magnetic field whose strength is normalized to be 1. Indeed, without the applied magnetic field, (??) would be a relativistic σ-model which is Lorentz invariant, and thus one would not expect any particular motion law for vortices.

Since we are interested in the dynamics of the topological solitons we introduce an appropriate space and time scaling and consider the system

(2.9)  \{ \begin{matrix}ϵ^{2} n_{tt} − Δn + \frac{n_{3} e^{⇀}_{3}}{ϵ^{2}} + e^{⇀}_{3} ∧ n_{t} = f, \\ f × n = 0,  (x, t) ∈ ℝ^{2} × ℝ_{+} , \end{matrix}

As for the ferromagnets problem, these equations are infinite dimensional Hamiltonian system. The energy density and the energy are given by

\begin{matrix}e_{ϵ}(n) =^{def} \frac{1}{2} (ϵ^{2}|n_{t}|^{2} + |∇n|^{2} + \frac{n_{3}^{2}}{ϵ^{2}} ) \\ E(n) = ∫_{ℝ^{2}} e_{ϵ}(n)dx. \end{matrix}

The symplectic form ω is easily derived from the Lagrangian of (??)

L = \frac{1}{2} [ϵ^{2}|n_{t}|^{2} − |∇n|^{2} − \frac{n_{3}^{2}}{ϵ^{2}} + e^{⇀}_{3} · (n ∧ n_{t})]

to obtain

ω = γ + ϵ_{ij} ϵ^{2} ∂_{x_{i}} (n_{t} · \frac{∂}{∂x_{j}} n)
γ =^{def} e^{⇀}_{3} · n_{x_{1}} ∧ n_{x_{2}} = \frac{1}{2}(∂_{x_{2}} e^{⇀}_{3} · n_{x_{1}} ∧ n + ∂_{x_{1}} e^{⇀}_{3} · n ∧ n_{x_{2}}).

The invariance of the symplectic form under the flow of (??) implies

(2.10)  \frac{∂}{∂t} ω = −ϵ_{ij} ∂_{x_{i}} ∂_{x_{j}} (σ_{ij}) ,

where σ_{ij} = e_{ϵ}(n) δ_{ij} − n_{x_{i}} · n_{x_{j}} is the stress tensor. From the above equation one can derive the following quantities for a classical solution n(x, t):

(i) Total topological vorticity ∫_{ℝ^{2}} γ dx;

(ii) Total linear momentum p_{i} =^{def} −ϵ_{ij} I_{j} where I_{j} = ∫_{ℝ^{2}} x_{j} ω(x, t) dx;

(iii) angular momentum a = \frac{1}{2} ∫_{ℝ^{2}} |x|^{2} ω(x, t) dx .

(iv) M = ∫_{ℝ^{2}} (e^{⇀}_{3} · (n ∧ n_{t}) − \frac{n_{3}^{2}}{ϵ^{2}} ) dx is also a constant in time though it has no physical meaning that we are aware of.

The linear and angular momentum can be written explicitly

p_{i} = − ∫_{ℝ^{2}} (ϵ^{2} n_{t} · ∂_{x_{i}} n + ϵ_{ij} x_{j} γ) dx

= − ∫_{ℝ^{2}} (ϵ^{2} n_{t} · ∂_{x_{i}} n + \frac{1}{2}n_{x_{i}} · (e^{⇀}_{3} ∧ n)) dx

a = ∫_{ℝ^{2}} (−ϵ_{ij} x_{i} ϵ^{2} n_{t} · ∂_{x_{j}} n + \frac{1}{2} |x|^{2} γ) dx .

Using these conservation laws, which at least hold under the assumption that we have a classical solution which tends to a constant unit vector at |x| ≃ ∞, we can state the following.

Theorem 2. Under the same assumptions as for Theorem ??, one has

\frac{ϵ^{2}|n_{t}|^{2} + |∇n|^{2} + \frac{n_{3}^{2}}{ϵ^{2}}}{2π log \frac{1}{ϵ}} ⇀ ∑_{j=1}^{2n} δ_{a_{j}(t)}

as Random measures when ϵ tends to zero. Moreover, a_{j}(t)’s are Lifschitz continuous for t ∈ [0, T] the initial data satisfies

2E(n_{0}^{ϵ}) = n2π log \frac{1}{ϵ} + W (a(0)) + o_{ϵ}(1) ,

then

\frac{d}{dt} a_{j}(t) = J ∇_{a_{j}} W (a(t)),  for t ∈ [0, T].

One should remark that neither (??) nor (??) have a general well-posedness theory for large finite energy data. However, there are natural ways to approximate systems (??) and (??) that lead to the existence of weak-solutions. One way of approximating (??) is to consider the following system instead

\begin{matrix}ϵ^{2} \frac{∂^{2}}{∂t^{2}} U_{ϵ}^{δ} − Δ U_{ϵ}^{δ} + \frac{U_{ϵ,e}^{δ} e^{⇀}_{3}}{ϵ^{2}} + e^{⇀}_{3} ∧ (\frac{∂}{∂t} U_{ϵ}^{δ}) \\ = \frac{(1 − |U_{ϵ}^{δ}|^{2})}{δ^{2}} U_{ϵ}^{δ} , U_{ϵ}^{δ}(x, t) ∈ ℝ^{3} , \end{matrix} ; (2.11)

which has global classical solution for a suitable initial data on ℝ^{2}. Note that various conservation laws for U_{ϵ}^{δ} analogous to those for equation (??) remain valid. Moreover, as δ → 0^{+}, u_{ϵ}^{δ} → n a weak solution of (??), see[?]. One then applies our proofs to the solutions {U_{ϵ}^{δ}} and let δ → 0^{+}, ϵ → 0^{+} at the end to obtain the desired conclusion. One point that we want to emphasize is that the dynamical law for these topological solitons are independent of how one approximates these systems. Similar process can be carried out for the system (??), however we wont elaborate about this here. Moreover Instead of presenting proofs for both Theorem ?? and Theorem ??, we shall simply sketch the proof for the case of Gross-Pitaevskii equation and make remarks how those arguments can be modified for ferromagnets and antiferromagnets systems.

3 Sketch of Proofs

The proofs are very similar to those presented in [?] for the case of Gross-Pitaevskii equation, which we shall recall below. Consider

(3.1)  \{ \begin{matrix}i U_{t}^{ϵ} = Δ U^{ϵ} + \frac{U^{ϵ}}{ϵ^{2}} (1 − |U^{ϵ}|^{2}) ,  x ∈ ℝ^{2} × ℝ \\ u^{ϵ}(x, 0) = U_{0}^{ϵ}(x) . \end{matrix}

where U_{0}^{ϵ}(x) is a complex valued function such that U_{0}^{ϵ} is a constant of magnitude one as |x| → +∞. Assume U_{0}^{ϵ} has n degree 1 vortices and n degree − 1 vortices, and

(∗)  E_{∗}(U_{0}^{ϵ}) = ∫_{ℝ^{2}} [\frac{1}{2}|∇U_{0}^{ϵ}|^{2} + \frac{1}{4ϵ^{2}} (1 − |U_{0}^{ϵ}|^{2})^{2}] dx

= 2π n log \frac{1}{ϵ} + O(1) .

The key conservation law used in [?] is the conservation of linear momentum

(3.2)  \frac{∂}{∂t} p(U_{ϵ}) = 2 div (∇U_{ϵ} ⊗ ∇U_{ϵ}) + ∇P_{ϵ} ,

for scalar valued function. Here p(U_{ϵ}) = ℑ(U∇Ū) = U_{ϵ} ∧ ∇U_{ϵ}. From (??), one derives also conservation of topological vorticity:

(3.3)  \frac{∂}{∂t} Jac(U_{ϵ}) = curl · div (∇U_{ϵ} ⊗ ∇U_{ϵ}) .

Under the assumption that U_{0}^{ϵ} has 2n-vortices satisfying (∗), one immediately derives (see [?] for details):

(i) From the structure of vortices and conservation of energy

(a)  \frac{e_{ϵ}(U_{ϵ})}{π log \frac{1}{ϵ}} dx ⇀ ∑_{j=1}^{2n} δ_{a_{j}(t)},  as ϵ = ϵ_{k} → 0^{+} ,

as Radon measures, for all 0 ≤ t ≤ T. (Note if one consider (??) on a bounded smooth domain Ω with U^{ϵ}(x, t) ≡ g(x) on ∂Ω, where g : ∂Ω → 𝕊^{1} is a degree d smooth map, and if E_{∗}(U_{0}^{ϵ}) = πd log \frac{1}{ϵ} + O(1), then T can be chosen to be infinity.)

∫_{ℝ^{2}} \frac{(1 − |U_{ϵ}|^{2})^{2}}{ϵ^{2}} dx + ∫_{ℝ^{2}} |∇|U_{ϵ}||^{2} ≤ C , (b)

∥∇U^{ϵ}∥_{L^{p}(ℝ^{2})}(t) ≤ C(p), 1 ≤ p < 2, 0 ≤ t ≤ T .

(ii) Using conservation of vorticity (??), one obtains

(c)  a_{j}(t) is continuous in t ∈ [0, T] .

(iii) Using conservation of momentum (??), one obtains

(d)  p(U_{ϵ}) ⇀ V in L^{1} (0, T; L_{loc}^{1}(ℝ_{a}^{2})) .
V_{t} = 2 div (V ⊗ V) + ∇P , div V = 0 in ℝ_{a}^{2}.

where L_{loc}^{1}(ℝ_{a}^{2}) denotes those L^{1}-vectors on ℝ^{2}\ ⋃_{j=1}^{2n} B_{ρ} (a_{j}(t)), for every ρ > 0.

(e) P is a single valued function on ℝ_{a}^{2}which is smooth away from {a_{1}(t), …, a_{2n}(t)} .

Moreover, ∇U_{ϵ} ⊗ ∇U_{ϵ} ⇀ V ⊗ V + μ, the matrix-valued Radon measure μ satisfies

div μ = ∇P_{μ} in ℝ_{a}^{2}.

The measure μ is called defect-measure that is responsible for the so-called sound wave effect, and P_{μ} can be viewed as sound wave pressure. The motion law for vortices under the assumption of (∗) follows from (??) to give

(3.4)  \frac{d}{dt} a_{j}(t) = J ∇_{a_{j}} W (a(t)) + f_{j}(μ) ,

where W is simply the Logrithium interaction potential between vortices,and all f_{j}(μ) are bounded in t,for j=1,2,...,2n. Under the additional assumption on the initial data

(3.5)  E_{∗}(U_{0}^{ϵ}) = 2π n log \frac{1}{ϵ} + W (a(0)) + o_{ϵ}(1) .

one can show

\frac{d}{dt} a_{j}(t) = J ∇_{a_{j}} W a(t), for 0 ≤ t ≤ T.

Turning our attention to equation (??) we can make similar conclusions. Under the assumption that m_{0}^{ϵ} has 2n, ±1 degree vortices and such that m_{0}^{ϵ}(x) ≃ a constant unit vector for |x| ≃ +∞, and E(m_{0}^{ϵ}) = n 2π log \frac{1}{ϵ} + O(1), we see that the function U =^{def} m_{1}^{ϵ} + i m_{2}^{ϵ} satisfies

(i)  U(x, t) ≃ a constant unit vector at |x| ≃ +∞;
(ii)  E_{∗}(U) = ∫_{ℝ^{2}} [\frac{1}{2} |∇U|^{2} + \frac{1}{4ϵ^{2}} (1 − |U|^{2})^{2}] dx

≤ ∫_{ℝ^{2}} (\frac{1}{2}|∇m^{ϵ}|^{2} + \frac{1}{4ϵ^{2}} |m_{3}^{ϵ}|^{2}) dx

≤ ∫_{ℝ^{2}} (\frac{1}{2}|∇m^{ϵ}|^{2} + \frac{|m_{3}^{ϵ}|^{2}}{2ϵ^{2}} ) dx

= E (m_{0}^{ϵ}) = 2π n log \frac{1}{ϵ} + O(1);

(iii)  U(x, 0) has 2n, degree ±1 vortices.

Thus from [?], we see that (a) and (b) are valid for m_{1}^{ϵ}(t) + i m_{2}^{ϵ}(t), 0 ≤ t ≤ T. In fact, (b) can be strengthened further, via (ii) above to give

∫_{ℝ^{2}} \frac{|m_{3}^{ϵ}|^{2}}{ϵ^{2}} dx + ∫_{ℝ^{2}} |∇|m_{1}^{ϵ} + i m_{2}^{ϵ}||^{2} dx (b′)

+ ∫_{ℝ^{3}} |∇m_{3}^{ϵ}|^{2} ≤ C, and
∥∇m^{ϵ}∥_{L^{p}(ℝ^{2})}(t) ≤ C(p), 1 ≤ p < 2, 0 ≤ t ≤ T,

while (e) should be rewritten as

(e′)  ∇m_{ϵ} ⊗ ∇m_{ϵ} ⇀ (\begin{matrix}V ⊗ V & 0 \\ 0 & 0 \end{matrix}) + μ̃, for a matrix-valued Radon measure μ̃.

Again in a manner similar to [?] we conclude (c), i.e. continuity (or Lifschitz continuity) of a_{j}(t), by using (??′) to obtain the following identity

(3.6)  \frac{d}{dt} ∫_{B_{R}(a_{j}(t)} x ω^{ϵ}(x, t) dx = ∫_{B_{R}(a_{j}(t)} −x curl div σ^{ϵ}(x, t) dx.
where ω^{ϵ}(x, t) = m^{ϵ} · m_{x_{1}}^{ϵ} ∧ m_{x_{2}}^{ϵ} ,
σ_{ij}^{ϵ}(x, t) = \frac{1}{2} (|∇m^{ϵ}|^{2} + \frac{|m_{3}^{ϵ}|^{2}}{ϵ^{2}} ) δ_{ij} − m_{x_{i}}^{ϵ} · m_{x_{j}}^{ϵ} ,

for t close to a given time τ ∈ [0, T], and R > 0 is a small but fixed constant. Integrating by parts on the right-hand side of (??) we obtain as ϵ → 0^{+} (as in [?])

(3.7)  \frac{d}{dt} a_{j}(t) = J ∇_{a_{j}} W (a_{j}(t)) + f_{j}(μ̃), for 0 ≤ t ≤ T.

Here we used the fact that

ω^{ϵ}(x, t) dx ⇀ ∑_{j=1}^{2n} ±δ_{a_{j}(t)}

in the sense of distributions, which follows from the structure of the vortices.

If the initial data satisfies the additional assumption:

(3.8)  E(m_{0}^{ϵ}) = 2π n log \frac{1}{ϵ} + W (a(t)) + o_{ϵ}(1), then

then as in [?], we denote by b_{j}(t) the solution of

\{ \begin{matrix}b′_{j}(t) & = J ∇_{b_{j}} W (b(t)),  j = 1, …, 2n, \\ b(0) & = a(0), \end{matrix}

and let ξ(t) = ∑_{j=1}^{2n} |a_{j}(t) − b_{j}(t)|. Then using (??) it is easy to verify that

(3.9)  \frac{d}{dt} ξ(t) ≤ ∑_{j=1}^{2n} |a′_{j}(t) − b′_{j}(t)| ≤ C ξ(t).

The last estimate follows from the following observation:

(3.10)  E(m^{ϵ}(t)) ≡ E(m^{ϵ}(0)) = 2π n log \frac{1}{ϵ} + W (a(0))
+0_{ϵ}(1) ≡ 2π n log \frac{1}{ϵ} + W (b(t)) + o_{ϵ}(1),

and

(3.11)  E(m^{ϵ}(t)) ≥ 2π n log \frac{1}{ϵ} + W (a(t)) + C_{0} ∑_{j=1}^{2n} |f_{j}(μ̃)|.

Hence ∑_{j=1}^{2n} |f(μ̃)| ≤ C|W (a(t))−W (b(t))| ≤ C m(t). Via the uniqueness of O.D.E. solution of

(3.12)  \{ \begin{matrix}\frac{d}{dt} ξ(t) & ≤ C ξ(t) \\ ξ(0) & = 0 \end{matrix}

we obtain a_{j}(t) ≡ b_{j}(t), and Theorem ?? is proved.

Remarks: (i) When μ̃(0) ≠ 0 (defect measure is nonzero) we do not know if Kirchhoff dynamical law

\frac{d}{dt} a_{j}(t) = J ∇_{a_{j}} W (a(t))

holds. In the case of the Gross-Pitaevskii equations, if either div μ = ∇P_{μ} in the whole of ℝ^{2} (instead of ℝ_{a}^{2}) or spt μ ⊆ ℝ_{a}^{2}, then we see that the same dynamical law is valid. The same remark also applies to solutions of (??).

(ii) We assumed that the initial data has exactly 2n vortices of degree ±1 and that the topological degree at |x| = +∞ is zero. If the winding number at |x| = +∞ is nonzero, then we may have the so-called radiation effect from the infinity. In the latter case, one always has E(m_{0}^{ϵ}) = +∞. For solution of the antiferromagnetic equations (??) the proofs are again very similar. The assumption on the initial energy

E(n_{0}^{ϵ}) = \frac{1}{2} ∫_{ℝ^{2}} (ϵ^{2}|n_{t}|^{2} + |∇n|^{2} + \frac{n_{3}^{2}}{ϵ^{2}} ) dx

= 2π n log \frac{1}{ϵ} + O(1),

implies that the function U = n_{1} + i n_{2} satisfies (∗). The dynamical law for vortices follows from the invariance of the symplectic form (??) which implies

\frac{d}{dt} ∫_{B_{R}(a_{j}(z))} x [e^{⇀}_{3} · (n_{x_{1}} ∧ n_{x_{2}}) + ϵ^{2} ϵ_{ij} ∂_{x_{j}} (n_{t} · \frac{∂}{x_{j}} n)] dx (3.13)

= ∫_{B_{R}(a_{j}(z))} x curl div σ_{ϵ} dx

with (σ_{ϵ})_{ij} = \frac{1}{2} (ϵ^{2}|n_{t}|^{2} + |∇n|^{2} + \frac{n_{3}^{2}}{ϵ^{2}} ) δ_{ij} − n_{x_{i}} · n_{x_{j}},

and the complex function (or 2-vector) U = n_{1} + i n_{2} satisfies

(3.14)  \frac{1}{2} (U ∧ ∇U)_{t} = div (∇n ⊗ ∇n) + ∇P + ϵ^{2} (n_{t} · ∇n)_{t} .

Note that e^{⇀}_{3} · n_{x_{1}}^{ϵ} ∧ n_{x_{2}}^{ϵ} ⇀ ∑_{j=1}^{2n} ±δ_{a_{j}(t)} as ϵ → 0^{+}. Thus under the assumption of Theorem ??, one may employ the arguments in [?] to establish the incompressible fluid limit from (??).

References

[BBH] F. Bethuel, H. Brezis, and F. Helein, Ginzburg-Landau vortices, Boston, Birkhauser, 1994

[D] T. H. O’Dell, Ferromagnetodynamics, the dynamics of magnetic bubbles, domain and domain walls, Wiley, New York, 1981.

[DeG] P. G. De Gennes and J. Prost, The physics of liquid crystals, second edition, (Oxford Science Publications), Clarendon Press, Oxford, 1993.

[FS] A.L.Fetter and A.A.Svidzinsky,Vortices in traped dilute Bose-Einstein condensates,preprint 2002.

[H] R. P. Huebener, Magnetic flux structures in superconductors, Springer series in solidstate sciences, Springer–Berlin, 1979.

[HL] Fengbo Hang and F. H. Lin, Static theory for planar ferromagnets and antiferromagnets, Acta Math. Sinica (Eng. ser.) 17 (2001), 541–580.

[HL2] , Travelling wave solutions of Schrödinger map equation, preprint.

[JT] A. Jaffe and C. H. Taubes, Vortices and Monopoles, Birkhäuser, Boston, 1980.

[KP] S. Komineas and N. Papanicolaou, Topology and dynamics in ferromagnetic media, Phys. D. 99 (1996), 81–107.

[LX] F. H. Lin and J. Xin, On the Incompressible fluid limit and vortex motion law of the Nonlinear Schrödinger equation, Comm. Math. Phys. 200 (1999), 249–274.

[MRS] V. G. Makhankov, Y. P Rybakov and V. I. Sanyuk, The skyrme model, Springer, Berlin, and Heidelberg, 1993.

[MS] A. P. Malozemoff and J. O. Slonzewski, Magnetic domain walls in bubble materials, Academic Press, New York , 1979.

[N] J. Neu, Vortices in the complex scalar fields, Phys. D 43 (1990), 385–406.

[PT] N. Papanicolaou and T. N. Tomaras, Dynamics of Magnetic Vortices, Nuclear Phys. B 360 (1991), 425–462.

[R] R. Rajaraman, Solitons and Instantons, North-Holland, Amsterdam, 1982.

[S] J. Shatah, Weak solutions and development of singularities of the SU(2) − ω-model, Comm. Pure and Appl. Math., 41 (1988), 459–469.