TRAVELING VORTEX HELICIES FOR SCHRÖDINGER MAP EQUATION

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1 TRAVELING VORTEX HELICIES FOR SCHRÖDINGER MAP EQUATION JUNCHENG WEI AND JUN YANG Abstract We construct traveling wave solutions with vortex helices for the Schrödinger map equation m t = m m m 3 e 3 in R 3 R, of the form ms, s, s 3 δ log ɛ ɛt with traveling velocity δ log ɛ ɛ along the direction of s 3 axis We use a perturbation approach which gives a complete characterization of the asymptotic behavior of the solutions Introduction The aim of this paper is to construct traveling wave solutions for a class of Landau-Lifshitz equations We shall concentrate on the traveling wave solutions of the Schrödinger map equation or equivalently the equation m t = m m m 3 e 3 in R N R, m m t = m m 3 e 3 m m 3m Here m : R N R S so that ms, t and where e 3 = 0, 0, R 3 The equation or equivalently is, in fact, the Landau-Lifshitz equation describing the planar ferromagnets, that is, ferromagnets with an easy-plane anisotropy 36, 40 The unit normal to the easy-plane is assumed to be e 3 in the equations, see for example 40 Despite some serious efforts see eg 7, 8, 3, 4, 5, 0, 8, 4, 6, 7, 3, 4, 5, 6 and the references therein, some basic mathematical issues such as local and global well-posedness and global in time asymptotics for the equation remain unknown If one is interested in onedimensional wave plane-wave solutions of, that is, m : R R S or S, a lot were known as becomes basically an integrable system see 4 and 9 The problem in -D or higher dimensions are much more subtle Even though it is possible to obtain weak solutions of see 9-0, 3-, 3-4, 38, 43, one does not know if such weak solutions are classical smooth or unique From the physical side of view, one expects topological solitons, which are half magnetic bubbles, exist in solutions of see 34 and 40 Indeed, in 3-33, F Hang and F Lin have established the corresponding static theory for such magnetic vortices They are very much like vortices in the superconductor described by the Ginzburg-Landau equation, see 7 and references therein Let us recall the essential features of vortex dynamics in a classical fluid or a superfluid modeled by the Gross-Pitaevskii equation, see 39 A single vortex or antivortex is always spontaneously pinned and hence can move only together with the background fluid However vortex motion relative to the fluid is possible in the presence of other vortices and it displays characteristics similar to the -D Hall motion of interacting electric charges in a uniform magnetic field In 99 Mathematics Subject Classification 35J5, 35J0, 35B33, 35B40 Key words and phrases Traveling Waves, Schrödinger Maps, Vortex Helices, Perturbation Corresponding author: Jun Yang, jyang@szueducn

2 J WEI AND J YANG 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 This latter fact had been obtained in a special case by Jones and Robert 35 They also asymptotically derived a three dimensional 3-D solitary wave that describe a vortex ring moving steadily along its symmetric axis For more precise mathematical proofs, there are some works for the case of Gross-Pitaevskii equation In two dimensional plane, F Bethuel and J Saut constructed a traveling wave with two vortices of degree ± in 3 In higher dimension, by minimizing the energy, F Bethuel, G Orlandi and D Smets constructed solutions with a vortex ring4, see also6 See also for another proof by Mountain Pass Lemma The reader can refer to the review paper 5 by F Bethuel, P Gravejat and J Saut and the references therein By the a completely different method, F Lin and J Wei3 obtained similar results as those of 3, 4 for the Landau-Lifshitz equation More precisely, if we look for a solution of the traveling wave ms, s N Ct ie, travel in the s N direction with the speed C = cɛ > 0 of the equation Then m must be a solution of After a proper scaling in the space, 3 becomes or The main results of the paper3 is the following cɛ m s N = m m m 3 e 3 3 c m s N = m m m 3 ɛ e 3, s R N, 4 c m m = m m 3 e 3 s N ɛ m m 3 m 5 ɛ Theorem 3 Let N and ɛ sufficiently small there is an axially symmetric solution m = m s, s N C R N, S of 4 such that E ɛ m = m m 3 ds < 6 ɛ R N and that m has exactly one vortex at s, s N = a ɛ, 0 of degree, where a ɛ If N =, the traveling velocity C ɛ, while C = N ɛ log ɛ for N 3 Naturally such solution m gives rise to a nontrivial two-dimensional traveling wave solution of with a pair of vortex and antivortex which undergoes the Kelvin Motion as described above Solutions constructed in Theorem are called traveling vortex rings for the case of the dimension N 3 We note that formal arguments as well as numerical evidences were already presented in the work 40 We should also note that in the case of the initial date of contains only one vortex one magnetic half-bubble, with its structure as described in the work of 3-33 very precisely, the above discussions imply that the vortex will simply stay at its center of mass, and a meaningful mathematical issue to examine would be its global stability It is, however, unknown to authors that whether such stability result is true or not, see 6, 7, 8 for relevant discussions On the other hand, it is relatively easy to generalize the work of 37 to the equation for the planar ferromagnets One may obtain the same Kirchhoff vortex dynamical law for these widely separated and slowly moving magnetic half-bubbles, see 9 as formally derived in 40 and also 39 for the Gross-Pitaevskii equation of solutions of In this note, we concern the existence of traveling wave solutions possessing vortex structures with rotating invariance for problem 3 In this direction, we refer the paper by D Chiron for the existence of vortex helices for Gross-Pitaevskii equation7 The main result reads

3 VORTEX HELICIES FOR SCHRÖDINGER MAP EQUATION 3 Theorem Let N = 3 and ɛ sufficiently small there is an axially symmetric solution m = m s, s N C R 3, S of 4 such that E ɛ m = m m 3 ds < ɛ R 3 Moreover u has a vortex helix directed along the curve in the form α R ˆd cos α, ˆd sin α, ˆλα R 3, and u is also invariant under the skew motion expressed by cylinder coordinate The traveling velocity C = ɛ log ˆd ˆλ ɛ Σ : r, θ, s 3 r, θ α, s 3 λα, α R For more precise asymptotic behaviors of the solutions, the reader can refer to the details in Sections 3 We end the introduction with some discussions In the papers 3-4 and - the authors constructed traveling wave solution for the Gross-Pitaevskii equation i u t = u u u u, in R N R 7 Their method is variational First they constructed a mountain-pass value for the energy functional in bounded domains and used variational method and fine estimates to prove the existence of traveling wave solutions to 7 with small speed However, in 3 the authors used a completely different and more direct approach to prove Theorem The method gives more precise asymptotic behavior of solutions as the speed approaches zero This may yield more information on the spectrum, uniqueness and dynamical properties of the traveling wave solutions This method can be easily adopted to give a different proof of the existence of traveling wave solutions in 3-4 and - to 7 Here we use the method to construct the solutions with vortex helices to problem 4 By using the rotational invariance of the solution, we transfer the problem to a two-dimensional case 39 with boundary condition 3 on the infinite strip S cf 30 The key point is then to construct a solution with a vortex of degree and its antipair of degree, which are directed along the x axis Additional to the computations for standard vortices in two dimensional case, there are several different ingredients: the traveling velocity becomes δɛ log ɛ with δ in 37, and there are two extra derivative terms u x x and γ λ λ u x d If we put w = ρle iθ as the approximate solution in the vortex core region, then there exist two singular terms cf 4 and 43 θ y d y d y, σ θ dγ y y 4σ yy dγ y 4, which were expressed in local translated coordinates y in 46 These two terms will play an important role in determining the dynamical behavior of the vortex helices Some preliminaries In this section, we collect some important facts which will be used later These include the asymptotic behaviors and nondegeneracy of degree one vortex From now on, we consider the high dimensional case N 3 Setting u = m im m 3 and c = δ log ɛ,

4 4 J WEI AND J YANG problem 3 becomes iδɛ log u ɛ s N u u u u = ū u u u Here and throughout the paper, we use ū to denote the conjugate of u The parameter δ is a constant to be chosen in 37 When ɛ = 0 and N =, problem 3 admits solutions of standard vortex of degree, ie u = w := ρle iθ in polar coordinate l, θ, where ρ satisfies ρ ρ l ρρ ρ ρ ρ = 0 l ρ Another solution w := ρle iθ will be of vortex of degree These two functions will be our block elements for future construction of approximate solutions Notations: For simplicity, from now on, we use w = ρle iθ to denote the degree vortex We also assume that σ 0, is a fixed and small constant The following properties of ρ are proved in 3 Lemma There hold the asymptotic behaviors: ρ0 = 0, 0 < ρl <, ρ > 0 for l > 0, ρl = c 0 e l l Ol 3/ e l as l, where c 0 > 0 Setting w = w iw and z = x ix, then we need to study the following linearized operator of S 0 in 37 around the standard profile w: L 0 φ = φ 4w w w w w φ 8 w, φ w w w w w w 4 w, φ w w 4 w, φ w w 4 w w, φ w w w w φ w w φ The nondegeneracy of w is contained in the following lemma3 Lemma Suppose that L 0 φ = 0, 3 where φ = iwψ, and ψ = ψ iψ satisfies the following decaying estimates for some 0 < ϱ < Then for certain real constants c, c ψ z ψ C z ϱ, ψ z ψ C z ϱ, φ = c w x c w 3 Symmetric formulation of the problem For simplicity of notations, we here only consider the case N = 3 As the statements in the introduction part, we can apply the method to generally high dimensional case Recall that by setting u = m im m 3 and c = δ log ɛ, 4

5 VORTEX HELICIES FOR SCHRÖDINGER MAP EQUATION 5 we have written problem 3 in the form cf iδɛ log u ɛ s 3 u u u u = ū u u 3 u Here and throughout the paper, we use ū to denote the conjugate of u The parameter δ is a constant to be chosen in 37 By using the cylinder coordinates s = r cos Ψ, s = r sin Ψ, s 3 = s 3, equation 3 is transferred to iδɛ log ɛ u s 3 u r r u r u r Ψ u s 3 u u u = ū u u 3 u For the problem 3, we want to find a solution u which has a vortex helix directed along the curve in the form with two parameters α R d cos α, d sin α, λα R 3, Moreover, u is also invariant under the skew motion Hence, u has the symmetry and also satisfies the problem d = ˆd/ɛ, λ = ˆλ/ɛ 33 Σ : r, Ψ, s 3 r, Ψ α, s 3 λα, α R 34 ur, Ψ, s 3 = ur, Ψ α, s 3 λα = ur, 0, s 3 λψ, 35 iδɛ log ɛ u u s 3 r r ū u = u r u r λ u r λ r u s 3 which will be defined on the region { r, s 3 0, λπ, λπ } Recall the parameters in 33 and then set σ = ˆλ ˆd, γ = σ, δ =, s 3 u u u 36 ˆd ˆλ = ˆdγ 37 It is worth to mention that these three positive parameters are of order O with respect to ɛ For further convenience of notations, we also introduce the rescaling r, s 3 = x, γx, z = x ix 38 Thus 36 becomes u iδɛ log ɛ γ = ū u u u x x u x λ x λ x γ u γ u u u u 39 By using the symmetries, in the sequel, we shall consider the problem on the region S = R λπ/γ, λπ/γ, 30

6 6 J WEI AND J YANG and then impose the boundary conditions uz as x u 0, x = 0, x x λπ/γ, λπ/γ, ux, λπ/γ = ux, λπ/γ, x R, u x x, λπ/γ = u x x, λπ/γ, x R 3 It is easy to see that the problem 39 is invariant under the following two transformations Thus we impose the following symmetry on the solution u Π := uz u z, uz u z 3 { } uz = u z, uz = u z 33 This symmetry will play an important role in our analysis In other words, if we write then u and u enjoy the following conditions: ux, x = u x, x iu x, x, u x, x = u x, x, u x, x = u x, x, u x, x = u x, x, u x, x = u x, x, u x 0, x = 0, u x, λπ/γ = u x, λπ/γ, u 0, x = 0, x u x, λπ/γ = u x, λπ/γ, u x, λπ/γ = u x, λπ/γ = 0, u x, λπ/γ = u x, λπ/γ = 0 34 Problem 39-3 becomes a two-dimensional problem with symmetries defined in 33 The key point is then to construct a solution with a vortex of degree and its antipair of degree, which are directed along the x axis Additional to the computations for standard vortices in two dimensional case, there are several different ingredients: the constant c becomes δɛ log ɛ, and there are two extra derivative terms u x x and γ λ u In the following we shall x mainly focus on how to deal with these terms by the methods in 3 Moreover, we shall improve the approximate solution to satisfy the boundary conditions in 34 By writing u λ x γ u = γ λ x λ d u, 35 u x λ x γ u = u u γ λ x λ u, 36 d

7 VORTEX HELICIES FOR SCHRÖDINGER MAP EQUATION 7 we denote, as before ū S 0 u : = u F u u u, u S u : = ū λ u γ x λ u, d S u : = u γ λ x x x λ u d, S 3 u : = iδγ ɛ log ɛ u, Su : = S 0 u S u S u S 3 u, 37 where we have denoted F u = u u u 4 Approximate solutions 4 First Approximate Solution and its Error Recall the vortex solutions w and w defined in For each fixed d := ˆd ɛ with ˆd 00, 00, we define the first approximate solution v 0 z := w z d e w z d e, 4 where e =, 0 A simple computation shows that In fact, for z >> d, we have v 0 z as z 4 v 0 e iθ d e iθ d e x d x ix d x d x x d x O d z d z Here θ ξ denotes the angle argument around ξ and r ξ = z ξ It is easy to see that r ξ = z ξ z ξ, θ ξ = z ξ z ξ, θ ξ = z ξ, 43 where we denote z = x, x By our construction and the properties of ρ, we have v 0 = e i θ de θ de O e min z d e, zd e 44 We divide the region S into two parts: { } = x > 0, λπ/γ < x < λπ/γ, { } S = x < 0, λπ/γ < x < λπ/γ 45 By our symmetry assumption, we just need to consider the region In the region, we consider two cases Firstly, if z d e > d z d e, then we have v 0 = wz d e e iθ d e Oe d/ z d e /, v 0 = ρ z d e Oe d/ z d e /

8 8 J WEI AND J YANG Secondly, if z d e < d z d e, in the rest of this paper, we often rescale the variable as follows z = d e y 46 We now obtain the following estimates w v 0 = w i w θ d e y d e Oe d/ y / e iθ d e, 47 and then also v v v 0 v 0 = w ρ e iθ d e, 48 v 0 v 0 v 0 = ρ ρ w Oe d/ y / e iθ d e w w iw w θ d e w θ d e θ d e Oe d/ y / Combining the estimates above, we have for z, S 0 v 0 = e iθ ρ d e ρ i w θ d e ρ w ρ y d e Oe d/ y / 49 The following term in S v 0 Ω v 0 v 0 γ λ x λ v0, d can be estimated by similar computations in 48, but it is small in the sense that Ω /v 0 = Oɛ On the other hand, we can estimate the term in S 3 v 0 as follows iδγ ɛ log ɛ v 0 = iδγ ɛ log ɛ y wye iθ d e Oe d/ y / = iδγ ɛ log ɛ w iw θ d e y y e iθ d e Oɛ log ɛ e d/ y /, where y θ d e = y d y d y = Od = Oɛ 40 The first component in S v 0 obeys the following asymptotic behavior Ω = x v 0 x = = d y y wye iθ d e Oe d/ y / d y w iw θ d e dd y y y e iθ d e Oɛe d/ y /, where we have y θ d e = x y d y = O d = Oɛ 4

9 Whence, by 43, the term d VORTEX HELICIES FOR SCHRÖDINGER MAP EQUATION 9 w y in Ω has a singularity in the form i d w θ d e / iv 0 y y d y 4 Similarly, the calculation for the second term in S v 0 is proceeded as Ω 3 = γ λ x λ d v 0 = γ λ d y λ d = σ dγ y O d y wye iθ d e Oe d/ y / w y i w θ d e iw θ d e y y y w θ d e e iθ d e Oɛe d/ y / y By the formulas in 43, there is a singularity in Ω 3 σ dγ y i w θ d e / iv 0 4σ yy dγ y 4 43 y In the next subsection, we will introduce a correction in the phase term to get rid of these two mentioned singularities In summary, we have obtained for z with z = d e y Sv 0 = S 0 v 0 S v 0 S v 0 S 3 v 0 = e iθ ρ d e ρ i w θ d e ρ w ρ y d e iδγ ɛ log ɛ e iθ w d e iw θ d e Oe d/ y / y y 3 Ω i i= 44 A similar and almost identical estimate also holds in the region z S A direct application of 49 and 44 yields the following decay estimates for the error Ẽ d = S 0v 0 S v S 3 v 0 iv 0 Lemma 4 It holds that for z B d e B d e c ReẼd ImẼd where ϱ 0, is a constant Proof: Since Ẽ d = S 0 v 0 Cɛ ϱ z d e 3 Cɛ ϱ z d e 3, 45 Cɛ ϱ z d e ϱ Cɛ ϱ, 46 z d e ϱ iv 0 S v 0 iv 0 S 3 v 0 iv 0, the estimates for the first term follows from 49 We just need to estimate the third term

10 0 J WEI AND J YANG Let us compute for z ɛi v / 0 iv 0 = iɛ ρz d e e iθ d e θ d e iv O ɛe z d e ɛe zd e = ɛ ρ z d e ρ z d e iɛ θ d e θ d e = O ɛe z d e ɛe zd e x d iɛ x d x O ɛe z d e ɛe zd e x d x d x The estimate 45 then follows Let us notice that for z, z d e < d, x d x d ɛ x d x x d x Cɛ z d e Cɛ ϱ z d e ϱ On the other hand, for z, z d e > d, we then have x d x d ɛ ɛ x d x x d x C z d e C ɛ ϱ z d e ϱ Thus 46 is proved 4 Further Improvement of the Approximation As we promised in previous subsection, we now define a new correction ϕ d in the phase term The phase function ϕ d will be decomposed into two parts: singular part and regular part Then we will define an improved approximation and estimate its error To cancel the singularities in 4 and 43 in the form y d y 4σ dγ y y y 4 = y dγ y σ 3y y y dγ y 4, we want to find a function Φy, y by solving the problem in the translated coordinates y, y Φ y Φ y = y dγ y σ 3y y y dγ y 4 in R 47 In fact, we can solve this problem by separation of variables and then obtain Φy, y = 4dγ y log y σ dγ y 3 y 3σ 8dγ y 48 Let χ be a smooth cut-off function in a way such that χz = for z B d d e and χ = 0 for 0 z B d d e c Let ϕd z = ϕ s z ϕ r z, where the singular part is defined by 5 ϕ s z := χz χz 4d γ x log z d e z d e χz σ x 3 dγ z d e σ x 3 dγ z d e 49 While by recalling the operator S in 37 and the region S in 30, we find the regular part ϕ r z by solving the problem θd e ϕ r z S ϕ r = S θ d e ϕ s z in S, 40 ϕ r = θ d e θ d e ϕ s z on S 4 Note that the function ϕ s is continuous but ϕ s is not The singularity of ϕ s comes from its derivatives

11 VORTEX HELICIES FOR SCHRÖDINGER MAP EQUATION By simple computations, we see that for z B d d e, 0 x = θd e θ d e ϕ s z x 4x x d x d x x d x x x x x d x d x x d x = Od = Oɛ For z B d d e c, it is easy to see that we also get Oɛ In fact for z B d d e c, ϕ s = and θd e S θ d e ϕ s z 4x = x d x x d x Going back to the original variable r, s 3 in 3 and letting ˆϕr, s 3 = ϕ r z we see that r,s 3 ˆϕ S ˆϕ C r s Thus we can choose ϕ r such that ˆϕ = O The regular term ϕr is C in the original r s 3 variable r, s 3 We observe also that by our definition, the function satisfies ϕ := θ d e θ d e ϕ s ϕ r, 43 ϕ S ϕ = 0, on S, ϕ = 0 on S 44 From the decomposition of ϕ d, we see that the singular term contains x log z d e which becomes dominant when we calculate the speed Finally, we define an improved approximation 43 Error Estimates Let So V d = ρe i ϕ Note that for x > 0, we have V d z := w z d e w z d e e iϕ d 45 ρ = ρ z d e ρ z d e, ϕ = θ d e θ d e ϕ d ρ z d e = Oe d z d e 46 Since the error between and ρ z d e is exponentially small, we may ignore ρ z d e in the computations below We shall check that V d is a good approximate solution satisfy the conditions in 34 and has a small error and Recall the boundary condition in 44 It is obvious that Im V d = ρ sin ϕ = 0 on S 47 Re V d = ρ cos ϕ = ρ cos ϕ on S, 48 which is of order Oɛ It can be checked that V d satisfies the conditions in 34 except u = 0 on S 49

12 J WEI AND J YANG Let us start to compute the errors: V d = ρ ϕ ρ i ρ ϕ i ρ S ϕ Here we have used the fact 44 We continue to compute other terms: We then obtain that V d V d = S 0 V d = e i ϕ In a small neighborhood of d e, we write S V d S V d = γ λ x = γ λ x γ λ ρ ρ ϕ i ρ ρ ϕ i ρ S ϕ ρ ρ ρ λ d λ d x e i ϕ 430 e i ϕ ϕ θ i ρ ρ ρ ϕ Oe d z d e λ d V d ρ ρ i ρ ϕ x x x x = γ λ x λ d V d V d γ λ x i ρ ϕ i ρ ϕ ρ ρ e i ϕ ρ i ρ ρ ρ ϕ = i ρ S ϕ Oɛ Vd λ d V d x x ρ ϕ e i ϕ ρ i ρ ρ ϕ ρ ϕ ρ ρ ρ ρ ϕ ρ ρ e i ϕ i ρ S ϕ ρ e i ϕ x x The estimate for the term S 3 V d = iδγ ɛ log ɛ V d is same as before The total error is E d = ρ ρ ϕ ρ θ i ρ ρ ρ ϕ Oe d iδγ w ɛ log ɛ iw θ d e y y Thus z d e e iθ d e Oɛ log ɛ e d/ y / e i ϕ Setting z = d e y, we then have ϕ s = dγ log d y Oɛ log y, ϕ r = Oɛ 433 ϕ θ = O ϕ s θ ϕ s Oɛ θ, ρ ϕ = Oɛ log ɛ ρ Oɛρ log y These asymptotic expression will play an important role in the reduction part

13 VORTEX HELICIES FOR SCHRÖDINGER MAP EQUATION 3 5 Setting up of the Problem Now we introduce the set-up of the reduction procedure We look for solutions of 39-3 in the form uz = ηv d iv d ψ ηv d e iψ, 5 where η is a function such that η = η z d e η z d e, 5 and ηs = for s and ηs = 0 for s This nonlinear decomposition 5 was introduced first in 0 for Ginzburg-Landau equation The conditions imposed on u in 3 and 33 can be transmitted to the symmetry on ψ ψz = ψ z, ψ 0, x = 0, x Vd iv d ψ x x, λπ/γ ψz = ψ z, ψx, λπ/γ = ψx, λπ/γ, Vd = iv d ψ x x,λπ/γ More precisely, by the computations in 47 and 48, for ψ = ψ iψ, there hold the conditions ψ x, x = ψ x, x, ψ x, x = ψ x, x, ψ x, x = ψ x, x, ψ x, x = ψ x, x, ψ x 0, x = 0, ψ x, λπ/γ = ψ x, λπ/γ = 0, ψ x, λπ/γ = ρ ρ x, λπ/γ, ψ x, λπ/γ = ψ x, λπ/γ ψ 0, x = 0, x ψ x, λπ/γ = ψ x, λπ/γ, ψ x, λπ/γ = ρ ρ x, λπ/γ, This symmetry will be important in solving the linear problems in that it excludes all but one kernel We may write ψ = ψ iψ with ψ, ψ real-valued and then set u = V d φ, φ = ηiv d ψ ηv d e iψ 55 In the sequel, we will derive the explicit form for the linearized problem In the inner region { z B d e B d e }, we have and the equation for φ becomes In the above, we have denoted linear operator by L d φ = φ x φ x u = V d φ, 56 L d φ N d φ = E d 57 4 V d V d V d φ V d V d φ Vd φ V d V d V d φ λ Vd V d γ λ d γ λ x λ d x φ F V d φ φ V d V d V d λ d 4 V d V d γ λ x V d V d V d φ Vd φ V d γ φ λ x λ d Vd

14 4 J WEI AND J YANG The nonlinear operator and the error are and N d φ = F V d φ F V d F V d φ O φ φ iδγ log ɛ ɛ φ, 58 E d = SV d 59 In the above, we have use the definition of F in 37 { In the outer region z B d e B d e } c, we have u = V d e iψ By simple computations we obtain SV d e iψ iv d e iψ = ψ ψ V d V d e ψ x x V d V d V d ψ e ψ V d Vd e ψ iv d V d V d e ψ V d V d i 4 i V d λ V d γ ψ e x λ V d d ψ V d V d i V d V d V d γ ψ e V d V d γ ψ e λ x λ x λ d λ d ψ ψ V d e ψ i V d e ψ V d γ λ x λ ψ d V d V d e ψ ψ ψ iδγ log ɛ ɛ ψ E d iv d In summary, we can also write the problem as an equation of ψ = ψ iψ with conditions in 54 In the above, we have denoted L 0 ψ = ψ V d V d γ L 0 ψ Ñψ Ñψ = Ẽd, 50 ψ V d x x V d V d ψ V d ψ 4 V d ψ i V d γ λ x λ d λ x λ d ψ, 4i V d ψ V d V d V d 4i V d ψ V d γ λ x λ d V d V d Ñ ψ = V d ψ V d Oψ O V d ψ ψ ψ io e ψ ψ, Ñ ψ = iδγ log ɛ ɛ ψ, Ẽ d = E d iv d Recall that ψ = ψ iψ Then setting z = d e y, we have for z R ψ ψ x x γ λ λ ψ x d Oe y ψ L 0 ψ = ψ ψ x x γ λ λ ψ x d 4 V d V d ψ O e y, 5 ψ

15 VORTEX HELICIES FOR SCHRÖDINGER MAP EQUATION 5 Note that Ñ ψ = O e y ψ ψ ψ y ψ y ψ O e y ψ ψ ψ ψ ψ ψ Ñ ψ = γ λ x x x λ d = Oɛ ψ y Oɛ ψ y γ, 5 53 x x λ γ x 54 Let us remark that the explicit form of all the linear and nonlinear terms will be very useful for later analysis in resolution theory Let E d = SV d, Ẽ d = E d 55 iv d Based on the form of the errors, we need to use suitable norms Let us fix two positive numbers p > 3, 0 < ϱ < Recall that φ = iv d ψ, ψ = ψ iψ Denote l = z d e and l = z d e, and define h = iv d h L p S l ϱ j h L S l ϱ j h L S, In the above, ψ = φ W,p S j= l ϱ j ψ L S l ϱ j ψ L S j= l ϱ j ψ L S l ϱ j ψ L S S = { z S : z d e < 3 or z d e < 3 }, S = { z S : z d e > and z d e > } 56 We remark that we use the norm L p,p loc or Wloc in the inner part due to the fact that the error term contains terms like ɛ log z d e which is not L -bounded Using the norms defined above, we can have the following error estimates Lemma 5 It holds that for z S and also ReẼd ImẼd where ϱ 0, is a constant As a consequence, there holds Cɛ ϱ z d e 3 Cɛ ϱ z d e 3, 57 Cɛ ϱ z d e ϱ Cɛ ϱ, 58 z d e ϱ E d Lp S Cɛ log ɛ, 59 Ẽd Cɛ ϱ 50

16 6 J WEI AND J YANG Let and the co-kernel L 0 ψ = ψ 6 Projected Linear and Nonlinear Problems γ ψ x ψ x V d V d V d ψ Z d := V d d z d e η R λ ψ γ x 4i V d ψ V d V d V d i η where η is defined at 5 before Then Z d satisfies the symmetry 33 4 V d ψ V d, z d e, 6 As the first step of finite dimensional reduction, we need to consider the following linear problem L 0 ψ = h in S, ψ satisfies the symmetry 53, Re φz S d dx 6 = 0 for φ = iv d ψ in B d e B d e We have the following a priori estimates Lemma 6 There exists a constant C, depending on ϱ only such that for all ɛ sufficiently small, d ɛ, and any solution of 6, it holds ψ C h 63 R Proof: The proof is exactly as in Lemma 5 in 3 We prove it by contradiction Suppose that there exists a sequence of ɛ = ɛ n 0, functions ψ n, h n which satisfy 6 with ψ n =, h n = o We derive inner estimates first We have the symmetries and boundary conditions for ψ and ψ in 54 Whence we may just need to consider the region Then we have Re Σ = { x > 0, 0 < x < λπ/ɛγ } R φn Z d = Re Σ φn Z d Let z Σ, z = d e y and φ n y = φ n z Then as n, = 0 V d = w ye iθ d e Oe d/ = w y o Since ψ n =, we may take a limit so that φ n φ 0 in R loc, where φ 0 satisfies L 0 φ 0 = 0 where L 0 is defined by 3 Observe that φ 0 satisfies the decay estimate 4 because of our assumption on ψ n By Theorem, we have φ 0 = c w y c w y Observe that φ 0 inherits the symmetries of φ and hence φ 0 = φ 0 z The other symmetry is not preserved under the transformation z = d e y But certainly w y does not enjoy the above w symmetry Hence φ 0 = c y On the other hand, taking a limit of the orthogonality condition Re Σ φn Z d = 0,

17 VORTEX HELICIES FOR SCHRÖDINGER MAP EQUATION 7 we obtain Re This implies that c = 0 and hence we have R w φ0 = 0 y φ n 0 in R loc which implies that for any fixed R > 0, φ L p l j<r φ L p l j<r φ L p l j<r φ L p l j<r = o 64 j= We use the L p -estimates in the inner part { z d e < R} By choosing p > N large we obtain the embedding W,p loc into C,α loc for any α 0, Next we shall derive outer estimates Again, by using the symmetries and boundary conditions for ψ and ψ in 54 we just need to consider the region Σ = { x R, 0 < x < λπ/γ } Let η be a cut-off function such that ηs = for s and ηs = 0 for s > We consider the new function z d e z d e ψ = ψ χz, where χz = η η 4 4 Using the explicit forms of L 0 in 5, the first equation becomes ψ ψ ψ x x γ λ ψ γ x = Oe y ψ O χ ψ Oψ χ h χ On the region Σ\B 4 d e B 4 d e, we have the conditions in 54 Moreover we have χ ψ = o z d e z d e σ For the outer part estimates, we use the following new barrier function where by the notations we have denoted l = 65 Bz : = B z B z, 66 x d k x, l = x d k x, B z = ηɛ z d e ηɛ z d e lβ xν l β xν, B z = x ϱ sinτɛx ν In the above β ν = ϱ, 0 < ϱ < ν < η is a smooth cut-off function with properties ηt = for t < c ϱ, where c ϱ is small, and ηt = 0 for t > c ϱ Note that c ϱ, τ and k are constants, which will be determined in the sequel Now, we do the computations for B B B k C l ϱ where C depends only on β and ν On the other hand, there holds B C xν lβ β x d l x d x x x ϱ l 67, 68 Thus for x d < c ϱ d or x d < c ϱ d, where c ϱ is small, we have B C c l ϱ ϱ ϱ l 69 x x

18 8 J WEI AND J YANG where C depends only on β and ν Furthermore, we can choose k such that there exists a positive constant C 3 depends only on β, ν, ˆλ and γ, for x d < c ϱ d or x d < c ϱ d there holds λ γ x γ B k ˆλ = γ ɛ x γ k βν ββ l x νν l x l β ν γx 60 ˆλ γ ɛ x γ k βν ββ l x νν l x l β ν γx C 3 c l ϱ ϱ ϱ l By choosing c ϱ small, we obtain B B B x x γ By trivial computations, we obtain that B B B x x γ = ϱ x ϱ sinτɛx ν λ B γ x λ B γ x C l ϱ sinτɛx ν νν γ τ ɛ x ϱ sinτɛx ν ν γ τ ɛ x ϱ ν ˆλ γ τ x ϱ sinτɛx ν ˆλ νν γ τ x ϱ sinτɛx ν By choosing τ small enough, we have 0 < sinτɛx < / for 0 < x < λπ/γ If x d < c ϱ d, there holds x cϱ x < d c ϱ On the other hand, x d > c ϱ d, there holds x cϱ x < d c ϱ We finally have ϱ l 6 6 B B B x x γ λ B γ x C z d e ϱ z d e ϱ 63 By comparison principle on the set Σ\B 4 d e B 4 d e, we get that ψ CB h o, z Σ\B 4 d e B 4 d e 64 Elliptic estimates then give l σ j ψ L l C h j>4 o 65 j= To estimate ψ, we perform the same cut-off and now the second equation becomes ψ x ψ x γ ψ λ ψ 4 V d γ x V d ψ = O y ψ Oe y ψ O χ ψ O ψ h χ 66

19 VORTEX HELICIES FOR SCHRÖDINGER MAP EQUATION 9 Note that we also have the conditions in 54 Since for z Σ\B 4 d e B 4 d e, there holds 4 V d V d 4, by standard elliptic estimates we have ψ L l j>4 C ψ L l j=4 ψ h z d e z d e ϱ, ψ C ψ L l j=r ψ h z d e z d e ϱ 67 Combining both inner and outer estimates in 64, and 67, we obtain that ψ = o, which is a contradiction By contraction mapping theorem, we conclude Proposition 6 There exists a constant C, depending on ν, ϱ only such that for all ɛ sufficiently small, d large, the following holds: there exists a unique solution φ ɛ,d, c ɛ d to SV d φ ɛ,d = c ɛ dz d 68 and φ ɛ,d satisfies Furthermore, φ ɛ,d is continuous in d φ ɛ,d Cɛ ϱ 69 Here, we do not give the proof to the last proposition The reader can refer to the arguments in 3 for details 7 Reduced Problem and the Proof of Theorem We now solve the reduced problem From Proposition 6, we deduce the existence of a solution φ, c = φ ɛ,d, c ɛ d to SV d φ ɛ,d = L d φ ɛ,d N d φ ɛ,d SV d = c ɛ dz d 7 Multiplying 7 by Z V d d and integrating, we obtain c ɛ Re R V d Z Z d d = Re Re V d Z d SV d R R V d Z d N d φ ɛ,d Re R V d Z d L d φ ɛ,d Using Proposition 6 and the expression in 58, we deduce that Re Zd N d φ ɛ,d = oɛ 7 R On the other hand, integration by parts, we have Re R V d Z d L d φ ɛ,d = Re R V d φ ɛ,dl d Z d Let us observe that and thus by Proposition 6 d S 0V d = L d V d d = L d Z d = Oɛ 73 Re R V d φ ɛ,d L d Z d = oɛ 74

20 0 J WEI AND J YANG As the strategy in standard reduction method, we are left to estimate the following integral Re V d Z d SV d = Re V d Z d S 0 V d Re V d Z d S V d ɛ log ɛ Re i V d Z V d d We will estimate the above integrals in the sequel On, recall that z = d e y Then we first have Re V d Z d S 0 V d ρ = ρ ρ ρ ϕs d y y ρ θ ϕ s ρ y ρ ρ θ ρ ρ ρ ϕ s Oɛ y Let us notice that Hence and also ρ ρ ρ ρ = dγ S log d ρ = π log d Oɛ 8dγ ϕ s = d log d y Oɛ log y ρ d y ρ y = Oɛ, ϕs θ ϕ s ρ ρ ρ θ y ρ ρ ϕ s Oɛ y ρρ ρ 3 dy Oɛ y Similarly, we can estimate Re V d Z d S V d = γ ρ ρ γ γ ρ ρ ρ ρ ρ ρ ρ λ ρ x θ θ ρ ρ ρ ρ ϕ s λ ρ d y λ ϕ s ϕ x λ d λ x θ Oɛ y λ ρ d y On the other hand, by the estimates in Section 6, we have V d Re i V d Zd = π o 76 y 4 Thus we obtain c ɛ d = c 0 75 π 8dγ log d π δɛ log ɛ Oɛ 4γ, 77

21 VORTEX HELICIES FOR SCHRÖDINGER MAP EQUATION where c 0 0 Therefore, we obtain a solution to c ɛ d = 0 with the following asymptotic behavior: This proves Theorem d ɛ o 78 ɛ Acknowledgment J Wei is partially supported by an Earmarked Grant from RGC of Hong Kong and a Focused Research Scheme of CUHK J Yang is supported by the foundations: NSFCNo09008, NSF of GuangdongNo Part of this work was done when J Yang visited the department of mathematics, the Chinese University of Hong Kong: he is very grateful to the institution for the kind hospitality We thank Prof T C Lin for useful disscussions References S Komineas and N Papanicolaou, Topology and dynamics in ferromagnetic media Phys D , no, 8-07 S Komineas and N Papanicolaou, Vortex dynamics in two-dimensional antiferromagnets, Nonlinearity 998, no, F Bethuel and J C Saut, Travelling waves for the Gross-Pitaevskii equation I Ann Inst H Poincaré Phys Theór , no, F Bethuel, G Orlandi and D Smets, Vortex rings for the Gross-Pitaevskii equation, J Eur Math Soc 6 004, no, F Bethuel, P Gravejat and J-G Saut, Travelling waves for the Gross-Pitaevskii equation, II, Comm Math Phys , no, F Bethuel, P Gravejat and J Saut, Existence and properties of travelling waves for the Gross-Pitaevskii equation, Stationary and time dependent Gross-Pitaevskii equations, 55-03, Contemp Math, 473, Amer Math Soc, Providence, RI, F Bethuel, H Brezis and F Hélein, Asymptotics for the minimization of a Ginzburg-Landau functional, Calc Var and PDE 993, no 3, F Bethuel, H Brezis and F Hélein, Ginzburg-Landau vortices, Boston: Birkhäuser, C Bardos, C Sulem and P Sulem, On the continuous limit for a system of classical spins, Comm math Phys , N H Chang, J Shatah and K Uhlenbeck, Schrödinger maps, Comm Pure Appl Math , no 5, D Chiron, Travelling waves for the Gross-Pitaevskii equation in dimension larger than two Nonlinear Anal , no -, D Chiron, Vortex helices for the Gross-Pitaevskii equation J Math Pures Appl , no, W Y Ding and Y D Wang, Schrödinger flow of maps into symplectic manifolds, Sci China Ser A 4 998, no 7, W Y Ding and Y D Wang, Local Schrödinger flow into Kähler manifolds, Sci China Ser A 44 00, no, W Y Ding, On the Schrödinger flows, Proceedings of the International Congress of Mathematicians, Vol II Beijing, 00, 83-9, Higher Ed Press, Beijing, 00 6 W Y Ding, H Y Tang and C C Zeng, Self-similar solutions of Schrödinger flows, Cal Var PDE , no, S Ding and B Guo, Initial boundary value problem for the Landau-Lifshitz system I-Existence and partial regularity, Prog in Natural Sci 8 998, no, -3 8 S Ding and B Guo, Initial boundary value problem for the Landau-Lifshitz system II-Uniqueness, Prog in Natural Sci 8 998, no, M del Pino, P Felmer and M Musso, Two-bubble solutions in the super-critical Bahri-Coron s problem, Calc Var Part Diff Eqn M del Pino, M Kowalczyk and M Musso, Variational reduction for Ginzburg-Landau vortices, J Funct Anal , M del Pino, P Felmer and M Kowalczyk, Minimality and nondegeneracy of degree-one Ginzburg-Landau vortex as a Hardy s type inequality Int Math Res Not 004, no 30, 5-57 C Gui and J Wei, Multiple interior peak solutions for some singularly perturbed Neumann problems, J Diff Equ , -7 3 Y Zhou and B Guo, Existence of weak solution for boundary problems of systems of ferro-magnetic chain Sci Sinica Ser A 7 984, no 8, Y Zhou, B Guo and S Tan, Existence and uniqueness of smooth solutions for system of ferromagnetic chain, Sci in China, Ser A 34 99, no 3, 57-66

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