Standing waves in near-parallel vortex filaments
aa r X i v : . [ m a t h . D S ] A ug Standing waves in near-parallel vortexfilaments
Walter Craig ∗ , Carlos Garc´ıa-Azpeitia † , Chi-Ru Yang ∗ March 13, 2018
Abstract
A model derived in [18] for n near-parallel vortex filaments in athree dimensional fluid region takes into consideration the pairwiseinteraction between the filaments along with an approximation formotion by self-induction. The same system of equations appears indescriptions of the fine structure of vortex filaments in the Gross –Pitaevski model of Bose – Einstein condensates. In this paper weconstruct families of standing waves for this model, in the form of n co-rotating near-parallel vortex filaments that are situated in a cen-tral configuration. This result applies to any pair of vortex filamentswith the same circulation, corresponding to the case n = 2. Themodel equations can be formulated as a system of Hamiltonian PDEs,and the construction of standing waves is a small divisor problem.The methods are a combination of the analysis of infinite dimensionalHamiltonian dynamical systems and linear theory related to Ander-son localization. The main technique of the construction is the Nash-Moser method applied to a Lyapunov-Schmidt reduction, giving riseto a bifurcation equation over a Cantor set of parameters. ∗ Department of Mathematics and Statistics, McMaster University, Hamilton, OntarioL8S 4K1, Canada † Departamento de Matem´aticas, Facultad de Ciencias, Universidad NacionalAut´onoma de M´exico, 04510 M´exico DF, M´exico ntroduction The dynamics of self-interacting infinitesimal vortex tubes in a three di-mensional fluid is in general a complex problem, involving the coherence ofstructures under time evolution of individual filaments, and possible filamentcollision and reconnection. However in certain limiting cases there are mod-els that are relatively straightforward to understand, which exhibit dynamicsthat are subject to rigorous analysis. In [18], a model system of equationswas derived for the interaction of n near-parallel vortex filaments, all sharingthe same circulation. In this model we consider points in R coordinatizedby ( x + ix , x ) ∈ C × R and give n -many curves ( u j ( t, s ) , s ) ∈ C × R thatdescribe the positions of n vertically oriented vortex filaments, each with unitcirculation γ = 1. From [18] the system of model equations for the dynamicsof n near-parallel vortex filaments is given by ∂ t u j = i ∂ ss u j + n X i =1 ,i = j u j − u i | u j − u i | ! , j = 1 , . . . n . (1)The case of exactly parallel vortex filaments in an incompressible inviscidfluid reduces to a problem of interactions of point vortices in R , whichis described by a finite dimensional Hamiltonian system. The model (1)represents an approximation of near - parallel vortex filament interactions,which is valid in an asymptotic regime in which vortex filaments have smalldeviation from being exactly parallel and they remain uniformly separated.The above model has been extensively studied. In [17] the long timeexistence of solutions of (1) is given for n = 2 and for certain near triangularconfigurations with n = 3. The article [1] gives a long time existence theoremfor the case n = 4 near the configuration of a square, and gives a global intime solution for central configurations consisting of rotating regular polygonsof an arbitrary number n ≥ Figure 1: Two vortex filaments that have large separation, that rotate uni-formly around the center with small frequency ω ∼
0. The perturbation fromstraight filaments oscillates approximately as u ( s, t ) ∼ r ( e − it ) cos s .confinement, the core of a higher index vortex filament separates into a finestructure described by the system of near-parallel index-1 filaments, whosepositions are described by the system (1). This asymptotic description hasbeen rigorously established in [6] in the stationary case, and in the case ofnontrivial time dependent evolution in [16]. To our knowledge, the rigorousanalytic justification of (1) as a model of vortex filaments for the Eulerequations of fluid dynamics is open.In this paper we consider families of time periodic solutions that addi-tionally are periodic in the spatial variable s . Our main result is a bifurcationtheory for periodic standing waves of this n vortex ensemble, giving rise tofamilies of solutions that bifurcate from n exactly parallel filaments that re-volve around an axis, with positions determined by a central configuration.For n = 2 this addresses the case of any two near-parallel filaments withthe same circulation. When n ≥ n vortex filament problem.3 Figure 2: Two vortex filaments with small separation, rotating with fre-quency ω ∼ ∞ . The perturbation from straight filaments oscillates approxi-mately as u ( s, t ) ∼ r ( i √ ω sin √ ωt ) cos s . Theorem 1
Let a j be a central configuration, satisfying ωa j = n X i =1( i = j ) a j − a i | a j − a i | .Then the n vortex filament problem (1) has solutions of the form u j ( t, s ) = a j e itω (1 + u (Ω( r ) t, s ; r )) ,where ω is a diophantine frequency, r is a small amplitude varying over aCantor set, and Ω( r ) = Ω + O ( r ) with Ω = √ ω and u ( t, s ) = r cos s (cos t − i Ω sin t ) + O ( r ) . Many PDE’s that describe physical phenomena exhibit the structure ofan infinite dimensional Hamiltonian system. Equations that model nonlin-ear wave phenomena are such, and they typically possess equilibria that areelliptic points in the sense of dynamical systems, for which one anticipatesfamilies of nearby periodic and quasi-periodic solutions. However for Hamil-tonian PDE’s, even the construction of periodic solutions often presents asmall divisor problem, due to the infinite number of degrees of freedom andthe spectral properties of the relevant linearized operators. Thus such con-structions are analogous to the analytic challenge of constructing invarianttori in KAM theory.In this paper the methods that are used to construct periodic solutionsinvolve a Lyapunov-Schmidt reduction along with a Nash-Moser procedure, a4trategy introduced in Craig and Wayne [7]. This technique is useful in prob-lems with multiple resonances, as it has the advantage that it does not requirethe third Melnikov condition that is a feature in the classical KAM meth-ods. This procedure was generalized by Bourgain to construct quasiperiodicsolutions for Hamiltonian PDEs, in for instance [3], and was further used animproved by Berti and Bolle [4] and a number of other authors.The strategy of Lyapunov-Schmidt reduction is used to solve the rangeequation through a Nash-Moser procedure, with smoothing operators givenby Fourier truncations of finite but increasing dimension. The convergenceof the procedure relies on estimates of the inverse of a sequence of finitedimensional matrices of asymptotically large dimension. Approximate inver-sion of linearized operators is the key to most applications of KAM theoryto Hamiltonian PDEs. Resolvent estimates that resemble Fr¨ohlich–Spencerestimates are used to control the inverse of the linearization, projected off ofthe kernel. However restrictions in the local coordinates of the kernel and thefrequency parameters have to be imposed in order to obtain these estimates.These in turn require the excision of certain near resonant subsets of param-eter space in an inductive procedure. The estimates for the projected inverseoperator diverge, due to the small divisors, a divergence which is overcomeby the rapid convergence of the Newton method. However, at the end of theprocess the set of parameters for which the scheme converges is reduced toa Cantor set of positive measure, and the bifurcation equation is solved onlyover this Cantor set.In this paper, the bifurcation equation is analyzed with the use of the sym-metries. These symmetries imply the existence of two bifurcation branches,one consisting of standing waves and one of traveling waves. The branchof standing waves corresponds to a symmetry-breaking phenomenon from aone-dimensional orbit to a three-dimensional orbit. Estimates on the mea-sure of the intersection of the branch of nontrivial standing waves with theCantor set finishes the proof. The branch of traveling wave solutions doesnot involve small divisors; it is discussed in the Appendix along with othersolutions.The Nash-Moser procedure has been previously applied to nonlinear waveequations and NLS equations in [7], [8], [3], [4] and [5], and for standing waterwaves in [23]. We have two purposes in developing the analysis of Hamilto-nian PDEs and small divisor problems in this paper. Firstly, the underlyingproblem of vortex filament dynamics has a relevance to fluid dynamics, andin particular to questions of Euler flows. Moreover, we have a secondary5bjective, which is to present a simplified and more straightforward proof ofthe relevant estimates of the inverse of the linearized operator, being a vari-ant and a conceptually simpler version of the classical Fr¨ohlich-Spencer esti-mates. We believe that these will be useful for other small divisor problems,whether for finite dimensional Hamiltonian systems in resonant situations forwhich the third Melnikov condition is not expected to hold, or for problemsof invariant tori for Hamiltonian PDEs.In section 1, we set up the question of existence of standing waves as aproblem of bifurcation theory in a space with symmetries, related to PDEanalogs of [12] and [15]. In section 2 we discuss the Lyapunov-Schmidt re-duction with the use of the Nash-Moser procedure. In section 3 we presentthe estimates for the projected inverses, assuming certain hypotheses on theseparation of singular sites and estimates of the spectra of the relevant linearoperators. This step is at the heart of the Nash-Moser procedure. In sec-tion 4 we estimate the measure of the set of parameters for which the abovehypotheses of separation and inversion are satisfied. Finally in section 5 weshow that solutions of the bifurcation equation intersect the above set of goodparameters, and show that the intersection has asymptotically full measure.In the Appendix we discuss the symmetries of the standing waves, and inaddition we give a result about the global bifurcation of periodic travelingwaves, which has a much different character than that of the standing waves.
A central configuration of exactly parallel vortex filaments is a family ofstraight and exactly parallel lines ( u j ( t ) , s ), whose dynamics satisfy u j ( t ) = e iωt a j , where the coordinates a j ∈ C satisfy ωa j = n X i =1 ,i = j a j − a i | a j − a i | (2)for all j = 1 , . . . n . Such configurations arise in studies of the n -body problem,for example [13], [21], [22] and the references contained in these papers.Homographic solutions for the vortex filament problem are solutions with u j ( t, s ) = w ( t, s ) a j , where a j ’s are complex numbers satisfying (2). In this class of solutions theshape of the intersections of the filaments with a horizontal complex plane is6omographic with the shape of their intersection with any other horizontalplane { x = c } for any x and at any time t . Homographic solutions of thisform satisfy the equations (2) and ∂ t w = i (cid:0) ∂ ss w + ω | w | − w (cid:1) . (3)The set of solutions w ( t, s ) a j foliate an invariant manifold. By rescaling onemay set ω = 1 in (2). In the case n = 2 of two filaments, the complement ofthis subspace, that is in center of circulation coordinates, the orbit space isfoliated by solutions of the linear Schr¨odinger equation, see [17].Solutions of equation (2) are known as central configurations of the n -vortex problem, and solutions of this form have been well studied in theliterature. For instance, in [17] and [14], a polygonal central configurationwith a central filament is discussed. Solutions to equation (3) also generatehomographic solutions when a central filament with different circulation isfixed at the central axis, see [17].Equation (3) can be formulated as a Hamiltonian system, given by ∂ t w = i∂ ¯ w H ( w )where the Hamiltonian for the system is H = Z π | ∂ s w | − ln( | w | ) ds .Since the Hamiltonian H ( w ) is autonomous and invariant under change ofphase and translation, the energy H , the angular momentum I = Z π | w | ds , and the momentum W = Z π ¯ w ( i∂ s w ) ds are conserved quantities.The simplest solutions to equation (3) are relative equilibria of the form w ( t, s ) = e iωt v ( s ). In these solutions the filaments turn around a central axisat a constant uniform speed. For this class of solutions equation (3) becomes ωv = ∂ ss v + | v | − v , (4)7ith ω being the angular frequency.One simple family of solutions of (4) in helical form is given by v ( s ) = ae iσs with ω = − σ + a − . Thus a continuum of solutions of equation (3) isgiven by w ( t, s ) = ae i ( ωt + σs ) with ω = − σ + a − (5)which is parametrized by the vortex filament separation a . The solutions inthis continuum have a one dimensional orbit ae iθ e i ( ωt + σs ) for θ ∈ S . Equation(3) is similar to the Kepler problem; other solutions to this equation arediscussed in the appendix.The equation (3) is invariant under the Galilean transformation e − iα t e iαs w ( t, s − αt ) . (6)Under the Galilean transformation, the solution (5) generates the family ae i (( ω − ασ − α ) t +( σ + α ) s ) .Since ω = − σ + a − , by the choice α = − σ , the solution (5) becomes ae iω with ω = a − . Hence, under the symmetry exhibited by the Galilean trans-formations, the different branches ae i ( ωt + σs ) are transformed to the branchwith σ = 0.From the previous remark, we may assume without loss of generalitythat the bifurcation branch of periodic solutions is close to the co-rotatingexactly parallel solutions ae iωt , with ω = 1 /a . Subsequently, using theGalilean transformation, any periodic solution near ae iωt can be reproducedas a solution that is a perturbation of ae i ( ωt + σs ) for an arbitrary choice of σ ∈ R .The equation (3) is invariant under the scaling τ − w ( τ t, τ s ) ,so that any P -periodic boundary condition in s may be fixed to P = 2 π .However, once the spatial period has been fixed, the amplitude a cannot bescaled further. It therefore suffices to consider the problem of bifurcationof solutions which have spatial period 2 π . The amplitude parameter a hasthe role of an external parameter in the problem, and solutions ae iωt havedifferent properties depending on a , due to patterns of resonance.8inally, we address the bifurcation of 2 π/ Ω-periodic solutions in timewhich are close to the exactly parallel central configuration e iωt a , using thechange of coordinates w ( t, s ) = ae ωit v (Ω t, s ) . (7)Under rescaling the time variable, the 2 π/ Ω-periodic solutions of the equation(3) satisfy the equation i Ω ∂ t v = − ∂ ss v + ω (1 − | v | − ) v . (8)Since the equation depends of the amplitude a only through the angularfrequency ω , in the subsequent analysis we choose to parametrize solutionfamilies of (8) through the frequency ω . By arguments of generic bifurcation, following the remarks in the appendix,it is expected that there is one bifurcating branch of traveling waves andone of standing waves. The traveling wave do not present a small divisorproblem, and we give a proof of their existence in the appendix.To prove the existence of families of standing wave solutions, we use aNash-Moser method in a subspace of symmetries (10), a reduction whichsimplifies several aspects of the proof; in particular in this setting singularregions consist of isolated sites, the linearization is not degenerate, and thekernel of the linear operator is one dimensional.Using the change of variables v = 1 + u in equation (8), where u is a smallperturbation, the bifurcation of periodic solutions are zeros of the map f ( u ; Ω) = − i Ω u t − u ss + ω ( u + ¯ u ) + ωg ( u, u ), (9)where Ω is the temporal frequency, and the nonlinearity is given by g ( u, ¯ u ) = ∞ X n =2 ( − n ¯ u n = ¯ u u . Our main result is stated in the following theorem.
Theorem 2
For ω diophantine, with only one exceptional value ω = ω ,there exists r > and a Cantor subset C ⊂ [0 , r ] with measure |C| ≥ (1 − r C β ) ( r ≪ ), such that the operator f ( u ; Ω) has a nontrivial bi-furcation branch of solutions of f ( u ; Ω) = 0 parametrized by r ∈ C . Thesolutions ( u ( s, t ; r ) , Ω( r )) are analytic and periodic in s and t , and they havethe following form; u ( t, s ; r ) = r cos s (cos t − i Ω sin t ) + O ( r ) ,where Ω( r ) = Ω + Ω r + O ( r ) with Ω = √ ω and Ω = 0 . Moreover,the solutions u ( s, t ) are Whitney smooth in r and exhibit the symmetries u ( t, s ) = u ( t, − s ) = ¯ u ( − t, s ) . (10)In fact, solutions of this theorem satisfy the additional symmetry u ( t, s ) = u ( t + π, s + π ).The solutions of Theorem 2 have an orbit which is a 3-torus, given by e iθ u j ( t + ϕ, s + ψ ) for ( θ, ϕ, ψ ) ∈ T .The exceptional value ω of the rotational frequency parameter is dueto the failure of the nondegeneracy condition of the amplitude - frequencymap in the construction of the bifurcation branch of periodic standing wavesolutions.Theorem 2 follows the basic structure of the Lyapunov center theorem,however it concerns the time evolution of a PDE and subsequently the anal-ysis must deal with the problem of small divisors. In addition the problemnaturally encounters a resonant situation corresponding to its translationinvariance in the variable s ∈ R , which presents a second difficulty. To over-come the latter, we seek solutions which are symmetric under reflections inspace and time. A posteriori we show that this set of time periodic solutionsare the unique standing wave solutions. In order to work in the setting ofsymmetric solutions, we define the Hilbert space L sym ( T ; C ) = { u ∈ L ( T ; C ) : u ( t, s ) = u ( t, − s ) = ¯ u ( − t, s ) } ,with the inner product h u , u i = 1(2 π ) Z T u ¯ u dtds .A function u ∈ L sym may be written in a Fourier basis u = X k ∈ N a ,k cos ks + X j ∈ N + ,k ∈ N ( a j,k cos jt + ib j,k sin jt ) cos ks a j,k , b j,k ∈ R . We are using the notation that N = { , , .. } and N + = { , , ... } . The linearization of the map f about u = 0 is L (Ω) u := f ′ (0; Ω) u = − i Ω u t − u ss + ω ( u + ¯ u ).Explicitly, L (Ω) is diagonal in the above basis, and the Fourier component j = 0 is L (Ω)( a ,k cos ks ) = ( k + 2 ω ) a ,k cos ks . For the Fourier component j ∈ N + , the linear map is L (Ω) (cid:18)(cid:18) a j,k b j,k (cid:19) · (cid:18) cos jti sin jt (cid:19) cos ks (cid:19) = M j,k (cid:18) a j,k b j,k (cid:19) · (cid:18) cos jti sin jt (cid:19) cos ks ,where M j,k is the matrix M j,k (Ω) = (cid:18) k + 2 ω − Ω j − Ω j k (cid:19) . The matrix M j,k has eigenvalues λ j,k, ± = k + ω ± p j Ω + ω ,and normalized eigenvectors v j,k, ± = 1 c j, ± (cid:18) ω ± p j Ω + ω − j Ω (cid:19) ,where c j, ± = √ (cid:16) ω + j Ω ± ω p j Ω + ω (cid:17) / .The orthonormal eigenbasis is given by e , , = 1, e ,k, = √ ks and e j,k,l = c k v j,k,l · (cid:18) cos jti sin jt (cid:19) cos ks for ( j, k, l ) ∈ N + × N ×{ , − } , where c = √ c k = 2 ( k = 0). That is,we have h e x , e y i = δ x,y and for every function u ∈ L sym , u = X x ∈ Λ h u, e x i e x , L (Ω) u = X x ∈ Λ λ x h u, e x i e x ,11here x ∈ Λ = N + × N ×{ , − } ∪ { }× N ×{ } .Define the analytic norm k u k σ = X ( j,k,l ) ∈ Λ h u, e j,k,l i e | ( j,k ) | σ h ( j, k ) i s ,where h ( j, k ) i = p j + k . Even though the eigenfunctions e x dependon the frequency parameter Ω, and the norm k u k σ is defined through thissystem of eigenfunctions, the norm is in fact independent of Ω since the basis { e x } x ∈ Λ is orthonormal. Lemma 3
The Hilbert space h σ of analytic functions given by h σ = { u ∈ L sym ( T ; C ) : k u k σ < ∞} . is an algebra for s > ; k uv k σ ≤ c σ,s k u k σ k v k σ . Proof.
The result is quite standard. Let u ∈ L ( T , C ), then the norm X ( j,k ) ∈ Z (cid:12)(cid:12)(cid:10) u, e i ( jt + ks ) (cid:11)(cid:12)(cid:12) e | ( j,k ) | σ h ( j, k ) i s (11)has the algebra property under the convolution for s >
1, see [8]. The resultfollows from the fact that the two norms k·k σ and (11) are equivalent in L sym . Lemma 4
The nonlinear operator g ( u, ¯ u ) satisfies k g ( u, ¯ u ) k σ < c σ k u k σ , for small k u k σ . Thus, the map f is well defined and continuous in (cid:8) u ∈ h σ : k u k σ < c − σ (cid:9) . roof. That the map is well defined follows from the equivariant prop-erty and the first statement. The first inequality follows from the algebraicproperty, k g ( u, ¯ u ) k σ = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X n =2 ( − n ¯ u n (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) σ ≤ ∞ X n =2 c n − σ,s k u k nσ = c σ,s k u k σ − c σ,s k u k σ < c σ k u k σ . The eigenvalue λ j,k, − (Ω) is zero when the frequency is equal toΩ j,k = j − √ k + 2 k ω .In the case that there are no additional resonances, and without loss ofgenerality, we may analyze the branch of bifurcation from the eigenvalue λ , , − ; all other being equivalent up to scaling (see the appendix). In thenext proposition we prove that there are no additional resonances if ω isirrational. Proposition 5
Let Ω = √ ω . For ω irrational, the kernel of the map L (Ω ) has dimension one, corresponding to the eigenfunction e , , − . Proof.
The eigenvalues of M j,k (Ω ) are λ j,k,l (Ω ) for l = ±
1. The determi-nant of the matrix M j,k (Ω ) is d j,k (Ω ) = λ j,k, λ j,k, − = 2 (cid:0) k − j (cid:1) ω + (cid:0) k − j (cid:1) .Since the frequency Ω is chosen such that M , is not invertible, then d , (Ω ) = 0. Since ω is irrational, then the determinant d j,k is zero onlyif both numbers k − j and k − j are zero. For ( j, k, l ) ∈ Λ, the condi-tion happens only when ( j, k ) = (1 ,
1) or ( j, k ) = (0 , λ , , = 2 ω is always positive, then the only zero eigenvalue in the lattice Λis λ , , − (Ω ).The eigenvalue λ , , − is always zero, and on L ( T ; C ) it contributes tothe null space of L (Ω). However on the space L sym that takes the symmetriesinto account, the (0 , , −
1) is not an element of the lattice Λ, and therefore e , , − / ∈ L sym does not contribute to the respective null space.For A ⊆ Λ a subset of lattice points, define P A as the projection onto theFourier components x ∈ A ⊆ Λ, P A u = X x ∈ A h u, e x i e x .13he null space of L (Ω ) is supported on only one lattice point N := { x =(1 , , − ∈ Λ } because of the symmetry reduction; hence L (Ω ) e x = 0.Thus, P N is the projection onto the one dimensional kernel, and P Λ \ N is theprojection onto its complement. In future studies of quasi-periodic solutionsthe set N will in general comprise a possibly large but finite number of latticesites. Lyapunov-Schmidt reduction
Define the decomposition u = v + w into its components in the kernel andthe range of L (Ω ); v = P N u, w = P Λ \ N u . Then the zeros of the nonlinear operator f ( u ; Ω) defined in (9) are the solu-tions of the pair of equations P N f ( v + w ; Ω) = 0 , P Λ \ N f ( v + w ; Ω) = 0 . Let r ∈ R be a parametrization of the kernel of f ′ (0; Ω ) given by v ( r ) = re , , − ∈ L sym ( T ; C ).The strategy of the construction consists in solving w ( r, Ω) in the rangeequation P Λ \ N f ( v ( r ) + w ; Ω) = 0, using a Nash-Moser procedure. The keyaspect of this step is that it is a small divisor problem, which implies thatthe function w ( r, Ω) can only be constructed in a Cantor set of parameters( r, Ω) in a neighborhood of (0 , Ω ). Initial Nash-Moser step
A first approximation of the solution is constructed projecting in the ball B of radius L , B = { ( j, k ) ∈ Λ \ N : | j | + | k | < L } .To solve the first step of the Nash-Moser iteration, we use the followingproposition; Lemma 6
For a diophantine number ω , if Ω is such that | Ω − Ω | < cγ/L τ +10 ,then | λ j,k, − (Ω) | > cγ/L τ +20 for all ( j, k ) with | ( j, k ) | < L . roof. Using the determinant of M j,k (Ω ), and the fact that | qω − p | ≥ γ/ | q | τ , we have | d j,k (Ω ) | ≥ γ/ (cid:12)(cid:12) (cid:0) k − j (cid:1)(cid:12)(cid:12) τ ≥ cγ/L τ .Since | λ j,k, | ≤ cL , then | λ j,k, − (Ω ) | ≥ cγ/L τ +20 . The result follows from the inequalities | λ j,k, − (Ω)) | ≥ | λ j,k, − (Ω ) | − | λ j,k, − (Ω) − λ j,k, − (Ω ) | & γ/L τ +20 − | j | | Ω − Ω | & γ/L τ +20 . By the implicit function theorem and the lemma above, there is a solution w ( r, Ω) of P B f ( v ( r ) + w , Ω) = 0, which is defined over the regime ofparameters ( r, Ω) with 0 ≤ r < r and | Ω − Ω | < cγ/L τ +10 . Subsequent Nash-Moser steps
Let P B n be the projection in the ball B n = { ( j, k ) ∈ Λ \ N : | j | + | k | < L n } with L n = 2 n L . The Nash-Moser method consists in providing a betterapproximation w n at the n th step, where w n ( r, Ω) = w n − ( r, Ω) + δw n .Here δw n is the correction given by the approximate Newton’s method δw n = − G B n P B n f ( w n − ( r, Ω); r, Ω) , (12)where the inverse operator to ∂ w f ( w n − ( r, Ω); r, Ω) on B n is given by G B n = ( P B n ∂ w f ( w n − ) P B n ) − .If the operator norm of the inverse has a tame estimate k G B n k σ n . γ − n d n , (13)15here σ n = σ n − − γ n , d n < d n − and ( r, Ω) ∈ N n , then we obtain aninductive estimate for δw n of the form k δw n k σ n < Cr e − κ n with κ ∈ (1 , Theorem 7 If d n = L − βn for β > / and P ∞ n =1 γ n → σ / , from theestimate (13), then w n ( r, Ω) converges to the solution w ( r, Ω) ∈ h σ / of P Λ \ N f ( v ( r ) + w ( r, Ω) , Ω) = 0 ,which is Whitney smooth in ( r, Ω) over the Cantor set N = ∩ ∞ n =1 N n (see theresult in [9]). Approximate inversion of the linearized operator ∂ w f ( v ( r ) + w ; Ω) is key toapplications of the Nash-Moser method. In this paper we follow an approachthat is motivated by techniques that have been developed to study Andersonlocalization . We use the basic approach developed by Craig and Wayne in [7]with innovations by Bourgain [3], and Berti and Bolle [4], but we introducecertain simplifications that clarify the method. We expect that this versionwill be useful in further applications of Nash-Moser methods to HamiltonianPDEs.In this paper we describe the situation in which singular sites occur inuniformly bounded clusters, and clusters are separated asymptotically in thelattice. This is typically the case of time periodic solutions for PDEs in onespace dimension.For the particular case of the problem of vortex filaments, we will bemaking certain assumption on the nature and the spectra of the relevantlinearized operators. These assumptions will be verified in the subsequentSection 4, where in order to do so we excise certain regions of parameterspace ( r, Ω) ∈ N .Given w ∈ P Λ \ N L sym , the projection P Λ \ N ∂ w f of the linearization of thenonlinear operator (9) at the point v ( r ) + w ∈ L sym is given by P Λ \ N ∂ w f ( v ( r ) + w ; Ω) = P Λ \ N ( L (Ω) + ω∂ w g ( v + w )) P Λ \ N . (14)16ecause (1) is a Hamiltonian PDE, this linear operator is Hermitian.In the Fourier basis { e x } the operator (14) is expressed by the matrix H ( w ; r, Ω) = D (Ω) + T ( w ; r, Ω),where, for x, y ∈ Λ \ N , the matrix D is a diagonal; D (Ω) = diag x ∈ Λ \ N ( λ x (Ω)),and T is a T¨oplitz linear operator T ( w ; r, Ω)( x, y ) = h ω∂ w g ( v ( r ) + w ) e y , e x i .Estimates of operators acting on analytic spaces h σ are realized with theoperator norm: k G k σ := max (cid:26) sup x P y | G ( x, y ) | e σ | x − y | h x − y i s sup y P x | G ( x, y ) | e σ | x − y | h x − y i s (cid:27) . Consider the restriction of the linearized operator H = P Λ \ N ∂ w f ( v ( r ) + w ; Ω)to ℓ ( E ), where E ⊆ Λ \ N is a region of lattice sites of Λ. Its inverse operatoris denoted by G E = ( P E HP E ) − Definition 8
Let
A, B ⊂ Λ . For the restriction of a matrix H , a linearmapping of ℓ (Λ) to a mapping ℓ ( B ) → ℓ ( A ) , we define H BA = P A HP B .With this notation we have ( G E ) BA = P A ( P E HP E ) − P B . Fix w = w n − the approximation of the ( n − th step of the Nash-Moseriteration. The main linear estimate is of the matrix G B n ( w n − ) = ( P B n H ( w n − ) P B n ) − ,where the region B n ⊆ Λ \ N is a ball centered at the origin of large radius L n . Definition 9
We say that a site x ∈ Λ is regular if | λ x | > d . The subsetof regular sites is denoted by A ⊂ Λ . We say that a site is singular if | λ x | ≤ d . We define S n ⊂ Λ to be theset of singular sites in the annulus B n \ B n − . A ∪ ∞ n =1 S n .The hypotheses we use to control the norm of G B n are: (h1) The non-diagonal T¨oplitz matrix T has the exponential decay property k T ( u ) k σ n ≤ Cr for k u k σ n < r . (h2) The set of singular points in B n \ B n − is the union of bounded regions S j that are pairwise separated. That is, S n = ∪ j S j , rad( S j ) < c ,and for S j , S i ⊂ S n ∪ S n − , we assume thatdist( S i , S j ) > ℓ n . (h3) For C ( S j ) the tubular neighborhood of radius ℓ n around S j , assumethat the spectra of H C ( S j ) are bounded away from zero by d n . That isspec( H C ( S j ) ) ⊂ R \ [ − d n , d n ] .These will be shown to hold inductively for parameters ( r, Ω) in the set N n . We defined recursively σ n = σ n − − γ n . Theorem 10
Assume hypothesis (h1)-(h3), and suppose that cr /d < / ,and c γ n r e − γ n ℓ n /d n < / , where c γ = c X x ∈ Z e − γ | x | . γ − .Let E n = B n − ∪ S n ∪ A n with A n consisting of regular sites in Λ \ B n − , thenwe have that k G E n ( w n − ) k σ n . c γ n /d n . (15)18he proof of Theorem 10 builds on a sequence of lemmas that serve toestimate the norms of local Hamiltonians about singular sites.Let C n ( S j ) be the neighborhood of radius ℓ n around S j ⊂ S n , C n ( S j ) = { x ∈ Λ : dist( x, S j ) < ℓ n } .Because of hypothesis (h2) the sets C n ( S j ) are disjoint for distinct S j . Weprove the following fact, omitting for clarity some instances of the indices n and j . Lemma 11
Assume cr /d < / . From hypotheses (h1)-(h3), for k u k σ n Since E is regular, (cid:13)(cid:13)(cid:13)(cid:0) D EE (cid:1) − T EE (cid:13)(cid:13)(cid:13) σ n ≤ cr /d < / 2, then k G E k σ n = (cid:13)(cid:13)(cid:13)(cid:13)(cid:16) I + (cid:0) D EE (cid:1) − T EE (cid:17) − (cid:0) D EE (cid:1) − (cid:13)(cid:13)(cid:13)(cid:13) σ n ≤ d .Since H C ( S ) is self-adjoint, by hypothesis (h3), the L -norm of G C ( S ) isbounded by 1 /d n , (cid:13)(cid:13) G C ( S ) (cid:13)(cid:13) ≤ /d n . Since the set S has radius bounded by c , then (cid:13)(cid:13) ( G C ( S ) ) SS (cid:13)(cid:13) σ n ≤ e σ n c c s (cid:13)(cid:13) ( G C ( S ) ) SS (cid:13)(cid:13) ≤ Cd n .Since singular regions are separated, then the set E = C ( S ) \ S is regular.From the self-adjoint property, only two more cases require verification, (cid:13)(cid:13) ( G C ( S ) ) SE (cid:13)(cid:13) σ n ≤ C/d n and (cid:13)(cid:13) ( G C ( S ) ) EE (cid:13)(cid:13) σ n ≤ C/d n . Let us define the connection matrixΓ := H C ( S ) − H E ⊕ H S = T SE + T ES . From resolvent identities we have that G C ( S ) = G E ⊕ G S − G E ⊕ G S Γ G C ( S ) ,19hen ( G C ( S ) ) SE = G E T SE ( G C ( S ) ) SS . (16)Therefore, for the first case we have (cid:13)(cid:13) ( G C ( S ) ) SE (cid:13)(cid:13) σ n ≤ C r d d n .From resolvent identities, we have that G C ( S ) = G E ⊕ G S − G E ⊕ G S Γ G E ⊕ G S + G E ⊕ G S Γ G C ( S ) Γ G E ⊕ G S , then ( G C ( S ) ) EE = G E + G E T SE ( G C ( S ) ) SS T ES G E . (17)Thus, for the second inequality we have (cid:13)(cid:13) ( G C ( S ) ) EE (cid:13)(cid:13) σ n ≤ d + C r d d n ≤ C d n . Lemma 12 Let P x G be the projection on the x th row of G , then for any σ > γ > k G k σ − γ ≤ c γ sup x k P x G k σ .Let A and B be two sets such that dist( A, B ) ≥ ℓ , then k P A GP B k σ − γ ≤ e − γℓ k G k σ . Proof. The first result follows from the inequalitiessup y X x | G ( x, y ) | e ( σ − γ ) | x − y | h x − y i s ≤ sup y X x k P x G k σ e − γ | x − y | ≤ c γ sup x k P x G k σ ,and sup x X y | G ( x, y ) | e ( σ − γ ) | x − y | h x − y i s ≤ sup x k P x G k σ . The second result follows from the long step ℓ between A and B thatgives the estimate e − γℓ in the exponential decay of the norm σ − γ . That is,since | x − y | > ℓ for x ∈ A and y ∈ B , thensup y ∈ B X x ∈ A | G ( x, y ) | e ( σ − γ ) | x − y | h x − y i s ≤ e − γℓ k G k σ ,200 G A n G C ( B n − ) G C ( S j ) S j C ( B n − ) A n S j B n − A n Figure 3: Preconditioner matrix L n and similarly for the supremum over x . Theorem 10. The main estimate on the inverse G B n = ( P B n H ( w n − ) P B n ) − is obtained by a approach distilled from [4]; we construct a preconditionermatrix L n from G C n ( B n − ) , G A n and inverses G C n ( S j ) of the local Hamiltonians H C n ( S j ) for S j ⊂ S n . These inverses satisfy estimates as follows: Since A n isregular, then k G A n k σ n . /d . (18)By (h2) and Lemma 13, then (cid:13)(cid:13) G C n ( B n − ) (cid:13)(cid:13) σ n − . c γ n − /d n − . (19)By (h1)-(h3) and Lemma 11, then (cid:13)(cid:13) G C n ( S j ) (cid:13)(cid:13) σ n − . /d n . (20)We define the preconditioner matrix L n as L n = G A n + P B n − G C n ( B n − ) + X S j ⊂S n P S j G C n ( S j ) . (21)21hus, we have that L n ( H E n E n ) = G A n ( H A n A n + H E n \ A n A n ) + P B n − G C n ( B n − ) ( H C n ( B n − ) C n ( B n − ) + H E n \ C n ( B n − ) C n ( B n − ) )+ X S j ⊂S n P S j G C n ( S j ) ( H C n ( S j ) C n ( S j ) + H E n \ C n ( S j ) C n ( S j ) ) = I E n + K n , where, using that G A n H A n A n = I A n etc., we have that K n = G A n H E n \ A n A n + P B n − G C n ( B n − ) H E n \ C n ( B n − ) C n ( B n − ) + X S j ⊂S n P S j G C n ( S j ) H E n \ C n ( S j ) C n ( S j ) .We conclude from (22) that as long as ( I E n + K n ) − exists, we have G E n = ( H E n E n ) − = ( I E n + K n ) − L n . (22)A bound on the operator norm k L n k σ n uses that k G A n k σ n ≤ c/d and (cid:13)(cid:13) P B n − G C n ( B n − ) (cid:13)(cid:13) σ n − ≤ c γ n − /d n − . Then referring to Lemma 12, we have (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X S j ⊂S n P S j G C n ( S j ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) σ n − − γ n ≤ c γ n sup x ∈S n (cid:13)(cid:13) G C n ( x ) (cid:13)(cid:13) σ n − . c γ n /d n .Since L n is the sum of the above three operators, we conclude that k L n k σ n ≤ c/d + c γ n − /d n − + c γ n /d n . c γ n /d n . There is a recursive relation among the constants γ n , ℓ n and d n , for whichwe will show that k K n k σ n ≤ / 4, hence k I + K n k σ n ≤ 4. Therefore, weobtain the result of the theorem k G E n k σ n ≤ (cid:13)(cid:13) L − n (cid:13)(cid:13) σ n . c γ n /d n . To estimate K n , we rewrite it as K n = G A n T E n \ A n A n + P B n − G C n ( B n − ) T P E n \ C n ( B n − ) + X S j ⊂S n P Sj G C n ( S j ) T P E n \ C n ( S j ) .The proof exploits the fact that dist( B n − , E n \ C n ( B n − )) and dist ( S j , E n \ C n ( S j ))are greater than ℓ n , i.e. the long step ℓ n gives the estimate in the exponential22ecay of the norm σ n − − γ n by e − γ n ℓ n . That is, by the first result of Lemma12 we have for S j that (cid:13)(cid:13)(cid:13) P Sj G C n ( S j ) T P E n \ C n ( S j ) (cid:13)(cid:13)(cid:13) σ n − − γ n ≤ c γ n d n r e − γ n ℓ n ,while for B n − we have (cid:13)(cid:13) P B n − G C n ( B n − ) T P E n \ C n ( B n − ) (cid:13)(cid:13) σ n − − γ n ≤ c γ n − d n − r e − γ n ℓ n .By the second result of Lemma 12, we can estimate row by row the matrixinvolving the singular sites S n as (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X S j ⊂S n P Sj G C n ( S j ) T P E n \ C n ( S j ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) σ n − − γ n ≤ c γ n d n r e − γ n ℓ n .Thus we conclude that k K n k σ n − − γ n ≤ cd r + c γ n − d n − r e − γ n ℓ n + c γ n d n r e − γ n ℓ n ≤ cr /d < / c γ n r e − γ n ℓ n /d n < / G B n ( w n − ) assumes the anal-ogous estimate for G B n − ( w n − ) holds. But in the proof, the estimate for G B n − ( w n − ) is used instead. In the next lemma we prove that this esti-mate is true also because w n − − w n − = δw n − is bounded for all n in theNash-Moser procedure. Lemma 13 For any set E n − = B n − ∪ A n − with A n − regular we have that (cid:13)(cid:13) G E n − ( w n − ) (cid:13)(cid:13) σ n − . c γ n − /d n − . Proof. We assume from the previous step that (cid:13)(cid:13) G E n − ( w n − ) (cid:13)(cid:13) σ n − . c γ n − /d n − .The difference of the Hamiltonians is defined as R n − := H E n − ( w n − ) − H E n − ( w n − )= T E n − ( w n − ) − T E n − ( w n − ) . δw n − = w n − − w n − , we have R n − = T ′ E n − ( w n − )[ θ ( δw n − )] . Since k T ′ ( w n − ) k σ n − . k w n − k σ n − | v | with w n − bounded for all n and | v | ≤ r , then k R n − k σ n − . k δw n − k σ n − . ε n − . By the inductive hypothesis, we have that (cid:13)(cid:13) G E n − ( w n − ) R n − (cid:13)(cid:13) σ n − . ε n − /d n − ≤ / G E n − ( w n − ) = G E n − ( w n − )( I E n − + G E n − ( w n − ) R n − ) − . In this section we prove the exponential decay of the T¨oplitz matrix T , andwe discuss the separation property of the singular regions. Then, we provethe estimate of the spectrum of the Hamiltonians in the singular regions forgood parameters ( r, Ω) in a subset N n . Lemma 14 If k u k σ < r , then k T ( u ) k σ < Cr . Proof. Let u be a function with k u k σ < r , by the algebra property of thenorm, we have that the function h ( u ) = ω ∞ X n =2 n ( − n ¯ u n − satisfy k h k σ < Cr . Then by definition |h h, e j,k,l i| < Cr e − σ | ( j,k ) | h j, k i − s . x n = ( j n , k n , l n ) ∈ Λ, then T ( x , x ) = h he x , e x i = X x ∈ Λ h h, e x i h e x e x , e x i Since h e x e x , e x i = 0 when j / ∈ {± j ± j } or k / ∈ {± k ± k } , and since |h e x e x , e x i| ≤ 1, then | T ( x , x ) | ≤ X l = ± , j ∈{± j ± j } , k ∈{± k ± k } |h h, e j ,k ,l i|≤ X j ∈{± j ± j } , k ∈{± k ± k } Cr e − σ | ( j ,k ) | h j , k i − s .Since there are four elements in the sum, and they satisfy j ≥ | j − j | and k ≥ | k − k | , then | T ( x , x ) | ≤ Cr e − σ | ( | j − j | , | k − k | ) | h| j − j | , | k − k |i − s . We say that ( j, k, l ) ∈ Λ is a singular site if | λ j,k,l (Ω) | ≤ d . Lemma 15 Let ( j , k ) and ( j , k ) be two different singular sites, then fora sufficiently small d , we have that | j − j | ≥ C | k + k | ,where C is a constant that only depends on Ω and ω . Furthermore, theconstant is uniform in ( j, k, l ) ∈ Λ for (Ω , ω ) in neighborhood of (Ω , ω ) . Proof. The sites of the form ( j, k, 1) are never singular if d << 1. Giventhat λ j,k, − = k + ω − p ( j Ω) + ω , then (cid:12)(cid:12) k − k (cid:12)(cid:12) − C Ω | j − j | ≤ | λ j ,k , − − λ j ,k , − | ≤ d . If k = k , taking d small enough such that d ≤ C Ω / 2, then j = j .Finally, if k = k , there exists a constant c such that | j − j | ≥ C Ω (cid:0)(cid:12)(cid:12) k − k (cid:12)(cid:12) − d (cid:1) ≥ c | k + k | . S = { ( j , k , − } be a singular site in the annulus B n +1 /B n . Bythe previous inequality, the neighborhood C n ( S ) = { ( j, k, l ) : | ( j, k ) − ( j , k ) | < ℓ n } contains only one singular site for ℓ n = CL / n . We analyze the spectrum of the local Hamiltonians H C ( S ) ( w n ; r, Ω) = P C ( S ) ( D (Ω) + T ( v ( r ) + w n ( r, Ω); Ω)) P C ( S ) .For | r | ≤ r , there is only one eigenvalue of H C ( S ) ( w n , Ω) with norm lessthan d / r << d / 2. Let e ( r, Ω) be that eigenvalue of H C ( S ) with | e ( r, Ω) | < d n << d / , then e ( r, Ω) is isolated from other eigenvalues, and as such it is analytic inthe parameters. Lemma 16 For | r | < r , there exists a constant C > such that ∂ Ω e ( r, Ω) ≤ − CL n . Proof. Since e ( r, Ω) is analytic, then ∂ Ω e ( r, Ω) = ∂ Ω e (0 , Ω) + O ( r ) = ∂ Ω λ j,k, − + O ( r ).By an explicitly calculation ∂ Ω λ j,k, − = − j Ω p j Ω + ω ≥ − C Ω L n ,because the site ( j, k ) is singular with | j | ≥ CL n .From the above lemma, for a fix r < r , the eigenvalue e ( r, Ω) is a mono-tone decreasing function of Ω. Since e ( r, Ω) is analytic, then there is anunique analytic function Ω z ( r ) such that e ( r, Ω z ( r )) = 0. Since e (0 , Ω) = λ j,k, − (Ω) for r = 0, thenΩ z (0) = Ω j,k = 1 j √ k + 2 k ω and ( j, k, − ∈ S j . 26 emma 17 We have that Ω z ( r ) = Ω j,k + 1 L n O ( r ) . Proof. Since e ( r, Ω) is analytic then e ( r, Ω) = e (0 , Ω j,k ) + ∂ Ω e (0 , Ω j,k )(Ω − Ω j,k ) + ∂ r e (0 , Ω j,k ) r + h.o.t By the Feynman-Hellman formula, we have that ∂ r e (0 , Ω j,k ) = h T ′ (0)[ ∂ r v, ψ ] , ψ i .In the space L sym , the functions are ψ = e j,k, − , ∂ r v = e , , − , and d g (0)[ w , w ] = ∂ u g (0) ¯ w ¯ w = 2 ¯ w ¯ w . Thus, for any j, k = 1 / 2, we have that h ψ , T ′ (0)[ ∂ r v, ψ ] i = h ω ¯ e , , − ¯ e j,k, − , e j,k, − i = 0 . Since ∂ Ω e (0 , Ω j,k ) < − CL n , then ∂ Ω e (0 , Ω j,k )(Ω z − Ω j,k ) + h.o.t. = 0Using the implicit function theorem, we have that Ω z − Ω j,k is a function of r , and Ω z − Ω j,k = 1 L n O ( r ).Let N j,k be the neighborhood of the curve Ω z ( r ) given by N j,k = { ( r, Ω) : | Ω z ( r ) − Ω | < C d n L n } , (23)by the previous lemma, and the mean value theorem, the eigenvalue satisfy | e ( r, Ω) | > d n if ( r, Ω) / ∈ N j,k . Thus, the hypothesis (h3) holds true in thecomplement of the set of parameters ∪ ( j,k ) ∈S n N j,k ,where the union is taken over all singular sites in the annulus B n \ B n − .27 Intersection property In this section, we present the arguments for the Whitney regularity of w ( r, Ω) for ( r, Ω) ∈ N = ∩ ∞ n =1 N n . Then, we prove that the intersectionof the curve C = { ( r, Ω( r )) : Ω( r ) = Ω + Ω r + O ( r ) } ,and the Cantor set N has positive measure in the case Ω = 0. Finally, weprove that the curve C corresponding to the bifurcation of standing waveshas the non-degeneracy property Ω = 0. At the n th Nash-Moser step excisions are made in parameter space ( r, Ω) ∈N n − consisting of the union of neighborhoods N j,k , each of width Cd n /L n .On the parameter set N n − \ ∪ j,k N j,k , after the excision of the N j,k , we solve(12) for the correction δw n ( r, Ω). We may now provide a smooth interpolantfor δw n ( r, Ω) across the excisions, in the usual way. Construct a cutoff func-tion ϕ n ( r, Ω) ∈ C ∞ , which is supported in N n − \ ∪ j,k N j,k and for which ϕ n ( r, Ω) = 1 on the new parameter set N n := N n − \ ∪ j,k N j,k , where2 N j,k := { ( r, Ω) : | Ω z ( r ) − Ω | < c d n L n } are excisions of just twice the width of the previous N j,k . This can bedone so that the cutoff function ϕ n has derivatives bounded by | ∂ α Ω ∂ βr ϕ n | ≤ C ( L n d n ) α + β . Then ϕ n δw n ( r, Ω) ∈ C ∞ ( N ) and ϕ n δw n = δw n on N n . Now w n = w n − + ϕ n δw n is C ∞ in the set of parameters ( r, Ω) ∈ N , w n = w n − + δw n on N n , and moreover, for ( r, Ω) ∈ N the sequence w n con-verges in h σ / along with all of its derivatives with respect to ( r, Ω); and thefollowing estimate holds k ∂ α Ω w k σ / ≤ Cr , k ∂ r ∂ α Ω w k σ / ≤ Cr . Lemma 18 Let r − and r + be the minimum and the maximum of { r : ( r, Ω) ∈C ∩ N j,k } , we have that (cid:12)(cid:12) r − − r (cid:12)(cid:12) ≤ C Ω d n L n . roof. Let r be such that Ω z ( r ) = Ω( r ), then the point ( r , Ω( r )) is theintersection of the curves Ω z ( r ) and Ω( r ). Since Ω( r ) = Ω + Ω r + O ( r ),then | Ω( r − ) − Ω( r ) | > Ω | r − − r | .By the previous lemma | Ω z ( r ) − Ω z ( r − ) | ≤ CL n (cid:12)(cid:12) r − − r (cid:12)(cid:12) .Since Ω( r − ) , Ω z ( r − ) ∈ N j,k , then Cd n L n ≥ | Ω( r − ) − Ω z ( r − ) | ≥ | Ω( r − ) − Ω z ( r ) | − | Ω z ( r ) − Ω z ( r − ) |≥ Ω | r − − r | − CL n (cid:12)(cid:12) r − − r (cid:12)(cid:12) ≥ Ω | r − − r | .Analogously, we have for the estimate of r + that (cid:12)(cid:12) r − r (cid:12)(cid:12) ≤ Cd n Ω L n . Thelemma follows from the triangle inequality. Lemma 19 If Ω = 0 , the measure of the set { r : ( r, Ω) ∈ C∩ N j,k } isbounded by |{ r : ( r, Ω) ∈ C∩ N j,k }| < C √ Ω d n √ L n . Proof. For a singular site Ω | j | ∼ k , thenΩ z (0) = Ω j,k = | k/j | √ k + 2 ω ∼ Ω √ ωk − . Thus, | Ω − Ω z (0) | = Ω (cid:12)(cid:12)(cid:12)(cid:12) − ωk − √ ωk − (cid:12)(cid:12)(cid:12)(cid:12) > Ω (cid:12)(cid:12) ωk − (cid:12)(cid:12) & CL n . From the definition of r − , we have | Ω z ( r − ) − Ω( r − ) | < Cd n /L n . By theproperties of Ω z we have | Ω z (0) − Ω z ( r − ) | < Cr − /L n , then CL n r − + Cd n L n > | Ω( r − ) − Ω z (0) | .Since the curve C is of the form Ω( r ) = Ω + Ω r + O ( r ), then | Ω( r − ) − Ω z (0) | > | Ω − Ω z (0) | − r − .29hus, (cid:18) + CL n (cid:19) r − > | Ω − Ω z (0) | − Cd n L n > CL n (1 − Cd n ) > C L n .For Ω = 0, we conclude that r + > r − > C/ √ Ω L n . By the above lemma,we have r + − r − ≤ C Ω d n L n ( C p Ω L n ) ≤ C √ Ω d n √ L n . Proposition 20 Let d n = L − βn . If Ω = 0 and β > / , then the measure ofgood parameters is positive. Moreover, |{ r ∈ [0 , r ) : ( r, Ω) ∈ N ∩ C}| > r (1 − r C β ) , where C β = C √ Ω ∞ X n =1 L / − βn < ∞ . Proof. There are at most cL n singular sites at the n step, then the previouslemma implies |{ r ∈ [0 , r ) : ( r, Ω( r )) / ∈ N n ∩ C}| ≤ r L n C √ Ω d n √ L n .Thus |{ r ∈ [0 , r ) : ( r, Ω) ∈ N ∩ C}| ≥ r − r C √ Ω ∞ X n =1 L / − βn = r (1 − r C β ). In this section we prove that for all parameters ω with only one exceptionalvalue ω = ω we have Ω = 0; namely, the bifurcation branch has nondegen-erate curvature at the bifurcation point. For the asymptotic expansion u = ru + r u + O ( r ) , Ω( r ) = Ω + r Ω + r Ω + O ( r ),30e have that f ( w ; Ω) = Lu + i (Ω( r ) − Ω ) u t + ω ¯ u − ω ¯ u + O ( | u | ) = 0,where L = L (Ω ) is the linear map at Ω .At order r , we have Lu = 0, and then u = e , , − (Ω ). Thus, at order r we have Lu − i Ω ∂ t u + ωu = 0.Multiplying by u , integrating by parts, and using that L is self-adjoint with Lu = 0, we get that Ω h i∂ t u , u i = ω (cid:10) ¯ u , u (cid:11) . The basis e j,k,l at Ω = Ω is given by e , , = 1, e , , = √ s , e , , − = 2( a cos t + ib sin t ) cos s,e , , ± = √ a ± cos 2 t + ib ± sin 2 t ), e , , ± = 2( a ± cos 2 t + ib ± sin 2 t ) cos 2 s, where ( a, b ) T = v , , − and ( a ± , b ± ) T = v , ,, ± . Since h i∂ t u , u i = 14 π Z T i∂ t u ¯ u dtds = − ab ,then (cid:10) u , u (cid:11) = 14 π Z T e , , − dtds = 0 . (24)Since Ω = 0 and u = − ωL − u , then we conclude that¯ u = − ωL − u .At order r , we obtain Lu − i Ω ∂ t u + 2 ωu u − ω ¯ u = 0.Multiplying by u and integrating by parts, then − Ω h i∂ t u, u i = ω (cid:10) ωu L − ( u ) + ¯ u , u (cid:11) .31hus, we have Ω = ω ab (cid:0)(cid:10) ¯ u , u (cid:11) + 2 ω (cid:10) L − u , u (cid:11)(cid:1) .To calculate the first product, we use that Z π cos θ dθ = 3 π/ , Z π cos θ sin θ dθ = π/ , then (cid:10) ¯ u , u (cid:11) = 4 π (cid:18) a π − a b π b π (cid:19) π 4= 94 (cid:0) a − a b + b (cid:1) = 94 (cid:0) a − b (cid:1) Since (cid:18) ab (cid:19) = 1 √ √ ω (cid:18) √ ω (cid:19) ,then (cid:0) a − b (cid:1) = ω (1 + ω ) and 2 ab = √ ω ω . (25)We conclude from the next proposition thatΩ = 16 ω ( ω + 1) ( ω + 2) √ ω + 1 (cid:0) ω + 29 ω + 33 ω − (cid:1) . Since Ω = 0 at only one point ω > 0, the curve has the property for theintersection with the Cantor set N except for ω . Proposition 21 We have that (cid:10) L − u , u (cid:11) = 124 ( ω + 1) ( ω + 2) (cid:0) ω + 31 ω + 12 ω − (cid:1) . Proof. To calculate h L − u , u i , we use the expression for u given by u = e , , − = ( a − b + cos 2 t + i ab sin 2 t )(1 + cos 2 s ).32rojecting the vector (1 , ab ) in the orthonormal components ( a + , b + ) and( a − , b − ), we have that u = (cid:0) a − b (cid:1) (cid:18) e , , + 1 √ e , , (cid:19) + ( a + + 2 abb + ) (cid:18) √ e , , + 12 e , , (cid:19) + ( a − + 2 abb − ) (cid:18) √ e , , − + 12 e , , − (cid:19) . Thus, we have (cid:10) L − u , u (cid:11) = (cid:0) a − b (cid:1) (cid:18) λ − , , + 12 λ − , , (cid:19) + P , (26)where P is the polynomial P = ( a + + 2 abb + ) (cid:18) λ − , , + 14 λ − , , (cid:19) + ( a − + 2 abb − ) (cid:18) λ − , , − + 14 λ − , , − (cid:19) .For the first term we have (cid:18) λ − , , + 12 λ − , , (cid:19) (cid:0) a − b (cid:1) = 14 ω ( ω + 1) ( ω + 2) (3 ω + 4) .To calculate P , we define the polynomial Q = √ ω + 8 ω + 4,then a ± and b ± are given by (cid:18) a ± b ± (cid:19) = 1 √ p Q ± ωQ (cid:18) ω ± Q − √ ω (cid:19) ,and the eigenvalues are λ , , = 2 ω , λ , , = 2( ω + 2), λ , , ± = ω ± Q and λ , , ± = ω ± Q + 4.Thus, we have12 λ − , , ± + 14 λ − , , ± = 148 12 ω + 1 (cid:0) ω + 3 ω + 4 ± (5 − ω ) Q (cid:1) ,33nd ( a ± + 2 abb ± ) = 12( Q ± ωQ ) (cid:18) ω ± Q − ω ω (cid:19) .After a computations with Maple, and alternatively by hand, we concludethat P = 124 1( ω + 1) (cid:0) ω − ω − (cid:1) . Appendix Periodic solutions of equation (8) are zeros of the map f ( v ) = − i Ω ∂ t v − ∂ ss v + ω (1 − | v | − ) v .The map f is T -equivariant with the action of ( θ, ϕ, ψ ) ∈ T given by ρ ( θ, ϕ, ψ ) v = e iθ v ( t + ϕ, s + ψ ).In addition, the map is equivariant by the actions ρ ( κ ) v ( t, s ) = v ( t, − s ) , ρ (¯ κ ) v ( t, s ) = ¯ v ( − t, s ) . Since ρ ( θ, ϕ, ψ ) ρ ( κ ) = ρ ( κ ) ρ ( θ, ϕ, − ψ ) , ρ ( θ, ϕ, ψ ) ρ (¯ κ ) = ρ (¯ κ ) ρ ( − θ, − ϕ, ψ ),then the map f is equivariant by the action of the non-abelian groupΓ = O (2) × ( T ∪ ¯ κ T ) . The equilibrium v is fixed by the actions of κ , ¯ κ and (0 , ϕ, ψ ) ∈ T , thenthe isotropy group of v isΓ v = O (2) × ( S ∪ ¯ κ S ) . The orbit of v has dimension one, withΓ v = { e iθ : θ ∈ S } . f ′ ( v ) in coordinates u = ( v, ¯ v ) ∈ C is given by L (Ω) u = (cid:18) − i Ω ∂ t − ∂ ss + ω ωω − i Ω ∂ t − ∂ ss + ω (cid:19) u .The linear map L in Fourier components is L (Ω) u = X ( j,k ) ∈ Z M j,k (Ω) u j,k e i ( jt + ks ) ,where the matrix M j,k (Ω) = (cid:18) Ω j + k + ω ωω − Ω j + k + ω (cid:19) has eigenvalues λ j,k, ± (Ω) as before.For j = 1, the eigenvalue λ ,k, − (Ω) is zero atΩ = | k | √ k + 2 ω. Let ( a, b ) be the eigenvector of M ,k (Ω ) corresponding to the eigenvalue λ ,k, − . Thus, the periodic functions v ( z , z ) = z ae i ( ks + t ) + ¯ z be − i ( ks + t ) + z ae i ( − ks + t ) + ¯ z be − i ( − ks + t ) are in the kernel of f ′ ( v ; Ω k ) for ( z , z ) ∈ C .The action of Γ v in the parametrization of the kernel ( z , z ) is given by ρ ( ϕ, ψ )( z , z ) = e iϕ ( e ikψ z , e − ikψ z ) ,ρ ( κ )( z , z ) = ( z , z ) ,ρ (¯ κ )( z , z ) = (¯ z , ¯ z ) . The isotropy groups of the action are inherited by the bifurcating solu-tions. Using the elements κ and ϕ ∈ S , we may assume without loss ofgeneralization that the first coordinate is real and positive, z = a ∈ R ,unless both coordinates are zero.If z = 0, then the isotropy group is generated by(0 , ϕ, − ϕ/k ) ∈ T and κ ¯ κ .This isotropy group Γ ( a, is isomorphic to O (2), and the orbit of ( a, 0) con-tains a 2-torus. 35f z = 0, using the action of (0 , ϕ, − ϕ/k ) ∈ T , which fixes the firstcoordinate and acts by multiplying the second one by e ϕ , we may assumethat z is real.If z = z , then the isotropy group of ( a, a ) is generated by(0 , π, − π/k ), κ and ¯ κ .This isotropy group Γ ( a,a ) is finite, and its orbit contains a 3-torus. In othercases the isotropy group is generated by (0 , π, − π/k ). Traveling waves Functions in the fixed point space of Γ ( a, satisfy v ( t, s ) = ρ (0 , ϕ, − ϕ/k ) v ( t, s ) = v ( t + ϕ, s − ϕ/k ) , thus they are of the form v ( t, s ) = X l ∈ Z v l,lk e i ( lt + lks ) .In this case the non-zero Fourier components are in a line on the lattice.Moreover, from v ( t, s ) = ¯ v ( − t, − s ), we have that v l,lk ∈ R .Since f is Γ × S -equivariant, the map f sends the fixed point space ofΓ ( a, into itself. The restriction of L to this subspace is given by L ( u ) = X l ∈ Z M l,kl u l,lk e i ( lt + slk ) , where u j,k = ( v j,k , ¯ v j,k ). Moreover, for l = 1, if ω is irrational, the kernel atΩ = Ω := | k | √ k + 2 ω is one dimensional. Since λ ,k, − (Ω) changes sign at Ω , by arguments oftopological bifurcation [15], there is a global bifurcation of periodic solutionsnear ( v , Ω ). Theorem 22 The map f ( v ; Ω) has a global bifurcation of periodic travelingwave solutions from (1 , Ω ) . These are solutions for the filament problem ofthe form u j ( t, s ) = a j e iωt v (Ω t + ks ) , (27) where v is a π -periodic solution. 36n [1] there is a proof of existence of traveling waves which are asymp-totic to parallel filaments at infinity. Also, in [14] there is a proof of globalbifurcations of periodic traveling waves for a polygonal relative equilibrium.The proof in this reference uses equivariant degree theory to deal with thesymmetries. Standing waves The group Γ ( a,a ) is generated by the elements (0 , π, − π/k ), κ and ¯ κ . Thus,functions with the isotropy group Γ ( a,a ) satisfy v ( t, s ) = v ( t + π, s − π/k ) = v ( t, − s ) = ¯ v ( − t, s ) . These solutions are the standing waves constructed in this article.Since v = 1 is the only solution of the orbit e iθ v that satisfy the sym-metries (10), then f ′ ( v ) is not degenerate when restricted to the subspaceof functions (10). We have taken advantage of this fact in our proof of theexistence of standing waves.In order to solve the small divisor problem, the frequency ω has beenchosen to be diophantine. In this case there are no resonant Fourier com-ponents, and all branches from different k ’s can be derived from the samebranch k = 1 and the transformation τ − v ( τ t, τ s ) for some τ . Thus, thecase k = 1 that we have analyzed in the paper is the general one. Relative equilibria To find other relative equilibria we may solve equation (4) by the method ofquadratures, as in the Kepler problem. In polar coordinates, v = re iθ , theequation becomes ∂ s θ = c/r where c is the angular momentum, and ∂ s r − c r − + r − − ωr = 0. (28)If c = 0, then θ = θ is constant. In this case, each filament lies in a planethat contain the central axis. Since ω is positive, there is a unique equilibriumat r = ω − / , this corresponds to the 2 π -periodic solution (5) with σ = 0.There are more bounded solutions with r ( s ) → s → ±∞ , but thesesolutions are less relevant for modeling because the filaments approach eachother and therefore exit from the regime of validity of the approximation.There are other relevant solutions that are unbounded.37f c = 0, the dependency of equations on s can be eliminated as in theKepler problem. Changing variables as ρ = r − (see [21]), then ∂ s r = − c∂ θ ρ and ∂ ss r = − c ρ ∂ θθ ρ . Thus, the equation becomes c ∂ θθ ρ + c ρ − ρ − + ωρ − = 0 . (29)These equation may be integrated from c ( ∂ θ ρ ) + V ( ρ ) = E with the poten-tial V ( ρ ) = c ρ − ln ρ − ωρ − .Equation (29) have two equilibria for ω ∈ (0 , / c ) given by ρ ± = 12 c (cid:16) ± √ − c ω (cid:17) .Since ω = − c ρ ± + ρ ± and θ ( s ) = cρ ± s , then the equilibria ρ ± correspond tosolutions u = ρ − ± e i ( ωt + cρ ± s ) , which are the helix solutions (5).Moreover, the equilibrium ρ + is a minimum of V , and there are periodicsolutions close to ρ + . The continuum of periodic solutions near ρ + consistsof helices that are in a bounded annulus. Actually, the projection in theplane of these periodic solutions are asymptotically ellipses. More complexsolutions may be obtained using the Galilean transformation applied to thehelix-like solutions. Acknowledgement. The authors are partially supported by the CanadaResearch Chairs Program, the Fields Institute and McMaster University.Also, C. Garc´ıa-Azpeitia is supported by CONACyT through grant No.203926. W. 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