Standing waves in a counter-rotating vortex filament pair
aa r X i v : . [ m a t h . A P ] J un Standing waves in a counter-rotating vortexfilament pair ∗ Carlos Garc´ıa-Azpeitia † September 24, 2018
Abstract
The distance among two counter-rotating vortex filaments satisfiesa beam-type of equation according to the model derived in [15]. Thisequation has an explicit solution where two straight filaments travelwith constant speed at a constant distance. The boundary conditionof the filaments is 2 π -periodic. Using the distance of the filamentsas bifurcating parameter, an infinite number of branches of periodicstanding waves bifurcate from this initial configuration with constantrational frequency along each branch.MSC: 35B10, 35B32Keywords: Vortex filaments. Periodic solutions. Bifurcation. Introduction
In [15] is derived a model for the movement of almost-parallel vortex filamentsfrom the three-dimensional Euler equation. This model takes in considera-tion the interaction between different filaments and an approximation forthe self-induction of each filament. The paper [15] presents a first analysisof the finite time collapse of two filaments with negative circulations; closeto collapse, the model of vortex filaments as an approximation to the Eulerequation loses validity. Later, [11] proves that two filaments with positive ∗ This is a corrected version of the printed article. † Departamento de Matem´aticas, Facultad de Ciencias, Universidad NacionalAut´onoma de M´exico, 04510 M´exico DF, M´exico u j ( t, s ) , s ) ∈ C × R , j = 1 , , and the distance among the filaments w = u − u satisfies the beam-typeof equation ∂ t w = − ∂ s w + ∂ s (cid:0) | w | − w (cid:1) . (1)This equation has the explicit solution w ( t, s ) = a that corresponds to thesolution of two straight filaments traveling with speed a − at distance a .The aim is to construct 2 π/ν -periodic families of standing wave bifurcatingfrom this initial configuration, where the filaments have 2 π -periodic boundarycondition.The present paper adopts the strategy followed in [12] for the wave equa-tion, where bifurcation of periodic solutions is proven to exist using externalparameters such as the amplitude, while the frequency is a fixed rational. In[13] and [18] this result was improved to obtain global bifurcation of periodicsolutions in spherical domains. A main difference with our result is that theequation is semilinear and requires special estimates. Theorem 1
For each number q , there is an infinite number of non-resonant(Definition 7) amplitudes a ’s given by a − := 2 q − k q (2) for some k ∈ N . For each of these non-resonant a ’s, there is a local contin-uum of πq/p -periodic solution bifurcating from the straight filaments withdistance a , where p = qk − . The local bifurcation consists of standingwaves satisfying the symmetries w ( t, s ) = w ( − t, s ) = w ( t, − s ) = w ( t, s + 2 π/k )= ¯ w ( t + l ( qπ/p ) , s ) , (3)2 here l = 0 , and the estimate w ( t, s ) = a + i l b cos ( pt/q ) cos k s + O C ( b ) , (4) where b ∈ [0 , b ] gives a parameterization of the local bifurcation. The symmetries imply that the standing waves are even in t and evenand 2 π/k -periodic in s . Setting w = x + iy , for l = 0, the symmetry (3)implies that y ( t, s ) = 0, i.e. the orbits of the standing waves are orthogonalto the traveling direction of the filaments. While for l = 1, this symmetryimplies that x ( t, s ) = x ( t + ( qπ/p ) , s ) , y ( t, s ) = − y ( t + ( qπ/p ) , s ) , i.e. the orbits of the standing waves resemble eight figures. a a − a a − Figure 1: Illustration of the two kind of solutions bifurcating from the straightcounter-rotating vortex filaments, initially separated by a and traveling withspeed a − . Left: case l = 0. Rigth: case l = 1.In [7] the existence of standing waves for n vortex filaments of equal vor-ticities from a uniformly rotating central configuration is investigated. In thecase of two filaments, the distance w ( s, t ) satisfies the Schr¨odinger equation ∂ t w = i (cid:0) ∂ ss w + | w | − w (cid:1) , which has the explicit solution ae ia − t that cor-responds to the solution where the two filaments rotate with frequency a − atdistance a . This article proves that the co-rotating filament pair has familiesof standing waves with amplitudes varying over a Cantor set for irrationaldiophantine frequencies a − . In order to solve the small divisor problem thatappears due the fact that the standing waves have irrational frequencies,37] implements a Nash-Moser procedure. This result is different but com-plementary to the existence of standing waves with rational frequencies inthe counter-rotating filament pair. Indeed, the method in [7] can be usedto obtain standing waves with irrational frequencies in the counter-rotatingfilament pair. The method presented here can be used to obtain standingwaves with rational frequencies in the co-rotating filament pair.Nash-Moser methods for wave, Schr¨odinger and beam equations havebeen implemented in [6], [5], [8] and references therein. Different methodswhich do not involve small divisor problems have been developed to proveexistence of periodic solutions. In these methods, the frequency is fixed toa rational or a badly approximated irrational. For rational frequencies, thelinear operator has isolated point spectrum, but the kernel associated to thebifurcation problem may have infinite dimension, see [1], [12] and [17]. Forstrong irrational frequencies, the inverse of the linear operator is bounded,but the inverse lacks compactness, see [4], [5] and [9]. These methods havelimited applicability to semilinear beam equations [12, 4], which is the case ofour problem, and also in Schr¨odinger equations, which is the case of the co-rotating vortex filament pair. We recommend [5] for an overview of differentapplications of these methods to Hamiltonian PDEs.The proof of our theorem relies on the fact that the inverse operatorassociated to the bifurcation problem gains two spatial derivatives whichcompensates the derivatives appearing in the nonlinearity. More precisely,the bifurcation problem is equivalent to solve L ( a ) u + ∂ s g ( u ) = 0 , for a perturbation u , where L is the linearized operator and g ( u ) is an analyticnonlinear operator. In the Fourier basis given by e i ( jt + ks ) , the eigenvalues of L are λ j,k,l ( a ) = ( pj/q ) − k + ( − l a − k ,for ( j, k, l ) ∈ Z × Z . The eigenvalues λ ± , ± k ,l are zero at a and the otherssatisfy λ j,k,l ( a ) ∈ ( qk ) − Z .Thus, the operator L ( a ) can be inverted in the orthogonal complement of thekernel for a neighborhood of a . By choosing a − ∈ (0 , /q ), the projectedinverse ( P LP ) − gains two spatial derivatives due to the sharp estimate λ j,k,l ( a ) & k + | j | . P LP ) − does not gain extra derivatives and ∂ s ( P LP ) − lacks the necessary compactness to establish the global bifurcation by theclassical Rabinowitz approach.The paper is structured as follows. In Section 1, we present the equationthat describes the dynamics of the distance of two straight vortex filaments.In Section 2 the existence of standing waves is obtained by the Lyapunov-Schmidt reduction method. In Section 3 the range equation is solved by thecontracting mapping theorem. In Section 4 the bifurcation equation is solvedusing the symmetries of the problem and the Crandall-Rabinowitz theorem.Existence of traveling waves solutions is discussed in Section 5. The counter-rotating filament pair consists of two filaments with circulationsΓ = 1 and Γ = −
1. According to [15], the equations that describe thedynamics of two almost parallel filaments are ∂ t u = i (cid:18) ∂ s u − u − u | u − u | (cid:19) , ∂ t u = i (cid:18) − ∂ s u + 12 u − u | u − u | (cid:19) .The factor 1 / w = u − u , w = u + u , represent the distance and the center of mass of two filaments, respectively.In these coordinates, the equations are ∂ t w = i∂ s w , ∂ t w = i (cid:0) ∂ s w − | w | − w (cid:1) .Therefore, the distance w satisfies the equation ∂ t w = i∂ s ∂ t w = − ∂ s w + ∂ s ( | w | − w ), (5)and the center of mass w can be obtained from w by integration: w ( t, s ) = i Z t (cid:0) ∂ s w − | w | − w (cid:1) dt + w (0 , s ). (6)5he explicit solution w ( t, s ) = a w ( t, s ) = − ia − t ,corresponds to the solution where the filaments travel with constant speed.We look for bifurcation of solution from this initial configuration of the form w ( t, s ) = a (1 − u ( νt, s )) , where u is 2 π -periodic in t and s .The equation that satisfies the perturbation u is ν ∂ t u = − ∂ s u + 1 a ∂ s (cid:18) − ¯ u (cid:19) .Using a Taylor expansion, this equation is equivalent to ν ∂ t u = − ∂ s u + a − ∂ s ¯ u + ∂ s g (¯ u ), (7)where g (¯ u ) = a − ¯ u − ¯ u = a − ∞ X j =2 ¯ u j (8)is analytic for | u | < Hereafter the frequency ν is fixed to the rational ν = pq , where p and q are relative prime. In order to simplify the analysis of symme-tries, the equation is changed to the real coordinates given by u = ( x, y ) ∈ R .In real coordinates, the equation is given by Lu + ∂ s g ( u ) = 0 , (9)where L is the linear operator Lu := − ν ∂ t u − ∂ s u + a − R∂ s u , (10)6here R = diag (1 , − , and g ( u ) = O (cid:0) | u | (cid:1) is analytic for | ( x, y ) | < L ( T ; R ) given by h u , u i = 1(2 π ) Z T u · u dt ds .Functions u ∈ L ( T ; R ) have the Fourier representation u = X ( j,k ) ∈ Z u j,k e i ( jt + ks ) , u j,k = ¯ u − j, − k ∈ C . The Sobolev space H s is the subspace of functions in L with bounded norm k u k H s = X ( j,k ) ∈ Z | u j,k | (cid:0) j + k + 1 (cid:1) s .This space has the Banach algebra property for s > k uv k H s ≤ k u k H s k v k H s . The Banach algebra property implies that the nonlinear operator g ( u ) = O ( k u k H s ) is well defined and continuous for k u k H s <
1. The Lyapunov-Schmidt reduction is implemented in the Sobolev space of functions withzero average, H s ( T ; R ) = (cid:26) u ∈ H s ( T ; R ) : Z T u = 0 (cid:27) . The linear operator L : D ( L ) → H s is continuous when the domain D ( L ) = { u ∈ H s : Lu ∈ H s } , is completed under the graph norm k u k L = k Lu k H s + k u k H s .
7n Fourier basis, the operator L : D ( L ) → H s is given by Lu = X ( j,k ) ∈ Z (cid:0) ν j I − k I + a − k R (cid:1) u j,k e i ( jt + ks ) , where Z = Z \{ (0 , } .Then, the eigenvalues of L are λ j,k,l = ( νj ) − k + ( − l a − k , (11)for ( j, k, l ) ∈ Z × Z . The set of eigenfunctions of L , given by e l e i ( jt + ks ) with e = (1 ,
0) and e = (0 , Lu = X ( j,k,l ) ∈ Z × Z λ j,k,l (cid:10) u, e l e i ( jt + ks ) (cid:11) e l e i ( jt + ks ) .Choosing a such that a − = ( − l (cid:0) k − ( pj /qk ) (cid:1) (12)for a fixed ( j , k , l ) ∈ N × Z , we have λ ± j , ± k ,l ( a ) = 0; the othereigenvalues satisfy λ j,k,l ( a ) = ( qj/p ) − k + ( − l + l (cid:0) k − ( pj /qk ) (cid:1) k . (13) Definition 2
Let N ⊂ Z × Z be the subset of all lattice points correspondingto zero eigenvalues, N = (cid:8) ( j, k, l ) ∈ Z × Z : λ j,k,l ( a ) = 0 (cid:9) . By definition we have that the kernel of L ( a ) is generated by eigenfunc-tions e l e i ( jt + ks ) with ( j, k, l ) ∈ N . Notice that additional sites to ( ± j , ± k , l )may be present in N due to resonances.The Lyapunov-Schmidt reduction separates the kernel and the rangeequations using the projections Qu = X ( j,k,l ) ∈ N u j,k,l e l e i ( jt + ks ) , P u = ( I − Q ) u. u = v + w, v = Qu, w = P u ,equation (9) is equivalent to the kernel equation
QLQv + Q∂ s g ( v + w ) = 0 , (14)and the range equation P LP w + P ∂ s g ( v + w ) = 0 . (15) Proof of Theorem 1.
The proof is split in three propositions. In Propo-sition 5 we use the contraction mapping theorem to prove that the rangeequation has a unique solution w ( v, a ) ∈ H s defined in a neighborhood of(0 , a ), where w = O ( k v k H s ). Using this solution in the kernel equation weobtain the bifurcation equation QLQv + Q∂ s g ( v + w ( v, a )) = 0, (16)which is defined in a neighborhood of (0 , a ) ∈ ker L ( a ) × R .Proposition 8 proves that for each fixed positive q there is an infinitenumber of non-resonant amplitudes a − ∈ (0 , /q ) with j = 1 and l = 0. InProposition 10, using the symmetries and a non-resonant amplitude a , thebifurcation equation is reduced to a subspace of dimension one within the ker-nel. Then, the existence of the local bifurcation is obtained by the Crandall-Rabinowitz theorem, which gives the estimates v ( t, a ) = be l cos t cos k s + O ( b ) and a = a + O ( b ) for b ∈ [0 , b ].Estimates in Propositions 5 and 10 imply that w ( t, s ) = a + ( v + w )( pt/q, s ) = a + bi l cos( pt/q ) cos k s + O H s ( b ). (17)The regularity of the solutions is obtained by the embedding H s ⊂ C for s ≥
6. The symmetries of u = v + w follow from the symmetries of v inPropositions 10, i.e. u ( t, s ) = u ( − t, s ) = u ( t, − s ) = u ( t, s + 2 π/k )= Ru ( t + l π, s ).Finally, the symmetries of w in the theorem follow from the symmetries of u after rescaling the period. 9 The range equation
In this section, the range equation is solved as a fixed point w ( a, v ) ∈ H s ofthe operator Kw = − ( P LP ) − ∂ s g ( w + v, a ).The key element in the proof consists in showing that( P LP ) − ∂ s : H s → H s is well defined and bounded. Once this result is established, the solutionis obtained by an application of the contraction mapping theorem to thenonlinear operator Kw = O ( ε − k w k H s ) : B ρ ⊂ H s → H s . Lemma 3
Assume that ε < a − / < /q − ε and (cid:12)(cid:12) a − − a − (cid:12)(cid:12) . ε . Then,we have the estimate | λ j,k,l ( a ) | & ε (cid:0) k + | j | (cid:1) for ( j, k, l ) ∈ N c . (18) Proof.
The inequality | pj/q − k | ≥ /q is true unless pj/q = k . In thecase that pj/q = k , then λ j,k,l ( a ) = ± a − k and | λ j,k,l ( a ) | & εk & ε (cid:0) k + | j | (cid:1) . We may assume that j ≥
0, since the case j ≤ | pj/q − k | ≥ /q and j ≥
0, we have | λ j,k,l ( a ) | = | pj/q + µ k | | pj/q − µ k | , where µ k = p k ± a − k . Since lim k →∞ | k − µ k | = a − /
2, then | pj/q − µ k | ≥ (cid:12)(cid:12) pj/q − k (cid:12)(cid:12) − (cid:12)(cid:12) µ k − k (cid:12)(cid:12) ≥ q − (cid:18) a − + ε (cid:19) > ε , for | j | + | k | ≥ M with M big. Therefore, we have the estimate | λ j,k,l ( a ) | ≥ ε | pj/q + µ k | ≥ cε (cid:0) k + | j | (cid:1) We can adjust the constant ε such that the estimate | λ j,k,l ( a ) | ≥ cε (cid:0) k + | j | (cid:1)
10s true for all ( j, k, l ) ∈ N c . Since (cid:12)(cid:12) a − − a − (cid:12)(cid:12) < cε and L ( a ) = L ( a ) ± (cid:0) a − − a − (cid:1) ∂ s ,we conclude that | λ j,k,l ( a ) | ≥ | λ j,k,l ( a ) | − cεk ≥ cε (cid:0) k + | j | (cid:1) . Lemma 4
Assume that ε < a − / < /q − ε and (cid:12)(cid:12) a − − a − (cid:12)(cid:12) . ε . Thelinear operator ( P LP ) − ∂ s is continuous with (cid:13)(cid:13) ( P LP ) − ∂ s w (cid:13)(cid:13) H s . ε − k w k H s . (19) Proof.
By the previous lemma | λ j,k,l ( a ) | & εk for ( j, k, l ) ∈ N c . Then, theestimate k ( P LP ) w k H s & ε (cid:13)(cid:13) ∂ s P w (cid:13)(cid:13) H s holds true with w ∈ H s . Applying this estimate to ( P LP ) − w ∈ D ( L ) ⊂ H s ,we obtain (cid:13)(cid:13) ∂ s ( P LP ) − w (cid:13)(cid:13) H s . ε − k P w k H s .Since H s ⊂ C for s ≥
6, then ∂ s and ( P LP ) − commute. Therefore, theoperator ( P LP ) − ∂ s : P H s → P H s is well define and bounded by O ( ε − ). Proposition 5
Assume a − ∈ (0 , /q ) . There is a unique continuous solu-tion w ( v, a ) ∈ H s of the range equation defined for ( v, a ) in a small neigh-borhood of (0 , a ) ∈ ker L ( a ) × R such that k w ( v, a ) k H s . ε − k v k , (20) for small ε . Proof.
By the Banach algebra property of H s , the operator g ( w ) = O ( k w k H s ) : B ρ → H s
11s well define in the domain B ρ = { w ∈ H s : k w k H s < ρ } for ρ <
1. Since a − ∈ (0 , /q ), we can chose a small enough ε such that the hypothesis of theprevious lemma hold true. Therefore, Kw = − ∂ s ( P LP ) − g ( w + v, a ) = O ( ε − k w k H s ): B ρ ⊂ P H s → P H s ,is well defined and continuous. Moreover, it is a contraction for ρ of order ρ = O ( ε ). By the contraction mapping theorem, there is a unique continuousfixed point w ( v, a ) ∈ B ρ . The estimate k w ( v, a ) k H s ≤ ε − k v k is obtainedfrom k Kw k H s . ε − (cid:16) k w k H s + k v k (cid:17) . Remark 6
Since λ j,k ≥ ε ( k + j ) , the domain D ( L ) is compactly containedin H s . However, we cannot prove the global bifurcation by the classical Rabi-nowitz theorem because ( P LP ) − ∂ s g ( u ) is not compact, but only continuous.This lack of compactness is the reason why we cannot obtain the regularity bybootstrapping arguments. Instead, the regularity of the solutions is obtainedusing the Sobolev embedding H s ⊂ C for s ≥ . In this section, the bifurcation equation is solved by an application of theCrandall-Rabinowitz theorem to the case of non-resonant a ’s with a − ∈ (0 , /q ). Definition 7 An a is non-resonant for the lattice point ( j , k , l ) ∈ N × Z if N ∩ ( j Z × k Z × Z ) = { ( ± j , ± k , l ) } . (21) Proposition 8
For each q , there is an infinite number of non-resonant a ’sfor k ∈ N , j = 1 and l = 0 , given by a − = 2 q − k q , p = qk − . (22)12 roof. First we fix positive numbers p and q . The condition a − = ( − l (cid:0) k − ( pj/qk ) (cid:1) ∈ (0 , /q )holds for the infinite number of lattice points ( j, k, l ) with jp = qk − l = 0. Then a − = k − (cid:18) k − qk (cid:19) = 2 q − k q . By Proposition (5), there is a finite number of elements ( j m , k m , l m ) cor-responding to a non-resonant amplitude a . That is, a − = ( − l m (cid:0) k m − ( pj m /qk m ) (cid:1) for m ∈ { , ..., M } . Therefore, there is an infinite number of a − ∈ (0 , /q )with a finite number of resonances.We say that ( j , k ) is a maximal latticepoint if j m < j or k m < k when j m = j for m = 0. Let ( j , k ) be amaximal lattice point such that a − = ( − l (cid:0) k − ( pj /qk ) (cid:1) , then one has that N ∩ ( j Z × k Z × Z ) = { ( ± j , ± k , l ) } .Therefore, there is an infinite number of non-resonant a ’s with a − ∈ (0 , /q ).The choice of a maximal j is equivalent to choose a maximal p = pj .That is, we have a − = ( − l (cid:0) k − ( p /qk ) (cid:1) for the numbers p and q and N ∩ ( Z × k Z × Z ) = { ( ± , ± k , l ) } . Therefore, for each fixed q , and possibly different numbers p , there is aninfinite number of non-resonant amplitudes a − ∈ (0 , /q ) with j = 1 and l = 0. Remark 9
The choice of maximal p leads to the choice of a minimal period πq/p for the bifurcation. This argument is similar to the argument used in[14] for the wave equation.
13o apply the Crandall-Rabinowitz theorem we need to reduce the bifurca-tion equation to a subspace of dimension one. This is attained by exploitingthe equivariance of the problem. The equation is equivariant under the actionof the group G = Z × O (2) × O (2) given by ρ ( τ, σ ) u ( t, s ) = u ( t + τ, s + σ ) , for the abelian part, and ρ ( κ ) u ( t, s ) = u ( − t, s ) , ρ ( κ ) u ( t, s ) = u ( t, − s ) , ρ ( κ ) u ( t, s ) = Ru ( t, s ) , for the reflections. By the uniqueness of w ( v, a ), the bifurcation equation hasthe same equivariant properties that the differential equation. This propertyis used in the following proposition to reduce the bifurcation equation to asubspace of dimension one. Proposition 10
Let a − ∈ (0 , /q ) be a non-resonant amplitude for the lat-tice point (1 , k , l ) ∈ N × Z . The bifurcation equation has a local continuumof πq/p -periodic solution bifurcating from the initial configuration with am-plitude a . These solutions satisfy the estimates v ( t, s ) = be l cos t cos k s + O ( b ) , a = a + O ( b ) , (23) and symmetries v ( t, s ) = v ( − t, s ) = v ( t, − s ) = v ( t, s + 2 π/k ) = Rv ( t + l π, s ) . (24) Proof.
In the Fourier basis, the action of G is given by ρ ( ϕ ) u j,k = e ijϕ u j,k , ρ ( θ ) u j,k = e ikθ u j,k , for the abelian part and ρ ( κ ) u j,k = u − j,k , ρ ( κ ) u j,k = u j, − k , ρ ( κ ) u j,k = Ru j,k , for the reflections. Setting u j,k = ( u j,k, , u j,k, ), the irreducible representa-tions correspond to the subspaces generated by ( u j,k,l , u j, − k,l ) ∈ C . Indeed,the linear operator L has blocks λ j,k,l I in these irreducible representations,which is predicted by Schur’s lemma.Set the irreducible representation( u , u ) = ( u ,k ,l , u , − k ,l ).14he action of the group in this representation is ρ ( ϕ )( u , u ) = e iϕ ( u , u ) , ρ ( θ )( u , u ) = ( e ik θ u , e − ik θ u ) , and ρ ( κ )( u , u ) = (¯ u , ¯ u ) , ρ ( κ )( u , u ) = ( u , u ) , ρ ( κ )( u , u ) = ( − l ( u , u ) . Therefore, the group S = h κ , κ , ( l π, κ ) , ( π, π/k ) i has fixed point space ( u , u ) = ( b, b ) for b ∈ R in this representation.Set ker L S ( a ) := ker L ( a ) ∩ Fix ( S ).The bifurcation equation QLQw + Q∂ s g ( v + w ( v, a )) : ker L S ( a ) × R → ker L S ( a ) (25)is well defined by the equivariant properties. Since for a non-resonant ampli-tude a , the kernel consist of the subspace ( u , u ) = ( b, b ) for b ∈ R , then thekernel in the fixed point space of S is generated by the simple eigenfunction X ( j,k,l ) ∈ N e l e i ( jt + ks ) = 4 e l cos j t cos k s .Therefore, ker L S ( a ) = { be l cos j t cos k s : b ∈ R } .Since ker L S ( a ) has dimension one, the local bifurcation for a close to a follows from the Crandall-Rabinowitz theorem applied to the bifurcationequation (25). It is only necessary to verify that ∂ a L ( a ) ( e l cos j t cos k s ) isnot in the range of L . This follows from ∂ a L ( a ) ( e l cos j t cos k s ) = − a − k Re l (cos j t cos k s ) ∈ ker L ( a ) . The estimates a = a + O ( b ) and v ( t, s ) = be l cos j t cos k s + O ( b )are consequence of the Crandall-Rabinowitz theorem. Moreover, the S -action of the element ϕ = π/j in the kernel generated by e l cos j t cos k s is given by ρ ( ϕ ) = −
1. This symmetry implies that the bifurcation equationis odd and a = a + O ( b ). 15 Traveling waves
The irreducible representation ( u ,k ,l , u , − k ,l ) has another isotropy groupgiven by T = h κ κ , ( l π, κ ) , ( ϕ, − ϕ/k ) i . This isotropy group has a one dimensional fixed point space correspondingto ( u , u ) = ( b,
0) for b ∈ R . Solutions with isotropy group T are travelingwaves of the form u ( νt + s ) for k = 1.For these traveling waves, the PDE becomes the ODE − u ′′ − ν u + a − Ru + g ( u ) = 0. (26)The spectrum of the linear operator associated to the bifurcation problem is λ j = j − ν + ( − l a − .Actually, the global bifurcation of traveling waves for filaments has beenproven in [10] applying equivariant degree theory to the reduced ODE. In asimilar manner, one can prove the following theorem. Theorem 11
The equation (1) has a global bifurcation of traveling wavesstarting from the initial configuration u = a with frequency ν = p − l a − ∈ R + .The local bifurcation can be parameterized by b with the estimate ν ( b ) = ν + O ( b ) and u ( νt + s ) = a + bi l cos ( νt + s ) + O C ( b ) . Observe that the set of traveling waves forms a two-dimensional familyparameterized by amplitude a and frequency ν , while standing waves existfor an infinite number of local and continuous curves that are parameterizedby amplitude a and have fixed rational frequency ν . Acknowledgement.
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