Standing waves of fixed period for n+1 vortex filaments
SSTANDING WAVES OF FIXED PERIOD FOR n + 1 VORTEX FILAMENTS
WALTER CRAIG AND CARLOS GARC´IA-AZPEITIA
Abstract.
The n +1 vortex filament problem has explicit solutions con-sisting of n parallel filaments of equal circulation in the form of nestedpolygons uniformly rotating around a central filament which has circula-tion of opposite sign. We show that when the relation between temporaland spatial periods is fixed at certain rational numbers, these configura-tions have an infinite number of homographic time dependent standingwave patterns that bifurcate from these uniformly rotating central con-figurations. Introduction
In reference [16], a model system of equations was derived for the interac-tion of near-parallel vortex filaments. The model considers vortex filamentsin R to be coordinatized by curves ( u j ( t, s ) , s ) ∈ C × R for j = 0 , . . . , n thatdescribe the positions of n + 1 vertically oriented vortex filaments. Differentaspects of this problem have been investigated in [3, 4, 8, 11, 12, 13, 17] andreferences therein. In this article we study central configurations of n + 1vortex filaments with n filaments of equal circulation and one filament ofopposite circulation.Let u j ( t, s ) for j = 1 , ..., n be the positions of the n filaments of circulation1 and u ( t, s ) the filament of circulation − κ with κ >
0. A homographicstanding wave of the n + 1 vortex filament problem with fixed period is asolution of the form(1) u j ( t, s ) = ae iωt ( a j + a j u ( t/q, s )) , where ω = − a − is real, q is an integer and u ( t, s ) is a complex 2 π -periodicfunction in t and s .The complex numbers a j ∈ C for j = 0 , ..., n lie in a central configurationwith a = 0. That is, the complex numbers a j satisfy(2) 0 = n (cid:88) i =1 a i | a i | , − a j = n (cid:88) i =1( i (cid:54) = j ) a j − a i | a j − a i | − κ a j | a j | , Mathematics Subject Classification.
Key words and phrases.
Vortex filaments. Periodic solutions. Bifurcation.Walter Craig is deceased. a r X i v : . [ m a t h . A P ] M a r WALTER CRAIG AND CARLOS GARC´IA-AZPEITIA for j = 1 , ..., n . There are many configurations that satisfy (2), for examplein the form of nested polygons. In particular, an explicit solution of (2) isgiven by the regular polygon a j = ( κ − ( n − / − / e ijζ , ζ = 2 π/n, if κ > ( n − / u = 0 in equation (1) corresponds to the family of homographicsolutions for which n straight parallel filaments rotate around the centralfilament with uniform frequency ω and amplitude a . The standing waves ofthe title of this article correspond to non-trivial 2 π -periodic solutions u ( t, s )of the equation Lu + g ( u ) = 0, where L is the linear operator(3) L ( ω ) u := − ( i/q ) ∂ t u − ∂ s u + ω ( u + ¯ u ) ,and g is an analytic nonlinearity describing the horizontal vortex filamentinteraction. Our goal is to construct standing wave solutions that bifurcatefrom the initial configuration u = 0, for which the frequency ω is the bi-furcation parameter. The solution given by (1) with a 2 π -periodic function u has fixed spatial period s ∈ [0 , π ) and temporal period t ∈ [0 , πq ) ina frame of reference that is rotating with frequency ω , i.e. the solution isperiodic or quasiperiodic with the two temporal frequencies ω and 1 /q whenobserved in a stationary reference frame. The main theorem is as follows. Theorem 1.
Let q be an integer. For each k ∈ N , there is a local continuumof π -periodic solutions u bifurcating from the unperturbed configuration with u = 0 and initial frequency (4) ω = − q (cid:18) − k q (cid:19) .The local bifurcation ( u, ω ) consists of standing waves satisfying the esti-mates u ( t, s ) = b (cid:2) cos j t + i (cid:0) − k − /q (cid:1) sin j t (cid:3) cos k s + O ( b ) , with ω = ω + O ( b ) and j = qk − , where b ∈ [0 , b ] gives a local param-eterization of the bifurcation curve. Furthermore, these solutions satisfy thefollowing symmetries u ( t, s ) = ¯ u ( − t, s ) = u ( t, − s ) . Therefore, for any central configuration a j satisfying (2), the previoustheorem gives homographic solutions of the form (1). The periodic solutions u are special in that the ratio of their temporal and spatial periods arerational. In reference [8] we studied the case of irrational ratios, which isa small divisor problem for a nonlinear partial differential equation whichrequires techniques related to KAM theory even for the case of constructionsof periodic solutions. Our approach is parallel to that of the semilinear waveand beam equation in one dimension, where time periodic solutions withrational periods (free vibrations) were shown to exist in [1, 2, 14, 15, 18], TANDING WAVES OF FIXED PERIOD FOR n + 1 VORTEX FILAMENTS 3 Figure 1.
Illustration of the standing waves obtained inTheorem 1 for the case of n = 3 vortex filaments of equalcirculation (blue) and one vortex filament of opposite circu-lation (red).and later for irrational periods in [5, 10]. On the other hand, time periodicsolutions bifurcating from stationary solutions with irrational periods is asmall divisor problem, for which constructions of solutions by Nash-Mosermethods came much later in [6, 7, 9], and references therein.In the present analysis the ratio of the periodic solution is rational andthe small divisor problem does not occur. The key element of the proofconsists on the fact that for special temporal frequencies, given by 1 /q , theSchr¨odinger operator L ( ω ), when restricted to the orthogonal complement ofthe null space, has a bounded inverse in the set of frequencies ω ∈ ( − /q, L ( ω ) which doesnot have the regularity that is usually obtained in other equations, i.e. ourresult can be obtained only in a narrow set of parameters where L ( ω ) has anontrivial kernel. This is also the case of the counter-rotating vortex filamentpair studied in [11], but this is the first time that periodic solutions withoutsmall divisors are obtained in a genuine non-linear Hamiltonian PDE usingthis method.In section 1, we set up a Lyapunov-Schmidt reduction to prove the exis-tence of standing waves. In section 2 we solve the range equation for ω ∈ ( − /q,
0) using the contracting mapping theorem. In section 3 we use thesymmetries of the problem to solve the bifurcation equation by means of theCrandall-Rabinowitz theorem.
WALTER CRAIG AND CARLOS GARC´IA-AZPEITIA Setting the problem
From [16] the system of model equations for the dynamics of n + 1 near-parallel vortex filaments, with circulations Γ = − κ and Γ j = 1 for j =1 , ..., n , is given by(5) ∂ t u j = i Γ j ∂ ss u j + n (cid:88) i =0 ( i (cid:54) = j ) Γ j u j − u i | u j − u i | , j = 0 , ..., n. Homographic solutions of the n + 1 filaments are particular solutions of theform u j ( t, s ) = w ( t, s ) a j , where w ( t, s ) is a complex valued function and where a j ’s are complex num-bers satisfying the condition of a central configuration. In this class ofsolutions the shape of the intersections of the filaments with a horizontalcomplex plane is homographic with the shape of their intersection with anyother horizontal plane { x = c } for any c and at any time t .For a general central configuration(6) − a j = n (cid:88) i =0( i (cid:54) = j ) Γ i a j − a i | a j − a i | , j = 0 , ..., n, homographic solutions satisfy the system of equations (5) if w ( t, s ) solvesthe system of equations a j ∂ t w ( t, s ) = i (cid:18) Γ j a j ∂ ss w ( t, s ) − w ( t, s ) | w ( t, s ) | a j (cid:19) , j = 0 , ..., n, .In the particular case that a = 0 in the central configuration, the conditionfor the configuration a j becomes (2) and the system of equations is satisfiedby solutions of the simple equation,(7) ∂ t w = i (cid:18) ∂ ss w − w | w | (cid:19) .Therefore, u j ( t, s ) = w ( t, s ) a j is an homographic solution of the vortexfilament problem if the configuration a j satisfies (2) and w is a solution ofthe equation (7)A particular solution of (2) is given by a regular polygon a j = re ijζ withradius r = ( κ − ( n − / − / if κ > ( n − /
2, because n (cid:88) i =1( i (cid:54) = j ) a j − a i | a j − a i | − κ a j | a j | = − (cid:18) κ − n − (cid:19) a j r = − a j . Also, there are other solutions of (2) corresponding to nested polygons.Equation (7) has the set of solution w = ae iωt with ω = − a − < , TANDING WAVES OF FIXED PERIOD FOR n + 1 VORTEX FILAMENTS 5 that corresponds to n vortex filaments uniformly rotating in the centralconfiguration a j with amplitude a and frequency ω . We look for bifurcationof solutions of the equation (7) of the form(8) w ( t, s ) = ae iωt (1 + u ( t/q, s )) , where q is an integer and u ( t, s ) is 2 π -periodic in t and s . This is a solutionthat has fixed temporal and spatial periodicity when viewed in a coordinateframe rotating about the x -axis with frequency ω . When u = 0 the so-lution corresponds to n vortex filaments uniformly rotating in the centralconfiguration a j . The equation (7) for a perturbation from this configurationis(9) ( i/q ) ∂ t u = − u ss + ω ( u + ¯ u ) + g ( u, u ) ,where the nonlinearity g is given by g ( u, ¯ u ) = ω ¯ u u = ω ∞ (cid:88) n =2 ( − n ¯ u n . In order to simplify the analysis of symmetries, the equation is representedin real coordinates u ( τ, s ) = ( x ( τ, s ) , y ( τ, s )) ∈ R , i.e., the equation isequivalent to Lu + g ( u ) = 0 , where g ( u ) = O (cid:16) | u | (cid:17) is analytic for | ( x, y ) | < L is the linear operator(10) Lu := − (1 /q ) J ∂ t u − ∂ s u + ω ( I + R ) u ,where R = diag (1 , − L ( T ; R ), with the inner product (cid:104) u , u (cid:105) = 1(2 π ) (cid:90) T u · u dtds .A function u ∈ L can be written in a Fourier basis as u = (cid:88) ( j,k ) ∈ Z u j,k e i ( jt + ks ) , u j,k = ¯ u − j, − k ∈ C . The Sobolev space H s is the usual subspace of functions in L with boundednorm (cid:107) u (cid:107) H s = (cid:88) ( j,k ) ∈ Z | u j,k | (cid:0) j + k + 1 (cid:1) s .This space has the Banach algebra property for s > (cid:107) uv (cid:107) H s ≤ C (cid:107) u (cid:107) H s (cid:107) v (cid:107) H s . The Banach algebra property implies that the nonlinear operator g ( u ) = O ( (cid:107) u (cid:107) H s ) is well defined and continuous for (cid:107) u (cid:107) H s < L : D ( L ) → H s is continuous when the domain D ( L ) = { u ∈ H s : Lu ∈ H s } , WALTER CRAIG AND CARLOS GARC´IA-AZPEITIA is completed under the graph norm (cid:107) u (cid:107) L = (cid:107) Lu (cid:107) H s + (cid:107) u (cid:107) H s . In Fourier basis, the operator L : D ( L ) → H s is given by Lu = (cid:88) ( j,k ) ∈ Z M j,k u j,k e i ( jt + ks ) , where M j,k = (cid:18) k + 2 ω i ( j/q ) − i ( j/q ) k (cid:19) .Then, the eigenvalues and eigenvectors of L are λ j,k,l = k + ω + l (cid:113) ( j/q ) + ω , (11) e j,k,l = (cid:32) − ω − l (cid:113) ( j/q ) + ω i ( j/q ) (cid:33) , (12)for ( j, k, l ) ∈ Z × Z , where Z = { , − } is a group under the product.The eigenvalue λ j,k, always is positive, and λ j,k, − ( ω ) = 0 if ω = (cid:16) ( j/qk ) − k (cid:17) / < . Given that L ( ω ) has a nontrivial kernel, we expect bifurcation of solutionsof L ( ω ) u + g ( u ) = 0 as ω crosses ω . Definition 2.
We define N as the subset of all lattice points correspondingto zero eigenvalues, N ( ω ) = (cid:8) ( j, k, − ∈ Z × Z : λ j,k, − ( ω ) = 0 (cid:9) . By definition we have that the kernel of L ( ω ) is generated by eigen-functions e j,k,l e i ( jt + ks ) with ( j, k, l ) ∈ N . Notice that additional sites to( ± j , ± k , −
1) may be present in N ( ω ) due to resonances. The Lyapunov-Schmidt reduction separates the kernel and the range equations using theprojections Qu = (cid:88) ( j,k,l ) ∈ N u j,k,l e j,k,l e i ( jt + ks ) , P u = ( I − Q ) u . Setting u = v + w , v = Qu , w = P u ,the equation Lu + g ( u ) = 0 is equivalent to the kernel equation(13) QLQv + Qg ( v + w ) = 0 , and the range equation(14) P LP w + P g ( v + w ) = 0 . TANDING WAVES OF FIXED PERIOD FOR n + 1 VORTEX FILAMENTS 7 The range equation
In this section, the range equation is solved as a fixed point w ( ω, v ) ∈ H s of the operator Kw = − ( P LP ) − g ( w + v, ω ) .The local solution w = w ( ω, v ) is provided by an application of the con-traction mapping theorem, where we only need to prove that ( P LP ) − : P H s → P H s is well defined and bounded. For this, we will establish boundestimates in the eigenvalues λ j,k,l .For l = 1, we clearly have λ j,k, = k + ω + (cid:113) ( j/q ) + ω (cid:38) k + | j | .For l = −
1, we have the following estimate,
Lemma 3.
For ε < | ω | < /q − ε , we have (15) | λ j,k, − ( ω ) | (cid:38) ε for ( j, k, l ) ∈ N c .Proof. In the case | j | /q (cid:54) = k , the inequality (cid:12)(cid:12) k − | j | /q (cid:12)(cid:12) ≥ /q holds and | λ j,k, − ( ω ) | ≥ (cid:12)(cid:12) k − | j | /q (cid:12)(cid:12) − (cid:12)(cid:12)(cid:12)(cid:12) | j | /q + ω − (cid:113) ( j/q ) + ω (cid:12)(cid:12)(cid:12)(cid:12) . Since lim x →∞ (cid:16) x + ω − √ x + ω (cid:17) = ω , then (cid:12)(cid:12)(cid:12)(cid:12) | j | /q + ω − (cid:113) ( j/q ) + ω (cid:12)(cid:12)(cid:12)(cid:12) < | ω | + ε , for | k | + | j | ≥ M with M big enough. Therefore, | λ j,k, − ( ω ) | ≥ (cid:12)(cid:12) k − | j | /q (cid:12)(cid:12) − | ω | − ε ≥ q − | ω | − ε ≥ ε . In the case | j | /q = k , then | λ j,k, − ( ω ) | = (cid:12)(cid:12)(cid:12) k + ω − (cid:112) k + ω (cid:12)(cid:12)(cid:12) ≥ | ω | − ε ≥ ε . for | k | big enough. In both cases we have that | λ j,k, − ( ω ) | ≥ ε if | k | + | j | ≥ M with M big enough. We conclude that the estimate holds except by a finitenumber of points ( j, k ) ∈ Z . Therefore, there is a constant c such that theestimate | λ j,k, − ( ω ) | ≥ cε holds for all ( j, k, − ∈ N c . (cid:3) From the previous estimates we have that (
P LP ) − is a bounded operatorwith (cid:13)(cid:13)(cid:13) ( P LP ) − w (cid:13)(cid:13)(cid:13) H s (cid:46) ε − (cid:107) w (cid:107) H s . Proposition 4.
Assume ε < | ω | < /q − ε . There is a unique continuoussolution w ( v, ω ) ∈ H s of the range equation defined for ( v, ω ) in a smallneighborhood of (0 , ω ) ∈ ker L ( a ) × R such that (16) (cid:107) w ( v, ω ) (cid:107) H s (cid:46) ε − (cid:107) v (cid:107) ,for small ε . WALTER CRAIG AND CARLOS GARC´IA-AZPEITIA
Proof.
By the Banach algebra property of H s , the operator g ( w ) = O ( (cid:107) w (cid:107) H s ) : B ρ → H s is well define in the domain B ρ = { w ∈ H s : (cid:107) w (cid:107) H s < ρ } for ρ <
1. We canchose a small enough ε such that the hypothesis of the previous lemma holdtrue. Therefore, Kw = − ( P LP ) − g ( w + v, ω ) = O ( ε − (cid:107) w (cid:107) H s ) K : 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, ω ) ∈ B ρ . The estimate (cid:107) w ( v, ω ) (cid:107) H s ≤ ε − (cid:107) v (cid:107) is obtainedfrom (cid:107) Kw (cid:107) H s (cid:46) ε − (cid:16) (cid:107) w (cid:107) H s + (cid:107) v (cid:107) (cid:17) . (cid:3) Remark 5.
Since ( P LP ) − is continuous but not compact, we do not auto-matically obtain the regularity of the solutions by bootstrapping arguments.Instead,the regularity is obtained using the Sobolev embedding H s ⊂ C for s ≥ . The bifurcation equation
Proposition 6.
For k ∈ N , we define (17) ω = − q (cid:18) − qk (cid:19) , j = qk − . For these frequencies we have ω ∈ ( − /q, and N ( ω ) = { (0 , , − , ( ± j , ± k , − } . Proof.
Since λ j,k, − = k + ω − (cid:113) ( j/q ) + ω , then λ j, , − ( ω ) = 0 only if j = 0. For k ∈ N + , the condition λ j ,k , − ( ω ) = 0 is satisfied only if ω = (cid:16) ( j /qk ) − k (cid:17) / ω ∈ ( − /q,
0) holds if an only if the lattice point( j , k ) ∈ N satisfies j = qk −
1. In this case ω = 12 (cid:32)(cid:18) k − qk (cid:19) − k (cid:33) = − q (cid:18) − qk (cid:19) , then the frequency ω is determined uniquely for each point ( j , k ) ∈ N because ω is decreasing in k . Therefore, we have that (0 , , −
1) and( ± j , ± k , −
1) are the only elements in N ( ω ). (cid:3) TANDING WAVES OF FIXED PERIOD FOR n + 1 VORTEX FILAMENTS 9 Since ker L ( ω ) has dimension 5 for ω ∈ ( − /q, G = O (2) × O (2) givenby ρ ( τ, σ ) u ( t, s ) = u ( t + τ, s + σ ) , for the abelian components, and for the reflections, ρ ( κ ) u ( t, s ) = Ru ( − t, s ) , ρ ( κ ) u ( t, s ) = u ( t, − s ) , where R = diag (1 , − w ( v, ω ), the bifurcation equa-tion has the same equivariant properties as the differential equation. Thisproperty is used in the following proposition to reduce the bifurcation equa-tion to a subspace of dimension one. Proposition 7.
The bifurcation equation has a local continuum of π -periodic solution bifurcating from ( v, ω ) = (0 , ω ) with estimates (18) v ( t, s ) = b (cid:18) cos j t (cid:0) − k − /q (cid:1) sin j t (cid:19) cos k s + O ( b ) , ω = ω + O ( b ) , where b ∈ [0 , b ] gives a parameterization of the local bifurcation, and sym-metries (19) v ( t, s ) = Rv ( − t, s ) = v ( t, − s ) = v ( t + π/j , s + π/k ) . Proof.
In Fourier components v = (cid:88) ( j,k,l ) ∈ N u j,k,l e j,k,l e i ( jt + ks ) , u j,k,l = ¯ u − j, − k,l , the action of the abelian part of the group G is given by ρ ( ϕ ) u j,k,l = e ijϕ u j,k,l , ρ ( θ ) u j,k,l = e ikθ u j,k,l . Since e j,k, − = (cid:32) − ω − (cid:113) ( j/q ) + ω i ( j/q ) (cid:33) = (cid:18) k i ( j/q ) (cid:19) , then Re j,k, − = e − j,k, − and e j,k, − = e j, − k, − . Therefore, we have ρ ( κ ) v = (cid:88) ( j,k,l ) ∈ N u j,k,l e − j,k, − e i ( − jt + ks ) = (cid:88) ( j,k,l ) ∈ N u − j,k,l e j,k, − e i ( jt + ks ) , and ρ ( κ ) v = (cid:88) ( j,k,l ) ∈ N u j,k,l e j,k, − e i ( jt − ks ) = (cid:88) ( j,k,l ) ∈ N u j, − k,l e j,k, − e i ( jt + ks ) .Therefore, the action of the reflections in Fourier components is given by ρ ( κ ) u j,k,l = u − j,k,l = ¯ u j, − k,l , ρ ( κ ) u j,k,l = u j, − k,l . The irreducible representations under the action of O (2) × O (2) corre-sponds to the subspaces ( u j ,k , − , u j , − k , − ) ∈ C . The linear operator L is diagonal in these irreducible representations witheigenvalue λ j ,k , − of complex multiplicity two. The group S = (cid:104) κ , κ , ( π/j , π/k ) (cid:105) has fixed point space ( u j,k, − , u j, − k, − ) = ( b, b ) for b ∈ R in this representa-tion. By setting ker L S ( ω ) := ker L ( ω ) ∩ Fix ( S ) ,the bifurcation equation(20) QLQw + Qg ( v + w ( v, ω )) : ker L S ( ω ) × R → ker L S ( ω )is well defined by the equivariance properties. Moreover, since u , , − isnot fixed by the subgroup S , then ker L S ( ω ) is generated by the simpleeigenfunction (cid:88) j = ± j ,k = ± k e j,k, − e i ( jt + ks ) = 4 (cid:18) k cos j tj /q sin j t (cid:19) cos k s .Since ker L S ( ω ) has dimension one, the local bifurcation for ω close to ω follows from the Crandall-Rabinowitz theorem applied to the bifurcationequation (20). It is only necessary to verify that ∂ ω L ( ω ) f is not in the rangeof L for f ∈ ker L S ( ω ), which follows from the fact that ∂ ω L ( ω ) f = ( I + R ) f . The estimates ω = ω + O ( b ) and v ( t, s ) = b (cid:18) cos j t (cid:0) − k − /q (cid:1) sin j t (cid:19) cos k s + O ( b )are consequence of the Crandall-Rabinowitz estimates. Moreover, the S -action of the element ϕ = π in the kernel generated is given by ρ ( ϕ ) = − ω = ω + O ( b ). (cid:3) The main theorem follows from this proposition and the fact that u = v + w ( v, ω ) with (cid:107) w ( v, ω ) (cid:107) H s = O (cid:16) (cid:107) v (cid:107) (cid:17) = O (cid:0) b (cid:1) . Acknowledgements.
W.C. was partially supported by the CanadaResearch Chairs Program and NSERC through grant number 238452–16.C.G.A was partially supported by a UNAM-PAPIIT project IN115019. Weacknowledge the assistance of Ramiro Chavez Tovar with the preparation ofthe figure.
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