Stationary states of the cubic conformal flow on S 3
SSTATIONARY STATES OF THE CUBIC CONFORMAL FLOW ON S PIOTR BIZO ´N, DOMINIKA HUNIK-KOSTYRA, AND DMITRY PELINOVSKY
Abstract.
We consider the resonant system of amplitude equations for the conformallyinvariant cubic wave equation on the three-sphere. Using the local bifurcation theory, wecharacterize all stationary states that bifurcate from the first two eigenmodes. Thanksto the variational formulation of the resonant system and energy conservation, we alsodetermine variational characterization and stability of the bifurcating states. For the lowesteigenmode, we obtain two orbitally stable families of the bifurcating stationary states: oneis a constrained maximizer of energy and the other one is a constrained minimizer of theenergy, where the constraints are due to other conserved quantities of the resonant system.For the second eigenmode, we obtain two constrained minimizers of the energy, which arealso orbitally stable in the time evolution. All other bifurcating states are saddle points ofenergy under these constraints and their stability in the time evolution is unknown.
Contents
1. Introduction 12. Preliminaries 42.1. Single-mode states 52.2. Invariant subspace of stationary states 52.3. Other stationary states 73. Bifurcations from the lowest eigenmode 84. Variational characterization of the bifurcating states 135. Bifurcation from the second eigenmode 256. Variational characterization of the bifurcating states 297. Numerical approximations 33References 351.
Introduction
The conformally invariant cubic wave equation on S (the unit three-dimensional sphere)is a toy model for studying the dynamics of resonant interactions between nonlinear waveson a compact manifold. The long-time behavior of small solutions of this equation is wellapproximated by solutions of an infinite dimensional time-averaged Hamiltonian system, Date : July 3, 2018.This research was supported by the Polish National Science Centre grant no. 2017/26/A/ST2/00530. a r X i v : . [ m a t h - ph ] J u l TATIONARY STATES IN THE CONFORMAL FLOW 2 called the cubic conformal flow , that was introduced and studied in [2]. In terms of complexamplitudes { α n ( t ) } n ∈ N , this system takes the form i ( n + 1) dα n dt = ∞ (cid:88) j =0 n + j (cid:88) k =0 S njk,n + j − k ¯ α j α k α n + j − k , (1.1)where S njk,n + j − k = min( n, j, k, n + j − k ) + 1 are the interaction coefficients. The cubicconformal flow (1.1) is the Hamiltonian system with the symplectic form (cid:80) ∞ n =0 i ( n +1) d ¯ α n ∧ ( − dα n ) and the conserved energy function H ( α ) = ∞ (cid:88) n =0 ∞ (cid:88) j =0 n + j (cid:88) k =0 S njk,n + j − k ¯ α n ¯ α j α k α n + j − k . (1.2)The attention of [2] has been focused on understanding the patterns of energy transferbetween the modes. In particular, a three-dimensional invariant manifold was found onwhich the dynamics is Liouville-integrable with exactly periodic energy flows. Of specialinterest are the stationary states for which no transfer of energy occurs. A wealth of explicitstationary states have been found in [2], however a complete classification of stationary stateswas deemed as an open problem.The purpose of this paper is to study existence and stability of all stationary states thatbifurcate from the first two eigenmodes of the cubic conformal flow. Thanks to the Hamil-tonian formulation, we are able to give the variational characterization of the bifurcatingfamilies. Among all the families, we identify several particular stationary states which areorbitally stable in the time evolution: one is a constrained maximizer of energy and the otherones are local constrained minimizers of the energy, where the constraints are induced byother conserved quantities of the cubic conformal flow.The constrained maximizer of energy can be normalized to the form α n ( t ) = (1 − p ) p n e − it , (1.3)where p ∈ (0 ,
1) is a parameter. This solution was labeled as the ground state in our previouswork [4], where we proved orbital stability of the ground state in spite of its degeneracy withrespect to parameter p ∈ (0 , α n ( t ) = c (cid:20) (1 − p ) n −
12 (1 + 5 p ± (cid:112) − p + p ) (cid:21) p n e − iλt + inωt , (1.4)where p ∈ (0 , − √
3) and c ∈ (0 , ∞ ) are parameters, whereas ( λ, ω ) are expressed by λ = c (cid:32) − p − p ± (3 + 4 p ) (cid:112) − p + p (1 − p ) (cid:33) , ω = c
12 1 + p ± (cid:112) − p + p − p . (1.5)The cutoff in the interval (0 , −√
3) for p ensures that 1 − p + p >
0. Since the constrainedminimizers of energy are nondegenerate with respect to ( λ, ω ), their orbital stability followsfrom the general stability theory [12].
TATIONARY STATES IN THE CONFORMAL FLOW 3
By using the local bifurcation methods of this paper, we are able to prove that the sta-tionary states (1.3) and (1.4) are respectively maximizer and two minimizers of energy con-strained by two other conserved quantities in the limit of small p . For the ground state (1.3),we know from our previous work [4] that it remains a global constrained maximizer of energyfor any p ∈ (0 , p ∈ (0 , − √
3) and c > existence of another constrained minimizer of energy bifurcat-ing from the second eigenmode . The new minimizer is nondegenerate with respect to ( λ, ω ),hence again its orbital stability follows from the general stability theory in [12]. However, weshow numerically that this stationary state remains a local constrained minimizer only nearthe bifurcation point and becomes a saddle point of energy far from the bifurcation point.Bifurcation analysis of this paper for the first two eigenmodes suggests existence of otherconstrained minimizers of energy bifurcating from other eigenmodes. This poses an openproblem of characterizing a global constrained minimizer of energy for the cubic conformalflow (1.1). Another open problem is to understand orbital stability of the saddle pointsof energy, in particular, to investigate if other conserved quantities might contribute tostabilization of saddle points of energy.The cubic conformal flow (1.1) shares many properties with a cubic resonant systemfor the Gross-Pitaevskii equation in two dimensions [1, 11]. In particular, the stationarystates for the lowest Landau level invariant subspace of the cubic resonant system have beenthoroughly studied in [7] by using the bifurcation theory from a simple eigenvalue [6]. Incomparison with Section 6.4 in [7], where certain symmetries were imposed to reduce multi-plicity of eigenvalues, we develop normal form theory for bifurcations of all distinct familiesof stationary states from the double eigenvalue without imposing any a priori symmetries.Another case of a completely integrable resonant system with a wealth of stationary statesis the cubic Szeg˝o equation [8, 9]. Classification of stationary states and their stability hasbeen performed for the cubic and quadratic Szeg˝o equations in [15] and [16] respectively.One more example of a complete classification of all travelling waves of finite energy for anon-integrable case of the energy-critical half-wave map onto S is given in [13], where thespectrum of linearization at the travelling waves is studied by using Jacobi operators andconformal transformations. The half-wave map was found to be another integrable systemwith the Lax pair formulation [10]. It is unclear in the present time if the cubic conformalflow on S is also an integrable system with the Lax pair formulation. Organization of the paper.
Symmetry, conserved quantities, and some particular sta-tionary states for the cubic conformal flow (1.1) are reviewed in Section 2 based on theprevious works [2, 4]. Local bifurcation results including the normal form computations forthe lowest eigenmode are contained in Section 3. Variational characterization of the bifur-cating families from the lowest eigenmode including the proof of extremal properties for thestationary states (1.3) and (1.4) is given in Section 4. Similar bifurcation results and varia-tional characterization of the bifurcating states from the second eigenmode are obtained in
TATIONARY STATES IN THE CONFORMAL FLOW 4
Sections 5 and 6 respectively. Numerical results confirming local minimizing properties ofthe stationary states (1.4) for all admissible values of p are reported in Section 7. Notations.
We denote the set of nonnegative integers by N and the set of positive integersby N + . A sequence ( α n ) n ∈ N is denoted for short by α . The space of square-summablesequences on N is denoted by (cid:96) ( N ). It is equipped with the inner product (cid:104)· , ·(cid:105) and theinduced norm (cid:107) · (cid:107) . The weighted space (cid:96) , ( N ) denotes the space of squared integrablesequences with the weight (1 + n ) . We write X (cid:46) Y to state X ≤ CY for some universal(i.e., independent of other parameters) constant C >
0. Terms of the Taylor series in ( (cid:15), µ )of the order O ( (cid:15) p , µ p ) are denoted by O ( p ).2. Preliminaries
Here we recall from [2, 4] some relevant properties of the cubic conformal flow (1.1) and itsstationary states. The cubic conformal flow (1.1) enjoys the following three one-parametergroups of symmetries: Scaling: α n ( t ) → cα n ( c t ) , (2.1)Global phase shift: α n ( t ) → e iθ α n ( t ) , (2.2)Local phase shift: α n ( t ) → e inφ α n ( t ) , (2.3)where c , θ , and φ are real parameters. By the Noether theorem, the latter two symmetriesgive rise to two conserved quantities: Q ( α ) = ∞ (cid:80) n =0 ( n + 1) | α n | , (2.4) E ( α ) = ∞ (cid:80) n =0 ( n + 1) | α n | . (2.5)It is proven in Theorem 1.2 of [4] that H ( α ) ≤ Q ( α ) and the equality is achieved if andonly if α = cp n for some c, p ∈ C with | p | < Z ( α ) = ∞ (cid:88) n =0 ( n + 1)( n + 2) ¯ α n +1 α n . (2.6)This quantity is related to another one-parameter group of symmetries: α n ( t ) → (cid:2) e sD α ( t ) (cid:3) n := ∞ (cid:88) k =0 s k k ! (cid:2) D k α ( t ) (cid:3) n , (2.7)where s ∈ R is arbitrary and D is a difference operator given by[ Dα ] n := nα n − − ( n + 2) α n +1 . (2.8) TATIONARY STATES IN THE CONFORMAL FLOW 5
Note that the difference operator D is obtained from Df = { i ( Z − ¯ Z ) , f } acting on anyfunction f ( α, ¯ α ) on phase space, where the Poisson bracket is defined by { f, g } := ∞ (cid:88) k =0 i ( k + 1) (cid:18) ∂f∂ ¯ α k ∂g∂α k − ∂f∂α k ∂g∂ ¯ α k (cid:19) , thanks to the symplectic structure (cid:80) ∞ n =0 i ( n + 1) d ¯ α n ∧ ( − dα n ) of the conformal flow (1.1).The factor (2 i ) in the definition of D is chosen for convenience. There exists another one-parameter group of symmetries obtained from ˜ Df = { Z + ¯ Z ) , f } , which is not going to beused in this paper.Stationary states of the cubic conformal flow (1.1) are obtained by the separation ofvariables α n ( t ) = A n e − iλt + inωt , (2.9)where the complex amplitudes A n are time-independent, while parameters λ and ω arereal. Substituting (2.9) into (1.1), we get a nonlinear system of algebraic equations for theamplitudes: ( n + 1)( λ − nω ) A n = ∞ (cid:88) j =0 n + j (cid:88) k =0 S njk,n + j − k ¯ A j A k A n + j − k . (2.10)Some particular solutions { A n } n ∈ N of the stationary system (2.10) are reviewed below.2.1. Single-mode states.
The simplest solutions of (2.10) are the single-mode states givenby A n = c δ nN , (2.11)with λ − N ω = | c | , where c ∈ C is an amplitude and N ∈ N is fixed. Thanks to thetransformations (2.1) and (2.2), we can set c = 1 so that λ − N ω = 1.2.2.
Invariant subspace of stationary states.
As is shown in [2], system (2.10) can bereduced to three nonlinear equations with the substitution A n = ( β + γn ) p n , (2.12)where p, β, γ are complex parameters satisfying the nonlinear algebraic system − ωp (1 + y ) = p (cid:0) y | γ | + ¯ βγ (cid:1) , (2.13) λγ (1 + y ) = γ (cid:0) | β | + (18 y + 4 y ) | γ | + (6 y −
1) ¯ βγ + 10 y ¯ γβ (cid:1) , (2.14) λβ (1 + y ) = β (cid:0) | β | + (6 y + 2 y ) | γ | + 2 yβ ¯ γ (cid:1) + γ (cid:0) y | β | + (4 y + 2) y | γ | + y ¯ βγ (cid:1) , (2.15)and we have introduced y := | p | / (1 − | p | ) so that 1 / (1 + y ) = 1 − | p | . It is clear from thedecay of the sequence (2.12) as n → ∞ that p must be restricted to the unit disk: | p | < p , β , γ are real-valued, which can be done without loss of generality. To see this, notethat p can be made real-valued by the transformation (2.3). If β = 0, then equation (2.15) TATIONARY STATES IN THE CONFORMAL FLOW 6 implies γ = 0 and hence no nontrivial solutions exist. If β (cid:54) = 0, it can be made real bythe transformation (2.2). If β is real, then γ is real, because equation (2.13) implies thatIm( ¯ βγ ) = 0. Thus, it suffices to consider system (2.13)–(2.15) with p ∈ (0 ,
1) and β, γ ∈ R .There exist exactly four families of solutions to the system (2.13)–(2.15).2.2.1. Ground state.
Equation (2.14) is satisfied if γ = 0. This implies that ω = 0 fromequation (2.13) and λ (1 − p ) = β from equation (2.15) with p ∈ (0 ,
1) and β ∈ R .Parameterizing β = c and bringing all together yield the geometric sequence A n = c p n , λ = c (1 − p ) . (2.16)As p →
0, this solution tends to the N = 0 single-mode state (2.11). Thanks to thetransformation (2.1), one can set c = 1 − p , then λ = 1 yields the normalized state (1.3).As is explained above, the geometric sequence (2.16) is a maximizer of H ( α ) for fixed Q ( α ), therefore, we call it the ground state . Nonlinear stability of the ground state has beenproven in [4], where the degeneracy due to the parameter p has been controlled with the useof conserved quantities E ( α ) and Z ( α ).2.2.2. Twisted state. If γ (cid:54) = 0 but ω = 0, equation (2.13) is satisfied with γ = − β/ (2 y ).Then, equation (2.14) yields λ = β (1 + y ) / (4 y ), whereas equation (2.15) is identicallysatisfied with p ∈ (0 ,
1) and β ∈ R . Parameterizing β = − cp and bringing all together yieldthe twisted state A n = cp n − ((1 − p ) n − p ) , λ = c (1 − p ) . (2.17)As p →
0, this solution tends to the N = 1 single-mode state (2.11). Thanks to transforma-tion (2.1), one can set c = 1 − p , then λ = 1.2.2.3. Pair of stationary states. If γ (cid:54) = 0 and ω (cid:54) = 0, then λ can be eliminated from system(2.14) and (2.15), which results in the algebraic equation(2 γy + β )(12 γ y + 6 βγy + 6 γ y + β + βγ ) = 0 . (2.18)The first factor corresponds to the twisted state (2.17). Computing γ from the quadraticequation in the second factor yield two roots: γ = − (1 − p ) β p (1 + p ) (cid:104) p ∓ (cid:112) − p + p (cid:105) . (2.19)The real roots exist for 1 − p + p ≥ p ≤ − √
3, or equivalently, for p ∈ [0 , − √ γ ± = c (1 − p ) , β ± = − c (1 + 5 p ± (cid:112) − p + p ) , (2.20) TATIONARY STATES IN THE CONFORMAL FLOW 7 where c ∈ R is arbitrary. Using equations (2.13) and (2.14) again, we get ω ± = c
12 1 + p ± (cid:112) − p + p − p , λ ± = c (cid:32) − p − p ± (3 + 4 p ) (cid:112) − p + p (1 − p ) (cid:33) . (2.21)As p →
0, the branch with the upper sign converges to the N = 0 single-mode state p → A n → − c δ n , λ + → c , ω + → c , (2.22)while the branch with the lower sign converges to the N = 1 single-mode state p → A n → ˜ cδ n , λ − →
53 ˜ c , ω − →
23 ˜ c , (2.23)where ˜ c = cp = O (1) has been rescaled as p →
0. The solutions (2.12), (2.20), and (2.21)have been cast as the stationary states (1.4) and (1.5). By using our bifurcation analysis,we will prove that these stationary states are local minimizers of H ( α ) for fixed Q ( α ) and E ( α ) for small positive p . We also show numerically that this conclusion remains true forevery p ∈ (0 , − √ Other stationary states.
Other stationary states were constructed in [2] from thegenerating function u ( t, z ) given by the power series expansion: u ( t, z ) = ∞ (cid:88) n =0 α n ( t ) z n . (2.24)The conformal flow (1.1) can be rewritten for u ( t, z ) as the integro-differential equation i∂ t ∂ z ( zu ) = 12 πi (cid:73) | ζ | =1 dζζ u ( t, ζ ) (cid:18) ζu ( t, ζ ) − zu ( t, z ) ζ − z (cid:19) . (2.25)The stationary solutions expressed by (2.9) yield the generating function in the form u ( t, z ) = U (cid:0) ze iωt (cid:1) e − iλt , (2.26)where U is a function of one variable given by U ( z ) = (cid:80) ∞ n =0 A n z n . The family of stationarysolutions expressed by (2.12) is generated by the function U ( z ) = β − pz + γpz (1 − pz ) . (2.27)Additionally, any finite Blaschke product U ( z ) = N (cid:89) k =1 z − ¯ p k − p k z (2.28) TATIONARY STATES IN THE CONFORMAL FLOW 8 yields a stationary state with λ = 1 and ω = 0 for p , . . . , p N ∈ C . If N = 1 and p = p ∈ (0 , A = − p, A n ≥ = p − (1 − p ) p n , λ = 1 , ω = 0 . (2.29)This solution can be viewed as a continuation of the N = 1 single-mode state (2.11) in p (cid:54) = 0.Another family of stationary states is generated by the function U ( z ) = cz N − p N +1 z N +1 , λ = c (1 − p N +2 ) , ω = 0 , (2.30)where N ∈ N , p ∈ (0 , c ∈ R is arbitrary. When N = 0, the function (2.30) generatesthe geometric sequence (2.16). When N = 1, the function (2.30) generates another stationarysolution A n = c (cid:26) p n − , n odd , , n even , λ = c (1 − p ) , ω = 0 , (2.31)which is a continuation of the N = 1 single-mode state (2.11) in p (cid:54) = 0. Thanks to transfor-mation (2.1), one can set c = 1 − p , then λ = 1. The new solutions (2.29) and (2.31) appearin the local bifurcation analysis from the N = 1 single-mode state (2.11).3. Bifurcations from the lowest eigenmode
We restrict our attention to the real-valued solutions of the stationary equations (2.10),which satisfy the system of algebraic equations( n + 1)( λ − nω ) A n = ∞ (cid:88) j =0 n + j (cid:88) k =0 S njk,n + j − k A j A k A n + j − k . (3.1)Here we study bifurcations of stationary states from the lowest eigenmode given by (2.11)with N = 0. Without loss of generality, the scaling transformation (2.1) yields c = 1 and λ = 1. By setting A n = δ n + a n with real-valued perturbation a , we rewrite the system (3.1)with λ = 1 in the perturbative form L ( ω ) a + N ( a ) = 0 , (3.2)where L ( ω ) is a diagonal operator with entries[ L ( ω )] nn = (cid:26) , n = 0 ,n ( n + 1) ω − n + 1 , n ≥ , (3.3)and N ( a ) includes quadratic and cubic nonlinear terms:[ N ( a )] n = 2 ∞ (cid:88) j =0 a j a n + j + n (cid:88) k =0 a k a n − k + ∞ (cid:88) j =0 n + j (cid:88) k =0 S njk,n + j − k a j a k a n + j − k . (3.4)We have the following result on the nonlinear terms. TATIONARY STATES IN THE CONFORMAL FLOW 9
Lemma 1.
Fix an integer m ≥ . If a m(cid:96) +1 = a m(cid:96) +2 = · · · = a m(cid:96) + m − = 0 , for every (cid:96) ∈ N , (3.5) then [ N ( a )] m(cid:96) +1 = [ N ( a )] m(cid:96) +2 = · · · = [ N ( a )] m(cid:96) + m − = 0 , for every (cid:96) ∈ N . (3.6) Proof.
Let us inspect [ N ( a )] n in (3.4) for n = m(cid:96) + ı with (cid:96) ∈ N and ı ∈ { , , . . . , m − } .Since a j (cid:54) = 0 only if j is multiple of m , then a n + j = 0 in the first term of (3.4) for all n whichare not a multiple of m . Similarly, since a k (cid:54) = 0 only if k is multiple of m , then a n − k = 0 inthe second term of (3.4). Finally, since a j (cid:54) = 0 and a k (cid:54) = 0 only if j and k are multiple of m ,then a n + j − k = 0 in the third term of (3.4). Hence, all terms of (3.4) are identically zero for n = m(cid:96) + ı with (cid:96) ∈ N and ı ∈ { , , . . . , m − } . (cid:3) Bifurcations from the lowest eigenmode are identified by zero eigenvalues of the diagonaloperator L ( ω ). Lemma 2.
There exists a sequence of bifurcations at ω ∈ { ω m } m ∈ N + with ω m = m − m ( m + 1) , m ∈ N + , (3.7) where all bifurcation points are simple except for ω = ω = 1 / .Proof. The sequence of values (3.7) yield zero diagonal entries in (3.3). To study if thebifurcation points are simple, we consider solutions of ω n = ω m for n (cid:54) = m . This equationis equivalent to mn = m + n + 1, which has only two solutions ( m, n ) ∈ { (2 , , } .Therefore, all bifurcation points are simple except for the double point ω = ω = 1 / (cid:3) The standard Crandall–Rabinowitz theory [6] can be applied to study bifurcation fromthe simple zero eigenvalue.
Theorem 1.
Fix m = 1 or an integer m ≥ . There exists a unique branch of solutions ( ω, A ) ∈ R × (cid:96) ( N ) to system (3.1) with λ = 1 , which can be parameterized by small (cid:15) suchthat ( ω, A ) is smooth in (cid:15) and | ω − ω m | + sup n ∈ N | A n − δ n − (cid:15)δ nm | (cid:46) (cid:15) . (3.8) Proof.
Let us consider the decomposition: ω = ω m + Ω , a n = (cid:15)δ nm + b n , n ∈ N , (3.9)where ω m is defined by (3.7), (cid:15) is arbitrary and b m = 0 is set from the orthogonality condition (cid:104) b, e m (cid:105) = 0. Let L ∗ = L ( ω m ). The system (3.2) is decomposed into the invertible part[ F ( (cid:15), Ω; b )] n := [ L ∗ ] nn b n + n ( n + 1)Ω b n + [ N ( (cid:15)e m + b )] n = 0 , n (cid:54) = m, (3.10)and the bifurcation equation G ( (cid:15), Ω; b ) := m ( m + 1)Ω + (cid:15) − [ N ( (cid:15)e m + b )] m = 0 . (3.11) TATIONARY STATES IN THE CONFORMAL FLOW 10
Since [ L ∗ ] nn = (cid:26) , n = 0 , ( n − m )( nm − m − n − m ( m +1) , n ≥ , (3.12)we have [ L ∗ ] nn (cid:54) = 0 for every n (cid:54) = m and [ L ∗ ] nn → ∞ as n → ∞ . Therefore, the ImplicitFunction Theorem can be applied to F ( (cid:15), Ω; b ) : R × (cid:96) , ( N ) → (cid:96) ( N ) , where as is defined by (3.10), F is smooth in its variables, F (0 , Ω; 0) = 0 for every Ω ∈ R , ∂ b F (0 ,
0; 0) = L ∗ , and (cid:107) F ( (cid:15), Ω; 0) (cid:107) (cid:96) (cid:46) (cid:15) . For every small (cid:15) and small Ω, there exists aunique small solution of system (3.10) in (cid:96) ( N ) such that (cid:107) b (cid:107) (cid:96) (cid:46) (cid:15) (1 + | Ω | ). Denote thissolution by b ( (cid:15), Ω).Since [ N ( (cid:15)e m )] m = (cid:15) S mmmm , (3.13)one power of (cid:15) is canceled in (3.11) and the Implicit Function Theorem can be applied to G ( (cid:15), Ω; b ( (cid:15), Ω)) : R × R → R , where as is defined by (3.11), G is smooth in its variables, G (0 , b (0 , ∂ Ω G (0 , b (0 , m ( m + 1) (cid:54) = 0 . For every small (cid:15) , there is a unique small root Ω of the bifurcation equation (3.11) such that | Ω | (cid:46) (cid:15) thanks to (3.13). (cid:3) Remark . The unique branch bifurcating from ω = 0 coincides with the normalized groundstate (1.3) and yields the exact result Ω = 0. The small parameter (cid:15) is defined in terms ofthe small parameter p by (cid:15) := p (1 − p ). Remark . For the unique branch bifurcating from ω m with m ≥
4, we claim that A n (cid:54) = 0, n ∈ N if and only if n is a multiple of m . Indeed, by Lemma 1, the system (3.2) canbe reduced for a new sequence { a m(cid:96) } (cid:96) ∈ N whereas all other elements are identically zero.By Lemma 2, the value ω = ω m is a simple bifurcation point of this reduced system. ByTheorem 1, there exists a unique branch of solutions of this reduced system with the bounds(3.8). Therefore, by uniqueness, the bifurcating solution satisfies the reduction of Lemma 1,that is, A n (cid:54) = 0, n ∈ N if and only if n is a multiple of m . Remark . There are at least three branches bifurcating from the double point ω = ω .One branch satisfies (3.5) with (cid:96) = 2, the other branch satisfies (3.5) with (cid:96) = 3, and thethird branch is given by the explicit solution (1.4) with (1.5) for the upper sign. We showin Theorem 2 that these are the only branches bifurcating from the double point.As is well-known [5], normal form equations have to be computed in order to studybranches bifurcating from the double zero eigenvalue at the bifurcation point ω ∗ := ω = ω = 1 / TATIONARY STATES IN THE CONFORMAL FLOW 11
Theorem 2.
Fix ω ∗ := ω = ω = 1 / . There exist exactly three branches of solutions ( ω, A ) ∈ R × (cid:96) ( N ) to system (3.1) with λ = 1 , which can be parameterized by small ( (cid:15), µ ) such that ( ω, A ) is smooth in ( (cid:15), µ ) and | ω − ω ∗ | + sup n ∈ N | A n − δ n − (cid:15)δ n − µδ n | (cid:46) ( (cid:15) + µ ) . (3.14) The three branches are characterized by the following: (i) (cid:15) = 0 , µ (cid:54) = 0 ; (ii) (cid:15) (cid:54) = 0 , µ = 0 ; (iii) (cid:15) < , | µ − | (cid:15) | / | (cid:46) (cid:15) ,and the branch (iii) is double degenerate up to the reflection µ (cid:55)→ − µ .Proof. Let us consider the decomposition: ω = ω ∗ + Ω , a n = (cid:15)δ n + µδ n + b n , n ∈ N , (3.15)where ω ∗ := ω = ω = 1 /
6, ( (cid:15), µ ) are arbitrary, and b = b = 0 are set from the orthog-onality condition (cid:104) b, e (cid:105) = (cid:104) b, e (cid:105) = 0. The system (3.2) is decomposed into the invertiblepart (cid:26) b + [ N ( (cid:15)e + µe + b )] = 0 , ( n − n − b n + n ( n + 1)Ω b n + [ N ( (cid:15)e + µe + b )] n = 0 , n (cid:54) = { , , } , (3.16)and the bifurcation equations (cid:26) (cid:15) + [ N ( (cid:15)e + µe + b )] = 0 , µ + [ N ( (cid:15)e + µe + b )] = 0 . (3.17)By the Implicit Function Theorem (see the proof of Theorem 1), there exists a unique map R (cid:51) ( (cid:15), µ, Ω) (cid:55)→ b ∈ (cid:96) ( N ) for small ( (cid:15), µ, Ω) such that equations (3.16) are satisfied and (cid:107) b (cid:107) (cid:96) (cid:46) ( (cid:15) + µ )(1 + | Ω | ). Denote this solution by b ( (cid:15), µ, Ω).Compared to the proof of Theorem 1, it is now more difficult to consider solutions ofthe system of two algebraic equations (3.17). In order to resolve the degeneracy of theseequations, we have to compute the solution b ( (cid:15), µ, Ω) up to the cubic terms in ( (cid:15), µ ) underthe apriori assumption | Ω | (cid:46) ( (cid:15) + µ ). Substituting the expansion for b ( (cid:15), µ, Ω) into thesystem (3.17) and expanding it up to the quartic terms in ( (cid:15), µ ), we will be able to confirmthe apriori assumption | Ω | (cid:46) ( (cid:15) + µ ) and to obtain all solutions of the system (3.17) for( (cid:15), µ, Ω) near (0 , , | Ω | (cid:46) ( (cid:15) + µ ), then b n = O (3) for every n ≥
7, where O (3) denotesterms of the cubic and higher order in ( (cid:15), µ ). Thanks to the cubic smallness of b n for n ≥ N ( (cid:15)e + µe + b )] n for n ∈ { , , , , } up to and including the cubic order: [ N ( (cid:15)e + µe + b )] = 2( (cid:15) + µ ) + O (4) , [ N ( (cid:15)e + µe + b )] = 2 (cid:15)µ + 2 (cid:15)b + 2 µb + 2 (cid:15) µ + O (4) , [ N ( (cid:15)e + µe + b )] = (cid:15) + 2 (cid:15)b + 2 µb + 3 (cid:15)µ + O (4) , [ N ( (cid:15)e + µe + b )] = 2 (cid:15)µ + O (4) , [ N ( (cid:15)e + µe + b )] = µ + 2 (cid:15)b + O (4) . TATIONARY STATES IN THE CONFORMAL FLOW 12
By using the system (3.16) and the quadratic approximations for ( b , b , b ), we obtain( b , b , b , b , b ) up to and including the cubic order: b = − (cid:15) − µ + O (4) ,b = − (cid:15)µ + 48 (cid:15) µ + O (4) ,b = − (cid:15) + 30 (cid:15)µ + O (4) ,b = − (cid:15)µ + O (4) ,b = − µ + 3 (cid:15) + O (4) . Next, we compute [ N ( (cid:15)e + µe + b )] n for n ∈ { , } up to and including the quartic order: [ N ( (cid:15)e + µe + b )] = 4 (cid:15)b + 2 (cid:15)b + 2 µb + 2 µb + b + 2 b b + 4 (cid:15)µb + 3 (cid:15) + 6 (cid:15)µ + 3 µ b + O (5) , [ N ( (cid:15)e + µe + b )] = 2 (cid:15)b + 2 (cid:15)b + 4 µb + 2 µb + 2 b b + 2 (cid:15) b + 6 (cid:15) µ + 6 (cid:15)µb + 4 µ + O (5) . By using the quadratic and cubic approximations for ( b , b , b , b , b ), we rewrite the system(3.17) up to and including the quartic order: (cid:26) (cid:15) = (cid:15) (7 (cid:15) + 14 µ − (cid:15)µ ) + O (5) , µ = µ (14 (cid:15) + µ − (cid:15) ) + O (5) . (3.18)By Lemma 1, if (cid:15) = 0, then [ N ( µe + b (0 , µ, Ω))] = 0, whereas if µ = 0, then [ N ( (cid:15)e + b ( (cid:15), , Ω))] =0. Hence, the system (3.18) can be rewritten in the equivalent form: (cid:26) (cid:15) = (cid:15) (7 (cid:15) + 14 µ − (cid:15)µ + O (4)) , µ = µ (14 (cid:15) + µ − (cid:15) + O (4)) . (3.19)We are looking for solutions to the system (3.19) with ( (cid:15), µ ) (cid:54) = (0 , b (0 , , Ω) = 0follows from the system (3.16). There exist three nontrivial solutions to the system (3.18):(I) (cid:15) = 0, µ (cid:54) = 0, and Ω = µ + O (4);(II) (cid:15) (cid:54) = 0, µ = 0, and Ω = (cid:15) + O (4);(III) (cid:15) (cid:54) = 0, µ (cid:54) = 0, and (cid:26)
6Ω = 7 (cid:15) + 14 µ − (cid:15)µ + O (4) ,
12Ω = 14 (cid:15) + µ − (cid:15) + O (4) . (3.20)All the solutions satisfy the apriori assumption | Ω | (cid:46) ( (cid:15) + µ ). In order to consider persis-tence of these solutions in the system (3.19), we compute the Jacobian matrix: J ( (cid:15), µ, Ω) = (cid:20) −
6Ω + 21 (cid:15) + 14 µ − (cid:15)µ (cid:15)µ − (cid:15) µ (cid:15)µ − (cid:15) µ −
12Ω + 14 (cid:15) + 3 µ − (cid:15) (cid:21) + O (4) . We proceed in each case as follows:(I) The Jacobian is invertible for small ( (cid:15), µ ) admitting the expansion in (I). By theImplicit Function Theorem, there exists a unique continuation of this root in thesystem (3.18). By Lemma 1 with m = 3, the bifurcating solution corresponds to thereduction with a j +1 = a j +2 = 0 for every j ∈ N , hence (cid:15) = 0 persists beyond allorders of the expansion. This yields the solution (i). TATIONARY STATES IN THE CONFORMAL FLOW 13 (II) The Jacobian is invertible for small ( (cid:15), µ ) admitting the expansion in (II). By theImplicit Function Theorem, there exists a unique continuation of this root in thesystem (3.18). By Lemma 1 with m = 2, the bifurcating solution corresponds to thereduction with a j +1 = 0 for every j ∈ N , hence µ = 0 persists beyond all orders ofthe expansion. This yields the solution (ii).(III) Eliminating Ω from the system (3.20) yields the root finding problem:27 µ − (cid:15)µ + 108 (cid:15) + O (4) = 0 , which has only two solutions for ( (cid:15), µ ) near (0 ,
0) from the two roots of the quadraticequation µ + 4 (cid:15) + O (4) = 0. Computingdet( J ( (cid:15), µ, Ω)) = 3024 (cid:15) + O (6) (cid:54) = 0verifies that J ( (cid:15), µ, Ω) is invertible at each of the two roots. By the Implicit FunctionTheorem, there exists a unique continuation of each of the two roots in the system(3.18). For the root with µ >
0, this yields the solution (iii). Thanks to the symmetry(2.3) with φ = π , every solution with µ > µ < a k (cid:55)→ a k and a k +1 (cid:55)→ − a k +1 for k ∈ N . Hencethe two solutions with µ > µ < µ (cid:55)→ − µ .No other branches bifurcate from the point ω ∗ = 1 / (cid:3) Remark . The branch (iii) bifurcating from ω ∗ = 1 / λ = 1 and taking the limit p → c = − − p + O ( p ), β = 1 + O ( p ), and γ = − O ( p ).The parameter (cid:15) and µ are related to each other by means of the small parameter p withthe definitions (cid:15) := − p + O ( p ) and µ := − p + O ( p ), hence µ + 4 (cid:15) = O ( (cid:15) ). Remark . Branches (i) and (ii) bifurcating from ω ∗ = 1 / (cid:96) ( N ) by theconstraints (3.5) with (cid:96) = 2 and (cid:96) = 3 respectively. The zero eigenvalue is simple on theconstrained subspace of (cid:96) ( N ), which enables application of the theory, as it was done in [7]in a similar context.4. Variational characterization of the bifurcating states
Stationary states (2.9) with parameters λ and ω are critical points of the functional K ( α ) = 12 H ( α ) − λQ ( α ) − ω [ Q ( α ) − E ( α )] , (4.1)where H , Q , and E are given by (1.2), (2.4), and (2.5). Let α = A + a + ib , where A is areal root of the algebraic system (2.10), whereas a and b are real and imaginary parts of theperturbation. Because the stationary solution A is a critical point of K , the first variation of K vanishes at α = A and the second variation of K at α = A can be written as a quadratic TATIONARY STATES IN THE CONFORMAL FLOW 14 form associated with the Hessian operator. In variables above, we obtain the quadratic formin the diagonalized form: K ( A + a + ib ) − K ( A ) = (cid:104) L + a, a (cid:105) + (cid:104) L − b, b (cid:105) + O ( (cid:107) a (cid:107) + (cid:107) b (cid:107) ) , (4.2)where (cid:104)· , ·(cid:105) is the inner product in (cid:96) ( N ) and (cid:107) · (cid:107) is the induced norm. After straightforwardcomputations, we obtain the explicit form for the self-adjoint operators L ± : D ( L ± ) → (cid:96) ( N ),where D ( L ± ) ⊂ (cid:96) ( N ) is the maximal domain and L ± are unbounded operators given by( L ± a ) n = ∞ (cid:88) j =0 n + j (cid:88) k =0 S njk,n + j − k [2 A j A n + j − k a k ± A k A n + j − k a j ] − ( n + 1)( λ − nω ) a n . (4.3)The following lemma gives variational characterization of the lowest eigenmode at the bifur-cation points { ω m } m ∈ N + in Lemma 2. Lemma 3.
The following is true: • For ω = 0 , the N = 0 single-mode state (2.11) is a degenerate saddle point of K with one positive eigenvalue, zero eigenvalue of multiplicity three, and infinitely manynegative eigenvalues bounded away from zero. • For ω = ω = 1 / , the N = 0 single-mode state (2.11) is a degenerate minimizerof K with zero eigenvalue of multiplicity five and infinitely many positive eigenvaluesbounded away from zero. • For ω m with m ≥ , the N = 0 single-mode state (2.11) is a degenerate saddle pointof K with m − negative eigenvalues, zero eigenvalue of multiplicity three, andinfinitely many positive eigenvalues bounded away from zero.Proof. By the scaling transformation (2.1), we take λ = 1 and A n = δ n , for which theexplicit form (4.3) yields( L ± a ) n = (1 − n ) a n ± a δ n + n ( n + 1) ωa n , n ∈ N . (4.4)We note that L + = L ( ω ) given by (3.3) and L − is only different from L + at the first diagonalentry at n = 0 (which is 0 instead of 2). Because L ± in (4.4) are diagonal, the assertion ofthe lemma is proven from explicit computations: • If ω = 0, then σ ( L + ) = { , , − , − , . . . } and σ ( L − ) = { , , − , − , . . . } , whichyields the result. • If ω ∗ := ω = ω = 1 /
6, then[ L ( ω ∗ )] nn = 16 ( n − n − , n ≥ , which yields the result. • If ω m = ( m − / ( m ( m + 1)) with m ≥
4, then[ L ( ω m )] nn = 1 m ( m + 1) ( n − m )( mn − m − n − , n ≥ . For n = 1 and every n > m , [ L ( ω m )] nn >
0. For 2 ≤ n ≤ m −
1, [ L ( ω m )] nn <
0. For n = m , [ L ( ω m )] mm = 0. This yields the result. TATIONARY STATES IN THE CONFORMAL FLOW 15 (cid:3)
Remark . It is shown in Lemma 6.2 of [4] that the normalized ground state (1.3) bifurcatingfrom ω has the same variational characterization as the N = 0 single-mode state, hence itis a triple-degenerate constrained maximizer of H subject to fixed Q . The triple degeneracyof the family (2.16) is due to two gauge symmetries (2.2) and (2.3), as well as the presenceof the additional parameter p . As is explained in [3], the latter degeneracy is due to theadditional symmetry (2.7). Indeed, by applying e sD with s ∈ R to α n ( t ) = δ n e − it and usingthe general transformation law derived in [3], we obtain another solution in the form: α n ( t ) = (tanh s ) n cosh s e − it , which coincides with the ground state (1.3) after the definition p := tanh s . Remark . All branches bifurcating from ω m for m ≥ e sD with s ∈ R to the N = 0 single-mode state α n ( t ) = δ n e − it , because the latter state isindependent of the values of ω . Remark . All branches bifurcating from ω m for m ≥ H subject to fixed Q and E .It remains to study the three branches bifurcating from ω ∗ = ω = ω = 1 /
6. By Lemma3, there is a chance that some of these three branches represent constrained minimizers of H subject to fixed Q and E . One needs to consider how the zero eigenvalue of multiplicity fivesplits when ω (cid:54) = ω ∗ with | ω − ω ∗ | sufficiently small and how the eigenvalues change underthe two constraints of fixed Q and E . The following lemma presents the count of negativeeigenvalues of the operators L ± denoted as n ( L ± ) at the three branches bifurcating from ω ∗ . Lemma 4.
Consider the three bifurcating branches in Theorem 2 for ω ∗ := ω = ω = 1 / .For every ω (cid:54) = ω ∗ with | ω − ω ∗ | sufficiently small, the following is true: (i) n ( L + ) = 2 , n ( L − ) = 1 ; (ii) n ( L + ) = 1 , n ( L − ) = 1 ; (iii) n ( L + ) = 1 , n ( L − ) = 0 .For each branch, L − has a double zero eigenvalue, L + has no zero eigenvalue, and the restof the spectrum of L + and L − is strictly positive and is bounded away from zero.Proof. Substituting λ = 1 and ω = ω ∗ + Ω into (4.3) yields( L ± a ) n = ∞ (cid:88) j =0 n + j (cid:88) k =0 S njk,n + j − k [2 A j A n + j − k a k ± A k A n + j − k a j ] − a n + 16 ( n − n − a n + n ( n + 1)Ω a n , (4.5)where A n = δ n + (cid:15)δ n + µδ n + b n with (cid:104) b, e (cid:105) = (cid:104) b, e (cid:105) = 0 follows from the decomposition(3.15). The correction terms (Ω , b ) are uniquely defined by parameters ( (cid:15), µ ). In whatfollows, we consider the three branches in Theorem 2 separately. TATIONARY STATES IN THE CONFORMAL FLOW 16
Case (i): (cid:15) = 0 , µ (cid:54) = 0 . Here we have Ω = µ + O ( µ ), b = − µ + O ( µ ), b = b = b = 0, b = − µ + O ( µ ), and b n = O ( µ ) for n ≥
7. Substituting these expansions in(4.5), we obtain L ± = L (0) ± + µL (1) ± + µ L (2) ± + O ( µ ) , where ( L (0) ± a ) n = 16 ( n − n − a n ± a − n , ( L (1) ± a ) n = 2( a n +3 + a n − ) ± a − n , ( L (2) ± a ) n = 112 n ( n + 1) a n + 2(min(3 , n ) − a n − ( a n +6 + a n − ) ∓ a − n ± min(3 , n, − n ) a − n . Let us represent the first 7-by-7 matrix block of the operator L ± and truncate it by up toand including O ( µ ) terms. The corresponding matrix blocks denoted by ˆ L + and ˆ L − aregiven respectively byˆ L + = − µ µ − µ + µ µ µ µ
00 2 µ µ µ µ µ µ µ µ µ + µ µ µ µ − µ µ µ and ˆ L − = − µ + µ − µ µ − µ − µ µ − µ µ
00 0 0 2 µ µ µ − µ + µ − µ µ µ − µ µ µ Zero eigenvalue of L (0)+ is double and is associated with the subspace spanned by { e , e } .There are two invariant subspaces of ˆ L + , one is spanned by { e , e , e } and the other oneis spanned by { e , e , e , e } . This makes perturbative analysis easier. For the subspacespanned by { e , e , e } , the eigenvalue problem is given by (2 − µ ) x + 4 µx − µ x = λx , µx + 8 µ x + 2 µx = λx , − µ x + 2 µx + (2 + µ ) x = λx , The small eigenvalue for small µ is obtained by normalization x = 1. Then, we obtain x = − µ + O ( µ ) , x = − µ + O ( µ ) , TATIONARY STATES IN THE CONFORMAL FLOW 17 and λ = − µ + O ( µ ) . (4.6)For the subspace spanned by { e , e , e , e } , the eigenvalue problem is given by ( + µ ) x + 2 µx + 2 µx + µ x = λx , µx + µ x + 2 µ x + 2 µx = λx , µx + 2 µ x + ( + µ ) x = λx ,µ x + 2 µx + (1 + µ ) x = λx , The small eigenvalue for small µ is obtained by normalization x = 1. Then, we obtain x = − µ + O ( µ ) , x = 30 µ + O ( µ ) , x = − µ + O ( µ ) , and λ = − µ + O ( µ ) . (4.7)By the perturbation theory, for every µ (cid:54) = 0 sufficiently small, L + has two simple (small)negative eigenvalues. Other eigenvalues are bounded away from zero for small µ and byLemma 2, all other eigenvalues of L + are strictly positive. Hence, n ( L + ) = 2.Zero eigenvalue of L (0) − is triple and is associated with the subspace spanned by { e , e , e } .For every µ (cid:54) = 0, a double zero eigenvalue of L − exists due to the two symmetries (2.2)and (2.3). The two eigenvectors for the double zero eigenvalue of L − are spanned by { e , e , e , . . . } . There are two invariant subspaces of ˆ L − , one is spanned by { e , e , e } and the other one is spanned by { e , e , e , e } . Since we only need to compute a shift of thezero eigenvalue of L (0) − , we only consider the subspace of ˆ L − spanned by { e , e , e , e } . Forthis subspace, the eigenvalue problem for ˆ L − is given by ( + µ ) x − µx + 2 µx − µ x = λx , − µx + µ x − µ x + 2 µx = λx , µx − µ x + ( + µ ) x = λx , − µ x + 2 µx + (1 + µ ) x = λx , The small eigenvalue for small µ is obtained by normalization x = 1. Then, we obtain x = 6 µ + O ( µ ) , x = − µ + O ( µ ) , x = − µ + O ( µ ) , and λ = − µ + O ( µ ) . (4.8)By the perturbation theory, for every µ (cid:54) = 0 sufficiently small, L − has one simple (small)negative eigenvalue and the double zero eigenvalue. Other eigenvalues are bounded awayfrom zero for small µ and by Lemma 2, all other eigenvalues of L − are strictly positive.Hence, n ( L − ) = 1. TATIONARY STATES IN THE CONFORMAL FLOW 18
Case (ii): (cid:15) (cid:54) = 0 , µ = 0 . Here we have Ω = (cid:15) + O ( (cid:15) ), b = − (cid:15) + O ( (cid:15) ), b = b = 0, b = − (cid:15) + O ( (cid:15) ), b = 3 (cid:15) + O ( (cid:15) ), and b n = O ( (cid:15) ) for n ≥
7. Substituting these expansionsin (4.5), we obtain L ± = L (0) ± + (cid:15)L (1) ± + (cid:15) L (2) ± + (cid:15) L (3) ± + O ( (cid:15) ) , where ( L (0) ± a ) n = 16 ( n − n − a n ± a − n , ( L (1) ± a ) n = 2( a n +2 + a n − ) ± a − n , ( L (2) ± a ) n = 76 n ( n + 1) a n + 2(min(2 , n ) − a n − a n +4 + a n − ) ∓ a − n ± (min(2 , n, − n ) − a − n , ( L (3) ± a ) n = − a n +2 + a n − ) + 6( a n +6 + a n − ) − , n ) + 1) a n +2 − , n −
2) + 1) a n − ∓ a − n ∓ , n, − n ) a − n . Let us represent the first 7-by-7 matrix block of the operator L ± and truncate it by up toand including O ( (cid:15) ) terms. The corresponding matrix blocks denoted by ˆ L + and ˆ L − aregiven respectively by − (cid:15) (cid:15) − (cid:15) − (cid:15) (cid:15) + 2 (cid:15) + (cid:15) − (cid:15) (cid:15) − (cid:15) − (cid:15) − (cid:15) − (cid:15) (cid:15) − (cid:15) (cid:15) (cid:15) − (cid:15) − (cid:15) (cid:15) − (cid:15) − (cid:15) (cid:15) − (cid:15) (cid:15) − (cid:15) − (cid:15) (cid:15) − (cid:15) + (cid:15) (cid:15) − (cid:15) − (cid:15) − (cid:15) (cid:15) − (cid:15) (cid:15) (cid:15) − (cid:15) (cid:15) − (cid:15) (cid:15) and − (cid:15) − (cid:15) (cid:15) − (cid:15) + (cid:15) + 2 (cid:15) (cid:15) + 4 (cid:15) − (cid:15) − (cid:15) + 6 (cid:15) − (cid:15) (cid:15) (cid:15) − (cid:15) − (cid:15) (cid:15) + 4 (cid:15) − (cid:15) (cid:15) + 12 (cid:15) (cid:15) − (cid:15) − (cid:15) (cid:15) − (cid:15) + (cid:15) (cid:15) − (cid:15) − (cid:15) + 6 (cid:15) (cid:15) − (cid:15) (cid:15) (cid:15) − (cid:15) (cid:15) − (cid:15) (cid:15) Zero eigenvalue of L (0)+ is double and is associated with the subspace spanned by { e , e } .There are two invariant subspaces of ˆ L + , one is spanned by { e , e , e , e } and the otherone is spanned by { e , e , e } . For the subspace spanned by { e , e , e , e } , the eigenvalue TATIONARY STATES IN THE CONFORMAL FLOW 19 problem is given by (2 − (cid:15) ) x + (4 (cid:15) − (cid:15) ) x − (cid:15) x + 6 (cid:15) x = λx , (4 (cid:15) − (cid:15) ) x + 6 (cid:15) x + (2 (cid:15) − (cid:15) ) x − (cid:15) x = λx , − (cid:15) x + (2 (cid:15) − (cid:15) ) x + ( + (cid:15) ) x + (2 (cid:15) − (cid:15) ) x = λx , (cid:15) x − (cid:15) x + (2 (cid:15) − (cid:15) ) x + (2 + 51 (cid:15) ) x = λx . The small eigenvalue for small (cid:15) is obtained by normalization x = 1. Then, we obtain x = − (cid:15) + O ( (cid:15) ) , x = − (cid:15) + O ( (cid:15) ) , x = 9 (cid:15) + O ( (cid:15) ) , and λ = − (cid:15) + O ( (cid:15) ) . (4.9)For the subspace spanned by { e , e , e } , the eigenvalue problem is given by ( + 2 (cid:15) + (cid:15) − (cid:15) ) x + (2 (cid:15) − (cid:15) − (cid:15) ) x − (6 (cid:15) + 6 (cid:15) ) x = λx , (2 (cid:15) − (cid:15) − (cid:15) ) x + (16 (cid:15) − (cid:15) ) x + (2 (cid:15) − (cid:15) ) x = λx , − (6 (cid:15) + 6 (cid:15) ) x + (2 (cid:15) − (cid:15) ) x + (1 + 37 (cid:15) ) x = λx . The small eigenvalue for small (cid:15) is obtained by normalization x = 1. Then, we obtain x = − (cid:15) + 48 (cid:15) + O ( (cid:15) ) , x = − (cid:15) + O ( (cid:15) ) , and λ = 108 (cid:15) + O ( (cid:15) ) . (4.10)By the perturbation theory, for every (cid:15) (cid:54) = 0 sufficiently small, L + has one simple smallnegative eigenvalue and one simple small positive eigenvalue. Other eigenvalues are boundedaway from zero for small (cid:15) and by Lemma 2, all other eigenvalues of L + are strictly positive.Hence, n ( L + ) = 1.Zero eigenvalue of L (0) − is triple and is associated with the subspace spanned by { e , e , e } .For every (cid:15) (cid:54) = 0, a double zero eigenvalue of L − exists due to the two symmetries (2.2)and (2.3). The two eigenvectors for the double zero eigenvalue of L − are spanned by { e , e , e , e , . . . } . There are two invariant subspaces of ˆ L − , one is spanned by { e , e , e , e } and the other one is spanned by { e , e , e } . Since we only need to compute a shift of thezero eigenvalue of L (0) − , we only consider the subspace of ˆ L − spanned by { e , e , e } . For thissubspace, the eigenvalue problem for ˆ L − is given by ( − (cid:15) + (cid:15) + 2 (cid:15) ) x + (2 (cid:15) + 4 (cid:15) − (cid:15) ) x + ( − (cid:15) + 6 (cid:15) ) x = λx , (2 (cid:15) + 4 (cid:15) − (cid:15) ) x + (16 (cid:15) + 12 (cid:15) ) x + (2 (cid:15) − (cid:15) ) x = λx , ( − (cid:15) + 6 (cid:15) ) x + (2 (cid:15) − (cid:15) ) x + (1 + 37 (cid:15) ) x = λx . The small eigenvalue for small (cid:15) is obtained by normalization x = 1. Then, we obtain x = − (cid:15) − (cid:15) + O ( (cid:15) ) , x = − (cid:15) + O ( (cid:15) ) , and λ = − (cid:15) + O ( (cid:15) ) . (4.11)By the perturbation theory, for every (cid:15) (cid:54) = 0 sufficiently small, L − has one simple (small) neg-ative eigenvalue and the double zero eigenvalue. Other eigenvalues are bounded away from TATIONARY STATES IN THE CONFORMAL FLOW 20 zero for small (cid:15) and by Lemma 2, all other eigenvalues of L − are strictly positive. Hence, n ( L − ) = 1. Case (iii): (cid:15) < , µ = 2 | (cid:15) | / + O ( (cid:15) ) . Here we introduce δ := ( − (cid:15) ) / and writeΩ = δ + δ + O ( δ ), b = − δ − δ + O ( δ ), b = 12 δ + O ( δ ), b = − δ + O ( δ ), b = 4 δ + O ( δ ), b = − δ + O ( δ ), and b n = O ( δ ) for n ≥
7. Substituting these expansionsin (4.5), we obtain L ± = L (0) ± + δ L (2) ± + δ L (3) ± + δ L (4) ± + δ L (5) ± + δ L (6) ± + O ( δ ) , where( L (0) ± a ) n = 16 ( n − n − a n ± a − n , ( L (2) ± a ) n = − a n +2 + a n − ) ∓ a − n , ( L (3) ± a ) n = − a n +3 + a n − ) ∓ a − n , ( L (4) ± a ) n = 76 n ( n + 1) a n + 2(min(2 , n ) − a n − a n +4 + a n − ) ∓ a − n ± (min(2 , n, − n ) − a − n , ( L (5) ± a ) n = − a n +1 + a n − ) − a n +5 + a n − ) + 4 min(2 , n − a n − +4 min( n, a n +1 ∓ a − n ± , n, − n ) − a − n , ( L (6) ± a ) n = 283 n ( n + 1) a n + 8( a n +2 + a n − ) − a n +6 + a n − ) + 6 min(2 , n − a n − +6 min(2 , n ) a n +2 + 8(min(3 , n ) − a n ∓ a − n ± a − n ± , n, − n ) a − n ± , n, − n ) a − n . Let us represent the first 7-by-7 matrix block of the operator L ± and truncate it by up toand including O ( δ ) terms. The corresponding matrix blocks denoted by ˆ L + and ˆ L − aregiven respectively byˆ L + = − δ − δ − δ − δ + 10 δ − δ − δ − δ + δ + δ − δ − δ − δ − δ + 14 δ − δ + 10 δ − δ − δ δ + 64 δ − δ − δ − δ − δ + 14 δ − δ δ + 152 δ − δ − δ − δ + 40 δ − δ − δ − δ + 10 δ − δ − δ + 20 δ − δ − δ − δ − δ TATIONARY STATES IN THE CONFORMAL FLOW 21 − δ − δ − δ − δ − δ + 10 δ − δ − δ + 40 δ − δ − δ − δ − δ + 20 δ − δ + δ + δ − δ − δ + 20 δ − δ δ + 296 δ − δ − δ + 20 δ − δ δ + 408 δ and ˆ L − =
20 4 δ δ δ + 2 δ + δ + δ δ − δ − δ + 4 δ + 14 δ δ δ − δ δ + 64 δ − δ − δ + 4 δ + 14 δ − δ δ + 104 δ − δ − δ − δ − δ − δ − δ − δ − δ − δ + 20 δ − δ − δ − δ − δ − δ − δ − δ − δ − δ − δ − δ − δ − δ − δ − δ − δ + 20 δ − δ + δ + δ − δ − δ + 20 δ − δ δ + 296 δ − δ − δ + 20 δ − δ δ + 408 δ Zero eigenvalue of L (0)+ is double and is associated with the subspace spanned by { e , e } .Since no invariant subspaces of ˆ L + exist, we have to proceed with full perturbative expan-sions. As a first step, we express ( x , x , x , x , x ) for the subspace spanned by { e , e , e , e , e } in terms of { x , x } for the subspace spanned by { e , e } , λ , and δ . We assume that λ = O ( δ )for the small eigenvalues and neglect terms of the order O ( δ ) and higher. This expansionis given by x = (2 δ + 32 δ + λδ ) x + 4 δ x ,x = (12 δ + 192 δ ) x + (6 δ + 48 δ + 240 δ + 18 λδ ) x ,x = (6 δ − δ + 18 λδ ) x + 120 δ x ,x = 4 δ x + (2 δ − δ + 2 λδ ) x ,x = (9 δ − δ ) x + (2 δ − δ − δ ) x . Next, we substitute these expansions to the third and fourth equations of the eigenvalueproblem for ˆ L + and again neglect terms of the order O ( δ ) and higher: (cid:26) ( − δ − λ ) x − δ x = 0 , − δ x + ( − δ − λ ) x = 0 . The reduced eigenvalue problem has two eigenvalues with the expansions λ = − δ + O ( δ ) when x = (4 δ + O ( δ )) x (4.12) TATIONARY STATES IN THE CONFORMAL FLOW 22 and λ = 216 δ + O ( δ ) when x = ( − δ + O ( δ )) x . (4.13)By the perturbation theory, for every δ (cid:54) = 0 sufficiently small, L + has one simple smallnegative eigenvalue and one simple small positive eigenvalue. Other eigenvalues are boundedaway from zero for small δ and by Lemma 2, all other eigenvalues of L + are strictly positive.Hence, n ( L + ) = 1.Zero eigenvalue of L (0) − is triple and is associated with the subspace spanned by { e , e , e } .We proceed again with full perturbative expansions. First, we express ( x , x , x , x ) for thesubspace spanned by { e , e , e , e } in terms of { x , x , x } for the subspace spanned by { e , e , e } , λ , and δ . We assume that λ = O ( δ ) for the small eigenvalues and neglect termsof the order O ( δ ) and higher. This expansion is given by x = − δ x + ( − δ + 192 δ ) x + (6 δ − δ + 240 δ ) x ,x = 3 δ x + (6 δ − δ ) x + 120 δ x ,x = 4 δ x + 4 δ x + (2 δ − δ ) x ,x = 8 δ x + 9 δ x + 2 δ x . Next, we substitute these expansions to the first, third and fourth equations of the eigenvalueproblem for ˆ L − and again neglect terms of the order O ( δ ) and higher: λx = 0 ,λx = 0 , (216 δ − λ ) x = 0 . The double zero eigenvalue persists due to two gauge symmetries, whereas one eigenvalue isexpanded by λ = 216 δ + O ( δ ) . (4.14)By the perturbation theory, for every δ (cid:54) = 0 sufficiently small, L − has one simple (small)negative eigenvalue and the double zero eigenvalue. Other eigenvalues are bounded awayfrom zero for small δ and by Lemma 2, all other eigenvalues of L − are strictly positive.Hence, n ( L − ) = 0. (cid:3) In the remainder of this section, we consider the two constraints related to the fixedvalues of Q and E . The constraints may change the number of negative eigenvalues of thelinearization operator L + constrained by the following two orthogonality conditions[ X c ] ⊥ := (cid:8) a ∈ (cid:96) ( N ) : (cid:104) M A, a (cid:105) = (cid:104) M A, a (cid:105) = 0 (cid:9) , (4.15)where A denotes a real-valued solution of system (2.10) and M = diag(1 , , , . . . ). The con-strained space [ X c ] ⊥ is a symplectically orthogonal subspace of (cid:96) ( N ) to X = span { A, M A } ⊂ (cid:96) ( N ), the two-dimensional subspace associated with the double zero eigenvalue of L − re-lated to the phase rotation symmetries (2.2) and (2.3). Alternatively, the constrained spacearises when the perturbation a does not change at the linear approximation the conservedquantities Q and E defined by (2.4) and (2.5). Note that the constraints in (4.15) are onlyimposed on the real part of the perturbation a . TATIONARY STATES IN THE CONFORMAL FLOW 23
Let ˜ A denote the stationary state of the stationary equation (2.10) continued with respectto two parameters ( λ, ω ). Let n ( L + ) and z ( L + ) denote the number of negative and zeroeigenvalues of L + in (cid:96) ( N ) counted with their multiplicities, where L + is the linearizedoperator at ˜ A . Let n c ( L + ) and z c ( L + ) denote the number of negative and zero eigenvaluesof L + constrained in [ X c ] ⊥ . Assume non-degeneracy of the stationary state ˜ A in the sensethat z ( L + ) = 0. By Theorem 4.1 in [14], we have n c ( L + ) = n ( L + ) − p ( D ) − z ( D ) , z c ( L + ) = z ( D ) , (4.16)where p ( D ) and z ( D ) are the number of positive and zero eigenvalues of the 2 × D := (cid:20) ∂ Q ∂λ ∂ Q ∂ω∂ ( Q−E ) ∂λ ∂ ( Q−E ) ∂ω (cid:21) , (4.17)with Q ( λ, ω ) = Q ( ˜ A ) and E ( λ, ω ) = E ( ˜ A ) evaluated at the stationary solution ˜ A as afunction of the two parameters ( λ, ω ). Remark . For the pair of stationary states (1.4), it was computed in [2] that Q ( λ, ω ) = 67 ( λ + ω ) , E ( λ, ω ) = 6 ω, (4.18)where λ and ω are related to the parameters c and p in (1.5). Substituting (4.18) into (4.17)yields D = (cid:20) / / / − / (cid:21) , (4.19)hence, D has one positive and one negative eigenvalue. If z ( L + ) = 0 holds, then n c ( L + ) = n ( L + ) − z c ( L + ) = 0 by (4.16). We will show in Lemma 5 below that the same countis true for all three branches of Theorem 2 bifurcating from ω ∗ .By the scaling transformation (2.1), if the stationary state is given by (2.9) with real A ,then the stationary state is continued with respect to parameter c > α n ( t ) = cA n e − ic λt + inc ωt , hence ˜ A = cA , ˜ λ = c λ , and ˜ ω = c ω . Substituting these relations into ˜ Q (˜ λ, ˜ ω ) = Q ( ˜ A ) and˜ E (˜ λ, ˜ ω ) = E ( ˜ A ) for λ = 1 yields˜ Q (˜ λ, ˜ ω ) = c Q ( ω ) = ˜ λ Q (˜ ω ˜ λ − )and ˜ E (˜ λ, ˜ ω ) = c E ( ω ) = ˜ λ E (˜ ω ˜ λ − ) , where Q ( ω ) = Q (1 , ω ) and E ( ω ) = E (1 , ω ). Substituting these representations into (4.17),evaluating derivatives, and setting c = 1 yield the computational formula D = (cid:20) Q ( ω ) − ω Q (cid:48) ( ω ) Q (cid:48) ( ω ) Q ( ω ) − E ( ω ) − ω [ Q (cid:48) ( ω ) − E (cid:48) ( ω )] Q (cid:48) ( ω ) − E (cid:48) ( ω ) (cid:21) , (4.20)which can be used to compute D for the normalized stationary state A with λ = 1. Thefollowing lemma gives the variational characterization of the three branches in Theorem 2bifurcating from ω ∗ as critical points of H subject to fixed Q and E . TATIONARY STATES IN THE CONFORMAL FLOW 24
Lemma 5.
Consider the three bifurcating branches in Theorem 2 for ω ∗ := ω = ω = 1 / .For every ω (cid:54) = ω ∗ with | ω − ω ∗ | sufficiently small, the following is true: (i) The branch is a saddle point of H subject to fixed Q and E with n c ( L + ) = 1 and n ( L − ) = 1 ; (ii) The branch is a saddle point of H subject to fixed Q and E with n c ( L + ) = 0 and n ( L − ) = 1 ; (iii) The branch is a minimizer of H subject to fixed Q and E with n c ( L + ) = 0 and n ( L − ) = 0 .The critical points are degenerate only with respect to the two phase rotations (2.2) and (2.3)resulting in z ( L + ) = 0 and z ( L − ) = 2 .Proof. Case (i): (cid:15) = 0 , µ (cid:54) = 0 . We compute Q ( ω ) and E ( ω ) as powers of µ with thefollowing relation between ω and µ : ω = 16 + 112 µ + O ( µ ) . Then it follows that (cid:26) Q ( ω ) = A + 4 A + 7 A + · · · = 1 + 2 µ + O ( µ ) , E ( ω ) = A + 4 A + 7 A + · · · = 1 + 14 µ + O ( µ ) , so that D = (cid:20) − O ( µ ) 24 + O ( µ )24 + O ( µ ) −
144 + O ( µ ) (cid:21) has one positive and one negative eigenvalue. Since n ( L + ) = 2 and z ( L + ) = 0 by Lemma 4,we have n c ( L + ) = n ( L + ) − n ( L − ) = 1 and z ( L − ) = 2 byLemma 4. Case (ii): (cid:15) (cid:54) = 0 , µ = 0 . We compute Q ( ω ) and E ( ω ) as powers of (cid:15) with the followingrelation between ω and (cid:15) : ω = 16 + 76 (cid:15) + O ( (cid:15) ) . Then it follows that (cid:26) Q ( ω ) = A + 3 A + 5 A + · · · = 1 + (cid:15) + O ( (cid:15) ) , E ( ω ) = A + 3 A + 5 A + · · · = 1 + 7 (cid:15) + O ( (cid:15) ) , so that D = (cid:20) / O ( (cid:15) ) 6 / O ( (cid:15) )6 / O ( (cid:15) ) − / O ( (cid:15) ) (cid:21) has one positive and one negative eigenvalue. Since n ( L + ) = 1 and z ( L + ) = 0 by Lemma4), we have n c ( L + ) = n ( L + ) − n ( L − ) = 1 and z ( L − ) = 2by Lemma 4. TATIONARY STATES IN THE CONFORMAL FLOW 25
Case (iii): (cid:15) < , µ = 2 | (cid:15) | / + O ( (cid:15) ) . We compute Q ( ω ) and E ( ω ) as powers of (cid:15) withthe following relation between ω and (cid:15) : ω = 16 + 76 (cid:15) − (cid:15) + O ( (cid:15) ) . Then it follows that (cid:26) Q ( ω ) = A + 2 A + 3 A + 4 A + 5 A + · · · = 1 + (cid:15) + O ( (cid:15) ) , E ( ω ) = A + 2 A + 3 A + 4 A + 5 A + · · · = 1 + 7 (cid:15) + O ( (cid:15) ) . The only difference in these expansions compared to the case (ii) is the remainder term aslarge as O ( (cid:15) ) compared to O ( (cid:15) ). This changes the remainder terms in D to O ( (cid:15) ) comparedto O ( (cid:15) ) but does not affect the conclusion on D . Moreover, we can see that the computationof D agrees with the exact expression (4.19) in Remark 9. The count n c ( L + ) = n ( L + ) − n ( L − ) = 0 follows by Lemma 4. (cid:3) Remark . The presence of conserved quantity Z ( α ) in (2.6) does not modify the variationalcharacterization of the stationary states with ω (cid:54) = 0 because Z ( α ) = 0 if α is the stationarystate (2.9) with ω (cid:54) = 0. This follows from the fact that Z ( α ) is independent of t , whichis impossible if Z ( α ) (cid:54) = 0 and ω (cid:54) = 0. Hence, any stationary state (2.9) must satisfy theconstraint: ω (cid:54) = 0 : ∞ (cid:88) n =0 ( n + 1)( n + 2) A n A n +1 = 0 , which can be verified for all states in Theorem 1 and 2 bifurcating from ω m for m ≥ Bifurcation from the second eigenmode
Here we study bifurcations of stationary states in the system of algebraic equations (3.1)from the second eigenmode given by (2.11) with N = 1. Without loss of generality, thescaling transformation (2.1) yields c = 1 and λ − ω = 1. By setting A n = δ n + a n withreal-valued perturbation a , we rewrite the system (3.1) with λ = 1 + ω in the perturbativeform (3.2), where L ( ω ) is a block-diagonal operator with the diagonal entries[ L ( ω )] nn = − ω, n = 0 , , n = 1 , ω, n = 2 , ( n − ω − n + 3 , n ≥ , (5.1)and the only nonzero off-diagonal entries [ L ( ω )] = [ L ( ω )] = 1, whereas the nonlinearterms are given by[ N ( a )] n = 2 ∞ (cid:88) j =0 S nj,n + j − , a j a n + j − + n +1 (cid:88) k =0 S nk,n +1 − k a k a n +1 − k + ∞ (cid:88) j =0 n + j (cid:88) k =0 S njk,n + j − k a j a k a n + j − k . We have the following result on the nonlinear terms.
TATIONARY STATES IN THE CONFORMAL FLOW 26
Lemma 6.
Fix an integer m ≥ . If a = 0 and a m(cid:96) +2 = a m(cid:96) +3 = · · · = a m(cid:96) + m = 0 , for every (cid:96) ∈ N , (5.2) then [ N ( a )] = 0 and [ N ( a )] m(cid:96) +2 = [ N ( a )] m(cid:96) +3 = · · · = [ N ( a )] m(cid:96) + m = 0 , for every (cid:96) ∈ N . (5.3) Proof.
The argument repeats the proof of Lemma 1. Under the conditions (5.2), every termin [ N ( a )] n for n = m(cid:96) + ı with (cid:96) ∈ N , and ı ∈ { , , . . . , m } is inspected and shown to bezero. (cid:3) Bifurcations from the second eigenmode are identified by zero eigenvalues of the diagonaloperator L ( ω ). Lemma 7.
There exists a sequence of bifurcations at ω ∈ { ω m } m ∈ N + with ω = 0 , ω = 2 / ,and ω m = m − m − , m ∈ { , , . . . } . (5.4) All bifurcation points are simple except for the three double points ω = ω = 0 , ω = ω =1 / , and ω = ω = 1 / .Proof. The diagonal terms [ L ( ω )] nn for n ∈ { , , . . . } vanish at the sequence (5.4). Inaddition, the double block for n = 0 and n = 2 has zero eigenvalues if and only if ω = 0or ω = 2 /
3. Therefore, ω = ω = 0 is a double bifurcation point. To study other doublebifurcation points, we consider solutions of ω m = ω = 2 / m ≥ ω n = ω m for n (cid:54) = m ≥
4. Equation ω m = ω = 2 / m − m + 7 = 0, which has nointeger solutions. Equation ω n = ω m for n (cid:54) = m is equivalent to mn = 3( m + n ) −
1, whichcan be solved for m in terms of nm = M ( n ) := 3 n − n − n − . Since the right-hand side is monotonically decreasing in n and M (12) <
4, we can find allinteger solutions for n in the range from 4 to 11. There exists only two pairs of integersolutions in this range, which give two double points ω = ω = 1 /
15 and ω = ω = 1 / (cid:3) Simple bifurcation points can be investigated similarly to the proof of Theorem 1. Thisyields the following theorem.
Theorem 3.
Fix m = 2 . There exists a unique branch of solutions ( ω, A ) ∈ R × (cid:96) ( N ) to system (3.1) with λ − ω = 1 , which can be parameterized by small (cid:15) such that ( ω, A ) issmooth in (cid:15) and | ω − ω | + sup n ∈ N | A n − δ n − (cid:15) (3 δ n − δ n ) | (cid:46) (cid:15) . (5.5) Fix an integer m ≥ with m (cid:54) = 7 and m (cid:54) = 11 . There exists a unique branch of solutions ( ω, A ) ∈ R × (cid:96) ( N ) to system (3.1) with λ − ω = 1 , which can be parameterized by small (cid:15) such that ( ω, A ) is smooth in (cid:15) and | ω − ω m | + sup n ∈ N | A n − δ n − (cid:15)δ nm | (cid:46) (cid:15) . (5.6) TATIONARY STATES IN THE CONFORMAL FLOW 27
Proof.
The proof of the second assertion repeats the proof of Theorem 1 verbatim. Theproof of the first assertion is based on the block-diagonalization of the singular matrix for[ L ( ω )] jk with j, k ∈ { , } : (cid:20) / (cid:21) . The null space is spanned by the vector (3 , − T and the vector b ∈ (cid:96) ( N ) in the decompo-sition (3.9) must satisfy the constraint 3 b − b = 0. The rest of the proof repeats the proofof Theorem 1 after a simple observation that [ N ( (cid:15) (3 e − e ))] , = O ( (cid:15) ) as (cid:15) → (cid:3) Remark . The unique branch bifurcating from ω = 2 / λ − ω = 1 and taking the limit p → c = p − + O ( p ), β = − p + O ( p ), and γ = p − + O ( p ). Thesmall parameter (cid:15) is defined in terms of the small parameter p by (cid:15) := − p + O ( p ).The three double bifurcation points in Lemma 7 have to be checked separately. In orderto characterize branches bifurcating from the double point ω = ω = 1 /
12, we note thefollowing symmetry. If u ( t, z ) is a generating function for the conformal flow (1.1) given bythe power series (2.24), so is zu ( t, z ). If u ( t, z ) is a stationary state in the form (2.26) with { A n } n ∈ N satisfying the system (3.1) with parameters ( λ, ω ), then the transformed state˜ A n = (cid:40) A m , n = 2 m + 1 , , n = 2 m, n ∈ N (5.7)also satisfies the system (3.1) with parameters˜ λ = λ + ω , ˜ ω = ω . By Theorem 2, three branches of solutions bifurcate from the lowest eigenmode at the doublebifurcation point ω = ω = 1 /
6. Applying the transformation (5.7) yields three branchesbifurcating from the second eigenmode at the double bifurcation point ω = ω = 1 / Theorem 4.
Fix ω ∗ := ω = ω = 1 / . There exist exactly three branches of solutions ( ω, A ) ∈ R × (cid:96) ( N ) to system (3.1) with λ − ω = 1 , which can be parameterized by small ( (cid:15), µ ) such that ( ω, A ) is smooth in ( (cid:15), µ ) and | ω − ω ∗ | + sup n ∈ N | A n − δ n − (cid:15)δ n − µδ n | (cid:46) ( (cid:15) + µ ) . (5.8) The three branches are characterized by the following: (i) (cid:15) = 0 , µ (cid:54) = 0 ; (ii) (cid:15) (cid:54) = 0 , µ = 0 ; (iii) (cid:15) < , | µ − | (cid:15) | / | (cid:46) (cid:15) , TATIONARY STATES IN THE CONFORMAL FLOW 28 and the branch (iii) is double degenerate up to the reflection µ (cid:55)→ − µ . Branches bifurcating from the double point ω = ω = 1 /
15 can be investigated bycomputing the normal form. The following theorem represents the main result.
Theorem 5.
Fix ω ∗ := ω = ω = 1 / . There exist exactly two branches of solutions ( ω, A ) ∈ R × (cid:96) ( N ) to system (3.1) with λ − ω = 1 , which can be parameterized by small ( (cid:15), µ ) such that ( ω, A ) is smooth in ( (cid:15), µ ) and | ω − ω ∗ | + sup n ∈ N | A n − δ n − (cid:15)δ n − µδ n | (cid:46) ( (cid:15) + µ ) . (5.9) The two branches are characterized by the following: (i) (cid:15) = 0 , µ (cid:54) = 0 ; (ii) (cid:15) (cid:54) = 0 , µ = 0 .Proof. The proof follows the computations of normal form in Theorem 2. For the doublebifurcation point ω ∗ , we write the decomposition ω = ω ∗ + Ω , a n = (cid:15)δ n + µδ n + b n , n ∈ N , where ( (cid:15), µ ) are arbitrary and b = b = 0 are set from the orthogonality condition (cid:104) b, e (cid:105) = (cid:104) b, e (cid:105) = 0. By performing routine computations and expanding the bifurcation equationsat n = 4 and n = 11 up to and including the cubic order, we obtain (cid:26) (cid:15) = (cid:15) (7 (cid:15) + 14 µ + O (3)) , µ = µ (cid:0) (cid:15) + µ + O (3) (cid:1) . (5.10)We are looking for solutions to the system (5.10) with ( (cid:15), µ ) (cid:54) = (0 , (cid:15) = 0, µ (cid:54) = 0, and Ω = µ + O (3);(II) (cid:15) (cid:54) = 0, µ = 0, and Ω = (cid:15) + O (3);whereas no solution exists with both (cid:15) (cid:54) = 0 and µ (cid:54) = 0. The Jacobian matrix of the system(5.10) is given by J ( (cid:15), µ, Ω) = (cid:20) −
15Ω + 21 (cid:15) + 14 µ (cid:15)µ (cid:15)µ − (cid:15) + µ (cid:21) + O (3) . For both branches (I) and (II), the Jacobian is invertible for small ( (cid:15), µ ), hence the twosolutions are continued uniquely with respect to parameters ( (cid:15), µ ) and yield branches (i) and(ii). By Lemma 6 with m = 10, branch (i) corresponds to the reduction (5.2) with m = 10,hence (cid:15) = 0 persists beyond all orders of the expansion. By Lemma 6 with m = 3, branch(ii) corresponds to the reduction (5.2) with m = 3, hence µ = 0 persists beyond all ordersof the expansion. (cid:3) Remark . Branch (ii) of Theorem 5 can be obtained by the symmetry transformation (5.7)from the branch bifurcating from lowest eigenstate at ω = 2 / TATIONARY STATES IN THE CONFORMAL FLOW 29
For the remaining double point ω = ω = 0, bifurcation of stationary state is morecomplicated. If we compute the normal form up to and including the cubic order, we obtaina trivial normal form, which is satisfied identically if ω = 0. This outcome of the normalform computations suggests that there exists a two-parameter family of solutions A ∈ (cid:96) ( N )to the system (3.1) with λ = 1 and ω = 0. Indeed, the two eigenvectors for the null space of L (0) are given by e − e and e so that we can introduce the decomposition a n = (cid:15) ( δ n − δ n ) + µδ n + b n , n ∈ N , subject to the orthogonality conditions b − b = 0 and b = 0. By computing powerexpansions for small ( (cid:15), µ ) with MAPLE, we can extend it to any polynomial order with thefirst terms given by A = (cid:15) + (cid:15)µ + (cid:16) (cid:15)µ − (cid:15) (cid:17) + ( (cid:15) µ + 2 (cid:15)µ ) + (cid:16) − (cid:15) + (cid:15) µ + 3 (cid:15)µ (cid:17) + O (6) ,A = 1 − ( (cid:15) + µ ) + (cid:15) µ + + (8 (cid:15) µ + 3 (cid:15) µ ) + ( − (cid:15) − (cid:15) µ − µ ) + O (6) ,A = − (cid:15) + (cid:15)µ + (cid:16) − (cid:15) + (cid:15)µ (cid:17) + ( (cid:15) µ + 2 (cid:15)µ ) + (cid:16) − (cid:15) + (cid:15) µ + 3 (cid:15)µ (cid:17) + O (6) ,A = µ,A = − (cid:15)µ + ( (cid:15) + (cid:15)µ ) − (4 (cid:15) µ + (cid:15)µ ) + (cid:16) (cid:15) + (cid:15) µ + 3 (cid:15)µ (cid:17) + O (6) ,A = µ + (cid:15) µ + (3 (cid:15) µ + (cid:15) µ ) + ( − (cid:15) + (cid:15) µ + µ ) + O (6) ,A = − (cid:15)µ + (2 (cid:15) µ + (cid:15)µ ) − (cid:0) (cid:15) µ + 5 (cid:15)µ (cid:1) + O (6) . The three explicit solutions (2.17), (2.29), and (2.31) are particular solutions of the two-parameter family of stationary states for small p . Indeed, the twisted state (2.17) correspondsto (cid:15) = − p + O ( p ) and µ = 3 p + O ( p ), the Blaschke state corresponds to (cid:15) = − p + O ( p )and µ = p + O ( p ), and the additional state (2.31) corresponds to (cid:15) = 0 and µ = p + O ( p ). Remark . We have shown by using the general transformation law derived in [3] that thetwisted state (2.17) can be obtained from the N = 1 single-mode state (2.11) by applying e sD with s ∈ R in the symmetry transformation (2.7) with p = tanh s . In the present time,we do not know how to obtain the two-parameter branch of the stationary states above fromthe N = 1 single-mode state (2.11).6. Variational characterization of the bifurcating states
We shall now give variational characterization of the bifurcating states from the secondeigenmode. The second variation of the action functional K ( α ) in (4.1) is given by thequadratic forms in (4.2) with the self-adjoint operators L ± : D ( L ± ) → (cid:96) ( N ) given by(4.3). The following lemma gives variational characterization of the second eigenmode atthe bifurcation points { ω m } m ∈ N + in Lemma 7. Lemma 8.
The following is true: • For ω = ω = 0 , the N = 1 single-mode state (2.11) is a degenerate saddle point of K with three positive eigenvalues, zero eigenvalue of multiplicity three, and infinitelymany negative eigenvalues bounded away from zero. TATIONARY STATES IN THE CONFORMAL FLOW 30 • For ω = 2 / and ω = 3 / , the N = 1 single-mode state (2.11) is a degenerateminimizer of K with zero eigenvalue of multiplicity three and infinitely many positiveeigenvalues bounded away from zero. • For ω m with m ≥ and m (cid:54) = 6 , the N = 1 single-mode state (2.11) is a degeneratesaddle point of K with an even number of negative eigenvalues, zero eigenvalue ofodd multiplicity, and infinitely many positive eigenvalues bounded away from zero.Proof. By the scaling transformation (2.1), we take λ − ω = 1 and A n = δ n , for which theexplicit form (4.3) yields( L ± a ) n = [1 − n + 2 min( n, a n ± [1 + min( n, , − n ) a − n + ( n − ωa n , n ∈ N . (6.1)We note that L + = L ( ω ) given by (5.1) and L − is only different from L + at the diagonalentry at n = 1 (which is 0 instead of 4) and for the off-diagonal entries at n = 0 and n = 2(which are − L ( ω ), the 2 × n = 0 and n = 2, (cid:20) − ω
11 1 + 3 ω (cid:21) , is positive definite for all { ω m } m ∈ N + with a simple zero eigenvalue only arising for ω = ω = 0and ω = 2 /
3. The diagonal entries of L ( ω ) for n ≥ n − ω + 3 − n, n ≥ . These entries are negative if ω = ω = 0 with a simple zero eigenvalue and strictly positiveif ω = 2 /
3. For ω m = ( m − / ( m −
1) with m ≥
4, the diagonal entries are simplified inthe form: ( n − ω m + 3 − n = ( n − m )( nm − n − m + 1) m − , n ≥ . For m = 4 and m = 11, two eigenvalues are zero, six are negative, and all others are positive.For m = 5 and m = 7, two eigenvalues are zero, one is negative, and all others are positive.For m = 6, one eigenvalue is zero and all others are positive. For m = 8 , ,
10 and m ≥ L + = L ( ω ) by a factor of 2 due to the matrix operator L − and adding an additional zero entry of L − at n = 1 yields the assertion of the lemma. (cid:3) Among all stationary states bifurcating from the second eigenmode, we shall only considerthe potential minimizers of energy H subject to fixed Q and E . By Lemma 8, this includesonly two branches bifurcating from ω and ω . In both cases, we are able to compute thenumber of negative eigenvalues of the operators L ± denoted by n ( L ± ). Lemma 9.
Consider the two bifurcating branches in Theorem 3 for ω and ω . For everysmall (cid:15) (cid:54) = 0 , we have n ( L + ) = 1 and n ( L − ) = 0 . For each branch, L − has a double zeroeigenvalue, L + has no zero eigenvalue, and the rest of the spectrum of L + and L − is strictlypositive and is bounded away from zero. TATIONARY STATES IN THE CONFORMAL FLOW 31
Proof.
By the second item of Lemma 8, the corresponding operators L + and L − at thebifurcation point (cid:15) = 0 have respectively the simple and double zero eigenvalue, whereasthe rest of their spectra is strictly positive and is bounded away from zero. By the twosymmetries (2.2) and (2.3), the double zero eigenvalue of L − is preserved for (cid:15) (cid:54) = 0 and theassertion of the lemma for L − follows by the perturbation theory.On the other hand, the simple zero eigenvalue of L + is not preserved for (cid:15) (cid:54) = 0 and willgenerally shift to either negative or positive values. We will show that it shifts to the negativevalues for (cid:15) (cid:54) = 0, hence n ( L + ) = 1 in both cases and the assertion of the lemma for L + followsby the perturbation theory.For ω , the Lyapunov–Schmidt decomposition of Theorem 3 yields power expansion ω = 23 − (cid:15) + O ( (cid:15) )and A = 3 (cid:15) + O ( (cid:15) ), A = 1 − (cid:15) + O ( (cid:15) ), A = − (cid:15) + O ( (cid:15) ), A = (cid:15) + O (4), and A n = O ( (cid:15) ) for n ≥
4. Similarly to the proof of Lemma 4, we compute the 4-by-4 block ofthe operator L + at n ∈ { , , , } and truncate it up to and including O ( (cid:15) ) terms. Thecorresponding matrix block denoted by by ˆ L + is given byˆ L + = + (cid:15) (cid:15) − (cid:15) − (cid:15) (cid:15) − (cid:15) − (cid:15) (cid:15) − (cid:15) − (cid:15) − (cid:15) (cid:15) − (cid:15) (cid:15) (cid:15) − (cid:15) We are looking for a small eigenvalue of ˆ Lx = λx , where x = ( x , x , x , x ) T . Assuming λ = O ( (cid:15) ) and expressing { x , x } in terms of { x , x } yield (cid:26) x = − (cid:15) (5 x − x ) + O ( (cid:15) ) ,x = (cid:15) ( x − x ) + O ( (cid:15) ) . Substituting these expressions into the eigenvalue problem ˆ Lx = λx and truncating it up toand including the order of O ( (cid:15) ), we obtain (cid:26) (cid:0) − (cid:15) − λ (cid:1) x + (cid:0) − (cid:15) (cid:1) x = 0 , (cid:0) − (cid:15) (cid:1) x + (cid:0) − (cid:15) − λ (cid:1) x = 0 . The small eigenvalue of this reduced problem is given by the expansion λ = − (cid:15) + O ( (cid:15) ) . (6.2)Hence n ( L + ) = 1 due to the shift of the zero eigenvalue of L + at (cid:15) to λ < (cid:15) (cid:54) = 0.For ω , the Lyapunov–Schmidt decomposition of Theorem 3 yields power expansion ω = 335 + 970 (cid:15) + O ( (cid:15) ) , with A = 1 − (cid:15) + O ( (cid:15) ), A = (cid:15) , A = − (cid:15) + O ( (cid:15) ), A n = O ( (cid:15) ) for n ≥
16, where A n = 0for every n (cid:54) = 5 (cid:96) + 1, (cid:96) ∈ N by Lemma 6. Similarly to the proof of Lemma 4, we compute TATIONARY STATES IN THE CONFORMAL FLOW 32 the 3-by-3 block of the operator L + at n = 1, n = 6, and n = 11 and truncate it up to andincluding O ( (cid:15) ) terms. The corresponding matrix block denoted by ˆ L + is given byˆ L + = − (cid:15) (cid:15) − (cid:15) (cid:15) (cid:15) (cid:15) − (cid:15) (cid:15) + (cid:15) Looking at the small eigenvalue of ˆ Lx = λx , where x = ( x , x , x ) T , we can normalize x = 1 and obtain x = − (cid:15) + O ( (cid:15) ) , x = − (cid:15) + O ( (cid:15) ) , and λ = − (cid:15) + O ( (cid:15) ) . (6.3)Hence n ( L + ) = 1 due to the shift of the zero eigenvalue of L + at (cid:15) to λ < (cid:15) (cid:54) = 0. (cid:3) Finally, we consider the two constraints related to the fixed values of Q and E by usingthe same computational formulas (4.16) and (4.17). Let ˜ A denote the stationary state of thestationary equation (2.10) continued with respect to two parameters ( λ, ω ). By the scalingtransformation (2.1), if the stationary state is given by (2.9) with real A , then the stationarystate is continued with respect to parameter c > α n ( t ) = cA n e − ic λt + inc ωt , hence ˜ A = cA , ˜ λ = c λ , and ˜ ω = c ω . Substituting these relations into ˜ Q (˜ λ, ˜ ω ) = Q ( ˜ A ) and˜ E (˜ λ, ˜ ω ) = E ( ˜ A ) for λ − ω = 1 yields˜ Q (˜ λ, ˜ ω ) = c Q ( ω ) = (˜ λ − ˜ ω ) Q (˜ ω (˜ λ − ˜ ω ) − )and ˜ E (˜ λ, ˜ ω ) = c E ( ω ) = (˜ λ − ˜ ω ) E (˜ ω (˜ λ − ˜ ω ) − ) , where Q ( ω ) = Q (1 + ω, ω ) and E ( ω ) = E (1 + ω, ω ). Substituting these representations into(4.17), evaluating derivatives, and setting c = 1 yield the computational formula D = (cid:20) Q ( ω ) − ω Q (cid:48) ( ω ) −Q ( ω ) + (1 + ω ) Q (cid:48) ( ω ) Q ( ω ) − E ( ω ) − ω [ Q (cid:48) ( ω ) − E (cid:48) ( ω )] −Q ( ω ) + E ( ω ) + (1 + ω ) [ Q (cid:48) ( ω ) − E (cid:48) ( ω )] (cid:21) , which can be used to compute D for the normalized stationary state A with λ − ω = 1. Thefollowing lemma confirms that the two branches in Lemma 9 are indeed local minimizers of H subject to fixed Q and E . Lemma 10.
Consider the two bifurcating branches in Theorem 3 for ω and ω . For everysmall (cid:15) (cid:54) = 0 , the two branches are local minimizers of H subject to fixed Q and E with n c ( L + ) = 0 and n ( L − ) = 0 .Proof. For ω , we use power expansions in Lemma 9 and compute (cid:26) Q ( ω ) = A + 2 A + 3 A + · · · = 2 − (cid:15) + O ( (cid:15) ) , E ( ω ) = A + 4 A + 9 A + · · · = 4 − (cid:15) + O ( (cid:15) ) , TATIONARY STATES IN THE CONFORMAL FLOW 33 so that D = (cid:20) / O ( (cid:15) ) 6 / O ( (cid:15) )6 / O ( (cid:15) ) − / O ( (cid:15) ) (cid:21) . Note that the expression for D agrees with the exact computations in Remark 9. Therefore, D has one positive and one negative eigenvalue, so that n c ( L + ) = n ( L + ) − ω , we use power expansions in Lemma 9 and compute (cid:26) Q ( ω ) = 2 A + 7 A + · · · = 2 + 3 (cid:15) + O ( (cid:15) ) , E ( ω ) = 4 A + 49 A + · · · = 4 + 41 (cid:15) + O ( (cid:15) ) , so that D = (cid:20) O ( (cid:15) ) 70 / O ( (cid:15) )70 / O ( (cid:15) ) − /
81 + O ( (cid:15) ) (cid:21) has again one positive and one negative eigenvalue, hence n c ( L + ) = n ( L + ) − (cid:3) Numerical approximations
We confirm numerically that the stationary states (1.4) with (1.5) have the same varia-tional characterization in the entire existence interval for p in (0 , − √ H for fixed Q and E .Figure 1 shows the smallest eigenvalues of L ± computed at the upper branch of solution(1.4). In agreement with item (iii) in Lemma 4 for small p , we have n ( L + ) = 1 and n ( L − ) = 0for every p ∈ (0 , −√ p is related to the small parameter δ in Lemma4 by p = δ + O ( δ ), see Remark 4. The dashed line shows the asymptotic dependencies (4.12),(4.13), and (4.14) for the small eigenvalues of L + and L − . p σ + p σ - Figure 1.
The smallest eigenvalues of L + (left) and L − (right) for the upperbranch of the stationary state (1.4) with normalization λ = 1.Figure 2 shows the smallest eigenvalues of L ± computed at the lower branch of solution(1.4). In agreement with Lemma 9 for small p , we have n ( L + ) = 1 and n ( L − ) = 0 in theentire region of existence of the stationary state. The small parameter p is related to the TATIONARY STATES IN THE CONFORMAL FLOW 34 small parameter (cid:15) in Lemma 9 by p = − (cid:15)/ O ( (cid:15) ), see Remark 11. The dashed line showsthe asymptotic dependence (6.2) for the small eigenvalue of L + . p σ + Figure 2.
The smallest eigenvalues of L + (left) and L − (right) for the lowerbranch of the stationary state (1.4) with normalization λ − ω = 1.Figure 3 shows the smallest eigenvalues of L ± computed at the stationary state bifurcatingfrom the second eigenmode at ω = 3 /
35. We use here parameter (cid:15) for continuation of thestationary state as in Lemma 9. In agreement with Lemma 9, we have n ( L + ) = 1 and n ( L − ) = 0 for small (cid:15) . However, this result does not hold for larger values of (cid:15) far from thebifurcation point because additional eigenvalues of L + and L − become negative eigenvaluefor (cid:15) ≈ .
04. Therefore, the stationary state becomes a saddle point of H for fixed Q and E when (cid:15) (cid:38) .
04. The dashed line shows the asymptotic dependence (6.3) for the smalleigenvalue of L + . - ϵ σ + - ϵ σ - Figure 3.
The smallest eigenvalues of L + (left) and L − (right) for the branchbifurcating from the second eigenmode at ω = 3 /
35 with normalization λ − ω = 1. Remark . The presence of zero eigenvalue in the spectrum of L + at (cid:15) ≈ .
04 on Figure 3singles out new bifurcation of the stationary state along the branch. We have checked thatthe numerical results are stable with respect to truncation. In the present time, it is not
TATIONARY STATES IN THE CONFORMAL FLOW 35 clear how to identify new solution branches which may branch off at one or both sides of thebifurcation point.
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Institute of Physics, Jagiellonian University, Krak´ow, Poland
E-mail address : [email protected] Institute of Physics, Jagiellonian University, Krak´ow, Poland
E-mail address : [email protected] Department of Mathematics, McMaster University, Hamilton, Ontario L8S 4K1, Canada
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