aa r X i v : . [ m a t h . A P ] J a n BREATHER SOLUTIONS OF THE CUBIC KLEIN-GORDON EQUATION
DOMINIC SCHEIDER
Abstract.
We obtain real-valued, time-periodic and radially symmetric solutions of thecubic Klein-Gordon equation ∂ t U − ∆ U + m U = Γ( x ) U on R × R , which are weakly localized in space. Various families of such “breather” solutions are shownto bifurcate from any given nontrivial stationary solution. The construction of weakly lo-calized breathers in three space dimensions is, to the author’s knowledge, a new conceptand based on the reformulation of the cubic Klein-Gordon equation as a system of couplednonlinear Helmholtz equations involving suitable conditions on the far field behavior. Introduction and Main Results
We construct real-valued solutions U ( t, x ) of the cubic Klein-Gordon equation(1) ∂ t U − ∆ U + m U = Γ( x ) U on R × R where Γ ∈ L ∞ rad ( R ) ∩ C ( R ) and m > ∂ , , , ∇ , ∆ , D refer to differential operators acting on the space variables. The solutions weaim to construct are polychromatic, that is, they take the form U ( t, x ) = u ( x ) + ∞ X k =1 ωkt ) u k ( x ) = X k ∈ Z e i ωkt u k ( x )(2) where u k ∈ X = n u ∈ C rad ( R , R ) (cid:12)(cid:12) (cid:13)(cid:13)(cid:13) (1 + | · | ) u (cid:13)(cid:13)(cid:13) ∞ < ∞ o ⊆ L ( R ) , u − k = u k and (for simplicity) ω > m. Such solutions are periodic in time and localized as well as radially symmetric in space. Theyare sometimes referred to as breather solutions, c.f. the “Sine-Gordon breather” in [1], equa-tion (28). The construction of breather solutions is of particular interest since, as indicatedin a study [7] on perturbations of the Sine-Gordon breather, Birnir, McKean and Weinsteinconjecture that “for the general nonlinear wave equation [author’s note: in 1+1 dimensions],breathing [...] takes place only for isolated nonlinearities”, see [7, p.1044]. This conjectureis supported by recent existence results for breathers for the 1+1 dimensional wave equation
Date : January 14, 2020.2010
Mathematics Subject Classification.
Primary: 35L71, 35B32, Secondary: 35B10, 35J05.
Key words and phrases.
Klein-Gordon equation, breather, bifurcation, Helmholtz equation. with specific, carefully designed potentials which we comment on below. Our results, how-ever, indicate that the situation might be entirely different for weakly localized breathers forthe Klein-Gordon equation in 1+3 dimensions, in the sense that such breather solutions areabundant even in “simple” settings.We will find breather solutions of (1) with u k k ∈ N by rewriting it into an infinite system of stationary equations for the functions u k . Indeed,inserting (2), a short and formal calculation leads to − ∆ u + m u = Γ( x ) ( u ⋆ u ⋆ u ) , (3a) − ∆ u k − ( ω k − m ) u k = Γ( x ) ( u ⋆ u ⋆ u ) k for k ∈ Z \ { } . (3b)In fact, (3b) includes (3a), but we intend to separate the “Schr¨odinger” equation characterizedby 0 σ ( − ∆ + m ) from the infinite number of “Helmholtz” equations characterized by0 ∈ σ ( − ∆ − ( ω k − m )), k = 0. Our construction of breathers for (1) relies on newmethods for such Helmholtz equations introduced in [19] which exploit the so-called far fieldproperties of their solutions and lead to a rich bifurcation structure. These methods will besketched only briefly in the main body of this paper; more details will be given in Section 4at the end (which can be read independently).The solutions we obtain bifurcate from any given stationary (radial) solution w ∈ X , w w solves the stationary nonlinear Schr¨odingerequation − ∆ w + m w = Γ( x ) w on R ;(4)regarding existence of such w , cf. Remark 1 (b). Let us remark briefly that all (distribu-tional) solutions of (4) in X ⊆ L ( R ) are twice differentiable by elliptic regularity. Inorder to make bifurcation theory work, we impose the following nondegeneracy assumption: q ∈ X , − ∆ q + q = 3Γ( x ) w q on R implies q ≡ . (5)We comment on this assumption in Remark 1 (c) below. In particular, (5) and our mainresult presented next hold if Γ is constant and w is a (positive) ground state of (4). We nowpresent our main result. Theorem 1.
Let Γ ∈ L ∞ rad ( R ) ∩ C ( R ) , ω > m > and assume there is some stationarysolution U ( t, x ) = w ( x ) , w of the cubic Klein-Gordon equation (1) , i.e. w ∈ X solving (4) . Assume further that w is nondegenerate in the sense of (5) . Then for every s ∈ N there exist an open interval J s ⊆ R with ∈ J s and a family ( U α ) α ∈ J s ⊆ C ( R , X ) with the following properties:(i) All U α are time-periodic, twice continuously differentiable classical solutions of (1) of the polychromatic form (2) , U α ( t, x ) = u α ( x ) + ∞ X k =1 ωkt ) u αk ( x ) . REATHER SOLUTIONS OF THE CUBIC KLEIN-GORDON EQUATION 3 (ii) The map α ( u αk ) k ∈ N is smooth in the topology of ℓ ( N , X ) with dd α (cid:12)(cid:12)(cid:12)(cid:12) α =0 u αk if and only if k = s (“excitation of the s-th mode”). In particular, for sufficiently small α = 0 , these solu-tions are non-stationary. Moreover, for different values of s , the families of solutionsmutually differ close to U .(iii) If we assume additionally Γ( x ) = 0 for almost all x ∈ R , then every nonstationarypolychromatic solution U α possesses infinitely many nonvanishing modes u αk . Remark 1. (a) We require continuity of Γ since we use the functional analytic frameworkof [19]. The existence and continuity of ∇ Γ will be exploited in proving that U α istwice differentiable. This assumption as well as Γ = 0 almost everywhere in (iii)might be relaxed; however, this study does not aim at the most general setting for thecoefficients but rather focuses on the introduction of the setup for the existence result.(b) The existence of stationary solutions of the Klein-Gordon equation (1) resp. of solu-tions to (4) can be guaranteed under additional assumptions on Γ . We refer to [18],Theorem I.2 and Remarks I.5, I.6 by Lions for positive (ground state) solutions andto Theorems 2.1 of [5], [4] by Bartsch and Willem for bound states.(c) In some special cases, nondegeneracy properties like (5) have been verified, e.g. byBates and Shi [6] in Theorem 5.4 (6), or by Wei [25] in Lemma 4.1, both assumingthat w is a ground state solution of (4) in the autonomous case with constant positive Γ . It should be pointed out that, although the quoted results discuss nondegeneracy ina setting on the Hilbert space H ( R ) , the statements can be adapted to the topologyof X , as we will demonstrate in Lemma 1.(d) The assumption ω > m on the frequency ensures that the stationary system (3) con-tains only one equation of Schr¨odinger type. This avoids further nondegeneracy as-sumptions on higher modes, which would not be covered by the previously mentionedresults in the literature.(e) The above result provides, locally, a multitude of families of breathers bifurcating fromevery given stationary solution characterized by different values of s , ω and possiblycertain asymptotic parameters, see Remark 2 below.It would be natural, further, to ask for the global bifurcation picture given some trivialfamily T = { ( w , λ ) | λ ∈ R } . (Here λ ∈ R denotes a bifurcation parameter which inour case is not visible in the differential equation and thus will be properly introducedlater.) Typically, global bifurcation theorems state that a maximal bifurcating contin-uum of solutions ( U, λ ) emanating from T at ( w , λ ) is unbounded unless it returnsto T at some point ( w , λ ′ ) , λ ′ = λ . In the former (desirable) case, however, asatisfactory characterization of global bifurcation structures should provide a criterionwhether or not unboundedness results from another stationary solution w = w with { ( w , λ ) | λ ∈ R } belonging to the maximal continuum. Since it is not obvious at allwhether and how such a criterion might be derived within our framework, we focus DOMINIC SCHEIDER on the local result, which already adds new aspects to the state of knowledge about theexistence of breather solutions summarized next.
An Overview of Literature.
Polychromatic Solutions.
The results in Theorem 1 can and should be compared with recentfindings on breather (that is to say, time-periodic and spatially localized) solutions of thewave equation with periodic potentials V ( x ) , q ( x ) = c · V ( x ) ≥ V ( x ) ∂ t U − ∂ x U + q ( x ) U = Γ( x ) U on R × R . (6)Such breather solutions have been constructed by Schneider et al., see Theorem 1.1 in [8], andHirsch and Reichel, see Theorem 1.3 in [13], respectively. In brief, the main differences to theresults in this article are that the authors of [8], [13] consider a setting in one space dimensionand obtain strongly spatially localized solutions, which requires a comparably huge technicaleffort. We give some details: Both existence results are established using a polychromaticansatz, which reduces the time-dependent equation to an infinite set of stationary problemswith periodic coefficients, see [8], p. 823, resp. [13], equation (1.2). The authors of [8] applyspatial dynamics and center manifold reduction; their ansatz is based on a very explicitchoice of the coefficients q, V, Γ. The approach in [13] incorporates more general potentialsand nonlinearities and is based on variational techniques. It provides ground state solutions,which are possibly “large” - in contrast to our local bifurcation methods, which only yieldsolutions close to a given stationary one as described in Theorem 1, i.e. with a typically“small” time-dependent contribution.Periodicity of the potentials in (6) is explicitly required since it leads to the occurrence ofspectral gaps when analyzing the associated differential operators of the stationary equations.In contrast to the Helmholtz methods introduced here, the authors both of [8] and of [13]strive to construct the potentials in such way that 0 lies in the aforementioned spectral gaps,and moreover that the distance between 0 and the spectra has a positive lower bound. Thisis realized by assuming a certain “roughness” of the potentials, referring to the step potentialdefined in Theorem 1.1 of [8] and to the assumptions (P1)-(P3) in [13] which allow potentialswith periodic spikes modeled by Dirac delta distributions, periodic step potentials or somespecific, non-explicit potentials in H r rad ( R ) with 1 ≤ r < (see [13], Lemma 2.8).Let us summarize that the methods for constructing breather solutions of (6) outlined abovecan handle periodic potentials but require irregularity, are very restrictive concerning theform of the potentials and involve a huge technical effort in analyzing spectral propertiesbased on Floquet-Bloch theory. The Helmholtz ansatz presented in this article providesa technically elegant and short approach suitable for constant potentials; in the context ofbreather solutions, it is new in the sense that it provides breathers with slow decay, it providesbreathers on the full space R , and it provides breathers for simple (constant) potentials. The Klein-Gordon Equation as a Cauchy Problem.
Possibly due to its relevance in physics,there is a number of classical results in the literature concerning the nonlinear Klein-Gordonequation. The fundamental difference to the results in this article is that the vast majority
REATHER SOLUTIONS OF THE CUBIC KLEIN-GORDON EQUATION 5 of these concerns the Cauchy problem of the Klein-Gordon equation, i.e. ∂ t U − ∆ U + m U = ± U on [0 , ∞ ) × R U (0 , x ) = f ( x ) , ∂ t U (0 , x ) = g ( x ) on R (7)for suitable initial data f, g : R → R . Usually, the dependence of the nonlinearity on U ismuch more general (allowing also derivatives of U ) and the space dimension is not restrictedto N = 3. On the other hand, most results in the literature only concern the autonomouscase, which is why we set in this discussion Γ ≡ ± E B [ U ( t, · )] = 12 Z B | ∂ t U ( t, x ) | + |∇ U ( t, x ) | + m | U ( t, x ) | d x + 14 Z B | U ( t, x ) | d x, B ⊆ R is proved provided Γ ≡ −
1. Following a classical strategy for evolution problems, localexistence is shown by means of a fixed point iteration, and global existence can be obtainedby an iteration argument based on energy conservation. For Γ ≡ +1, Theorem 1.4 due toKeller and Levine demonstrates the existence of blow-up solutions.During the following decade, Klainerman [15,16] and Shatah [22,23] independently developednew techniques leading to significant improvements in the study of uniqueness questionsand of the asymptotic behavior of solutions as t → ∞ . These results work in settingswith high regularity and admit more general nonlinearities with growth assumptions forsmall arguments, which includes the cubic one as a special case. In particular, Klainermanand Shatah prove the convergence to solutions of the free Klein-Gordon equation and showuniform decay rates of solutions as t → ∞ . In the case of a cubic nonlinearity, these resultsonly apply if the space dimension is at least 2. This is why, more recently, the questionof corresponding uniqueness and convergence properties for cubic nonlinearities in N = 1space dimensions has attracted attention; we wish to mention at least some of the relatedpapers. For explicit choices of the cubic nonlinearity, there are results by Moriyama and byDelort, see Theorem 1.1 of [20] resp. Th´eor`emes 1.2, 1.3 in [9]. Only the latter result allowsa nonlinearity of the form ± U not containing derivatives (see [9], Remarque 1.4); however,the initial data are assumed to have compact support. Global existence, uniqueness, decayrates and scattering exclusively for the nonlinearity ± U can be found in Corollary 1.2 of [12]by Hayashi and Naumkin.The relation to our results is not straightforward since the bifurcation methods automaticallyprovide solutions U α which exist globally in time irrespective of the sign (or even of a possible x -dependence) of Γ and which do not decay as t → ∞ , and there is no special emphasis onthe role of the initial values U α (0 , x ) , ∇ U α (0 , x ) along the bifurcating branches. Our methodsinstead focus on several global properties of the solutions U α ( t, x ) such as periodicity in timeand localization as well as decay rates in space, i.e. the defining properties of breathers. DOMINIC SCHEIDER
Research Perspectives.
Apart from bifurcation methods, nonlinear Helmholtz equa-tions and systems can also be discussed in a “dual” variational framework as introduced byEv´equoz and Weth [10]. This might offer another way to analyze the system (3) leadingto “large” breathers in the sense that they are not close to a given stationary solution asthe ones constructed in Theorem 1. Furthermore, such an ansatz might be a promising steptowards extensions to non-constant, e.g. periodic potentials.2.
The Proof of Theorem 1
The Functional-Analytic Setting.
We look for polychromatic solutions as in (2)with coefficients u = ( u k ) k ∈ Z ∈ X where X := ℓ ( Z , X ) := ( ( u k ) k ∈ Z (cid:12)(cid:12)(cid:12)(cid:12) u k = u − k ∈ X , k ( u k ) k ∈ Z k X := X k ∈ Z k u k k X < ∞ ) . The Banach space X has been defined in (2); it prescribes a decay rate which is the naturalone for solutions of Helmholtz equations as in (3b), see also Section 4. Throughout, wedenote by w = ( δ k, w ) k ∈ Z = ( ..., , w , , ... ) the stationary solution with w ∈ X ∩ C ( R )fixed according to equation (4). We will find polychromatic solutions of (1) by solving thecountably infinite Schr¨odinger-Helmholtz system (3a), (3b), which is equivalent to (1), (2)on a formal level; for details including convergence of the polychromatic sum in (2), seeProposition 3.Our strategy is then as follows: ⊲ Intending to apply bifurcation techniques, we have to analyze the linearized versionof the infinite-dimensional system (3a), (3b), which resembles the one of the two-component system discussed by the author in [19]. We therefore summarize, forthe reader’s convenience, a collection of results concerning the linearized setting inProposition 2. ⊲ We then present a suitable setup for bifurcation theory; in particular, we introduce abifurcation parameter which is not visible in the differential equation but appears inthe so-called far field of the functions u k , more specifically a phase parameter in theleading-order contribution as | x | → ∞ . ⊲ The aforementioned fact that solutions of (3a), (3b) obtained in this setting providepolychromatic, classical solutions of the Klein-Gordon equation (1) will be proved asa part of Proposition 3 below. Indeed, regarding differentiability, we will see that thechoice of suitable asymptotic conditions will ensure uniform convergence and hencesmoothness properties of the infinite sums defining the polychromatic states. ⊲ Finally, in Proposition 4, we essentially verify the assumptions of the Crandall-Rabinowitz Bifurcation Theorem.After that, we are able to give a very short proof of Theorem 1. The auxiliary results will beproved in Section 3. The final Section 4 provides some more details on the theory of linearHelmholtz equations in X . REATHER SOLUTIONS OF THE CUBIC KLEIN-GORDON EQUATION 7
Throughout, we denote the convolution in R by the symbol ∗ and use ⋆ in the convolutionalgebra ℓ . Extending the notation defined above, for q ≥
0, we let X q := n u ∈ C rad ( R , R ) | k u k X q < ∞ o with k u k X q := sup x ∈ R (1 + | x | ) q/ | u ( x ) | , X q := ℓ ( Z , X q ) with k u k X q := k ( u k ) k k X q := X k ∈ Z k u k k X q . Proposition 1.
The convolution of sequences u (1) , u (2) , u (3) ∈ X is well-defined in a point-wise sense and satisfies u (1) ⋆ u (2) ⋆ u (3) ∈ X . Moreover, we have the estimate (cid:13)(cid:13) u (1) ⋆ u (2) ⋆ u (3) (cid:13)(cid:13) X ≤ (cid:13)(cid:13) u (1) (cid:13)(cid:13) X (cid:13)(cid:13) u (2) (cid:13)(cid:13) X (cid:13)(cid:13) u (3) (cid:13)(cid:13) X . We rewrite the system (3a), (3b) using u = w + v with w = ( ..., , w , , ... ); then, − ∆ v k − ( ω k − m ) v k = Γ( x ) · (cid:2) (( w + v ) ⋆ ( w + v ) ⋆ ( w + v )) k − δ k, w (cid:3) on R . (8)We will find solutions of this system of differential equations by solving instead a system ofcoupled convolution equations which, for k
6∈ { , ± s } , have the form v k = R τ k µ k [ f k ]. Here f k represents the right-hand side of (8), µ k := ω k − m , and the coefficients τ k ∈ (0 , π )will have to be chosen properly according to a nondegeneracy condition. The convolutionoperators R τµ = sin( | · |√ µ + τ )4 π sin( τ ) | · | ∗ : X → X ( µ > , < τ < π )can be viewed as resolvent-type operators for the Helmholtz equation ( − ∆ − µ ) v = f on R involving an asymptotic condition on the far field of the solution v , namely | x | v ( x ) ∼ sin( | x |√ µ + τ ) + O (cid:18) | x | (cid:19) as | x | → ∞ . Such conditions are required since the homogeneous Helmholtz equation ( − ∆ − µ ) v = 0 hassmooth nontrivial solutions in X (known as Herglotz waves), which are all multiples of˜Ψ µ ( x ) := sin( | x |√ µ )4 π | x | ( x = 0) . We refer to Section 4, more precisely Lemma 5, for details; the case τ = 0 requires a largertechnical effort and is presented in Lemma 6. This involves linear functionals α ( µ ) , β ( µ ) ∈ X ′ which, essentially, yield the coefficients of the sine resp. cosine terms in the asymptoticexpansion above. Relying on these tools and notations, we summarize the relevant facts onthe linearized versions of the Helmholtz equations (3b) in the following Proposition. Proposition 2.
Let w ∈ X be a solution of equation (4) with Γ ∈ L ∞ rad ( R ) ∩ C loc ( R ) and ω > m > ; define µ k := ω k − m . For every k ∈ Z \ { } , there exists (up to a multiplicativeconstant) a unique nontrivial and radially symmetric solution q k ∈ X of − ∆ q k − µ k q k = 3 Γ( x ) w ( x ) q k on R . (9a) DOMINIC SCHEIDER
It is twice continuously differentiable and satisfies, for some c k = 0 and σ k ∈ [0 , π ) , q k ( x ) = c k · sin( | x | √ µ k + σ k ) | x | + O (cid:18) | x | (cid:19) as | x | → ∞ . (9b) The equations (9a) , (9b) are equivalent to the convolution identities ( q k = 3 R σ k µ k [Γ w q k ] = 3 (cid:16) R µ k [Γ w q k ] + cot( σ k ) ˜ R µ k [Γ w q k ] (cid:17) if σ k ∈ (0 , π ) ,q k = 3 R π/ µ k [Γ w q k ] + (cid:0) α ( µ k ) ( q k ) + β ( µ k ) ( q k ) (cid:1) · ˜Ψ µ k if σ k = 0 . For all k ∈ Z , cos( σ k ) β ( µ k ) ( q k ) = sin( σ k ) α ( µ k ) ( q k ) . The existence statement and the asymptotic properties in (9) can be proved using the Pr¨ufertransformation, see [19], Proposition 6; the statements in the second part are consequencesof Lemmas 5 and 6 in the final Section 4. For these results to apply we have assumed initiallythat Γ is continuous and bounded, whence 3 Γ w ∈ X .We now present the general assumptions valid throughout the following construction and theproof of Theorem 1. We let σ k for k ∈ Z \ { } as in Proposition 2 above and fix s ∈ N ,recalling that we aim to “excite the s -th mode” in the sense of Theorem 1 (ii). With this,let us introduce(10) τ ± s := σ ± s , τ k := ( π if σ k = π , π if σ k = π for k ∈ Z \ { , ± s } , see also Remark 2 (b). Thus in particular τ k = σ k for k ∈ Z \ { , ± s } , and we conclude fromthe uniqueness statement in Proposition 2 the nondegeneracy property(11a) k ∈ Z \ { , ± s } , q ∈ X , q = 3 R τ k µ k [Γ w q ] ⇒ q ≡ P µ = ( − ∆ + µ ) − : X → X (see Lemma 7), thecorresponding property is assumed in (5):(11b) q ∈ X , q = 3 P µ [Γ w q ] ⇒ q ≡ . We now introduce a map the zeros of which provide solutions of the system (8). Throughout,we use the shorthand notation u = v + w for v ∈ X and the stationary solution w =( ..., , w , , ... ). As above, we have to distinguish the cases τ s ∈ (0 , π ) and τ s = 0. (In thefollowing, please recall that we consider some fixed s = 0.) For 0 < τ ± s < π , we introduce F : X × R → X via(12a) F ( v , λ ) k := v k − P µ [Γ ( u ⋆ u ⋆ u ) − Γ w ] k = 0 , R π/ µ s (cid:2) Γ ( u ⋆ u ⋆ u ) ± s (cid:3) +(cot( τ ± s ) − λ ) ˜Ψ µ s ∗ (cid:2) Γ ( u ⋆ u ⋆ u ) ± s (cid:3) k = ± s, R τ k µ k [Γ ( u ⋆ u ⋆ u ) k ] else . REATHER SOLUTIONS OF THE CUBIC KLEIN-GORDON EQUATION 9
Similarly, if σ s = 0, we define G : X × R → X by(12b) G ( v , λ ) k := v k − P µ [Γ ( u ⋆ u ⋆ u ) − Γ w ] k = 0 , R π/ µ s (cid:2) Γ ( u ⋆ u ⋆ u ) ± s (cid:3) +(1 − λ ) (cid:0) α ( µ s ) ( v ± s ) + β ( µ s ) ( v ± s ) (cid:1) ˜Ψ µ s k = ± s, R τ k µ k [Γ ( u ⋆ u ⋆ u ) k ] else . The following result collects some basic properties of the maps F and G and the polychromaticstates related to their zeros. Proposition 3.
Let s ∈ N and ( τ k ) k ∈ Z be chosen as in (10) . The maps F, G : X × R → X are well-defined and smooth with F ( , λ ) = G ( , λ ) = for all λ ∈ R . Further, if F ( v , λ ) = resp. G ( v , λ ) = for some v ∈ X , λ ∈ R , then v solves the stationary system (8) and U ( t, x ) := w ( x ) + v ( x ) + ∞ X k =1 ωkt ) v k ( x ) ( t ∈ R , x ∈ R ) defines a twice continuously differentiable, classical solution U ∈ C ( R , X ) of the Klein-Gordon equation (1) . Again, the proof can be found in Section 3. We will even show that U ∈ C ∞ ( R , X ). For thederivatives of F resp. G with respect to the Banach space component v ∈ X , we will verifythe following explicit formulas: Letting q ∈ X and abbreviating u := v + w ,( DF ( v , λ )[ q ]) k = q k − P µ [Γ ( q ⋆ u ⋆ u ) ] k = 0 , R π/ µ s (cid:2) Γ ( q ⋆ u ⋆ u ) ± s (cid:3) +3 (cot( τ s ) − λ ) ˜Ψ µ s ∗ (cid:2) Γ ( q ⋆ u ⋆ u ) ± s (cid:3) k = ± s, R τ k µ k [Γ ( q ⋆ u ⋆ u ) k ] else;(13a) ( DG ( v , λ )[ q ]) k = q k − P µ [Γ ( q ⋆ u ⋆ u ) ] k = 0 , R π/ µ s (cid:2) Γ ( q ⋆ u ⋆ u ) ± s (cid:3) +(1 − λ ) (cid:0) α ( µ s ) ( q ± s ) + β ( µ s ) ( q ± s ) (cid:1) ˜Ψ µ s k = ± s, R τ k µ k [Γ ( q ⋆ u ⋆ u ) k ] else . (13b) Remark 2. (a) As earlier announced, we now see that the bifurcation parameter λ ap-pears only in the asymptotic expansions of the s -th components v ± s of the solutionsand not in the differential equation (1) . This is different from [19] where the bifurca-tion parameter takes the role of a coupling parameter of the Helmholtz system.(b) The choice of the parameters τ k in equation (10) is far from unique. Indeed, one couldinstead consider any configuration satisfying τ k = τ − k = σ k for all k ∈ Z \ {± s } , { τ k | k ∈ Z \ {± s }} ⊆ ( δ, π − δ ) for some δ ∈ (cid:0) , π (cid:1) . The former condition is required for the nondegeneracy state-ment (11a) , and the latter will be used to obtain uniform decay estimates in the proofof Proposition 3, see Lemma 3. However, as in [19], the question whether another choice of τ k leads to different bi-furcating families is still open. Hence we discuss only the explicit choice in (10) . In the so-established framework, we intend to apply the Crandall-Rabinowitz BifurcationTheorem. The next result shows that its assumptions are satisfied.
Proposition 4 (Simplicity and transversality) . Let s ∈ N and ( τ k ) k ∈ Z be chosen as in (10) .The linear operator DF ( ,
0) : X → X is 1-1-Fredholm with a kernel of the form ker DF ( ,
0) = span { q } where q k = 0 if and only if k = ± s. Moreover, the transversality condition is satisfied, that is, ∂ λ DF ( , q ] ran DF ( , . A corresponding statement holds true for DG ( ,
0) : X → X . The Proof of Theorem 1.
Let us fix some s ∈ N , and choose ( τ k ) k ∈ Z as in (10). Weintroduce the trivial family T := { ( , λ ) ∈ X × R | λ ∈ R } . Step Proof of (i).
By Proposition 3, the maps F resp. G are smooth and vanish on the trivial family T . In viewof Proposition 4, the Crandall-Rabinowitz Theorem shows that ( , ∈ T is a bifurcationpoint for F ( v , λ ) = 0 resp. G ( v , λ ) = 0 and provides an open interval J s ⊆ R containing 0and a smooth curve J s → X × R , α ( v α , λ α ) = (( v αk ) k ∈ Z , λ α )of zeros of F resp. G (we do not denote its dependence on s ) with v = , λ = 0 as wellas dd α (cid:12)(cid:12) α =0 v α = q where q is a nontrivial element of the kernel of DF ( ,
0) resp. DG ( , u α := v α + w and define polychromatic states U α as in (i). Then U α is a classicalsolution of the cubic Klein-Gordon equation (1) due to Proposition 3 since F ( v α , λ α ) = 0resp. G ( v α , λ α ) = 0. By their very definition, the solutions U α are time-periodic with period2 π/ω (maybe less). This proves (i). Step Proof of (ii).
Since F resp. G are smooth, so is the map J s → X × R , α ( v α , λ α ). By Proposition 4, q k = 0 if and only if k = ± s , which implies that only the ± s -th components ofdd α (cid:12)(cid:12)(cid:12)(cid:12) α =0 u α = dd α (cid:12)(cid:12)(cid:12)(cid:12) α =0 v α = q REATHER SOLUTIONS OF THE CUBIC KLEIN-GORDON EQUATION 11 do not vanish. For sufficiently small nonzero values of α , the solutions U α are thus nonsta-tionary. In particular, the direction of bifurcation changes when changing the value of s , andthe associated bifurcating curves are, at least locally, mutually different. Step Proof of (iii).
We show finally that, under the additional assumption that Γ( x ) = 0 for almost all x ∈ R ,every non-stationary solution U α ( t, x ) = w ( x ) + v α ( x ) + ∞ X k =1 ωkt ) v αk ( x )in fact possesses infinitely many nontrivial coefficients v αk . Indeed, assuming the contrary,we can choose a maximal r > U α is non-stationary) with v αr u αr = v αr + w r = v αr
0. But then, v α r = X l + m + n =3 r R τ r µ r [Γ u αl u αm u αn ] = R τ r µ r [Γ ( v αr ) ] − ∆ v α r − µ r v α r = Γ( v αr ) , and Γ( v αr ) x ) = 0almost everywhere by assumption. This contradicts the maximality of r . (cid:3) The Proof of Remark 1 (c).
Finally, as announced in Remark 1 (c), we verify thenondegeneracy assumption (11b) resp. (5) for constant positive Γ.
Lemma 1 (Nondegeneracy, `a la Bates and Shi [6]) . Let Γ ≡ Γ for some Γ > , andassume that w ∈ C ( R ) is a radially symmetric solution of (4) the profile of which satisfies w ( r ) > , w ′ ( r ) < for all r > , and both w ( r ) and w ′ ( r ) decay exponentially as r → ∞ .Then the nondegeneracy property (5) holds, i.e. for any radial, twice differentiable q ∈ X − ∆ q + q = 3Γ w q on R implies q ≡ . This can be proved closely following the line of argumentation by Bates and Shi [6], Theo-rem 5.4 (6). The main difference is that they state the nondegeneracy result as a spectralproperty of the operator − ∆ + m + 3Γ w : H ( R ) → L ( R ) whereas we cannot use theHilbert space setting but discuss solutions in X . However, the technique of Bates and Shi(and also of Wei’s proof in [25]) is based on an expansion at a fixed radius r > L ( S ). This provides coefficientsdepending on r , and the conclusions are obtained from the analysis of these profiles on anODE level using results due to Kwong and Zhang [17]. These ideas apply in the topology of X in the very same way; for details, cf. [21], (proof of) Lemma 4.11.3. Proofs of the Auxiliary Results
Proof of Proposition 1.
Let u ( j ) = ( u ( j ) k ) k ∈ Z ∈ X for j = 1 , ,
3. We find the following chain of inequalities (cid:13)(cid:13) u (1) ⋆ u (2) ⋆ u (3) (cid:13)(cid:13) X = X k ∈ Z (cid:13)(cid:13) ( u (1) ⋆ u (2) ⋆ u (3) ) k (cid:13)(cid:13) X ≤ X k ∈ Z X l,m,n ∈ Z l + m + n = k (cid:13)(cid:13)(cid:13) u (1) l u (2) m u (3) n (cid:13)(cid:13)(cid:13) X ≤ X k ∈ Z X l,m,n ∈ Z l + m + n = k (cid:13)(cid:13)(cid:13) u (1) l (cid:13)(cid:13)(cid:13) X (cid:13)(cid:13) u (2) m (cid:13)(cid:13) X (cid:13)(cid:13) u (3) n (cid:13)(cid:13) X = (cid:13)(cid:13)(cid:13)(cid:13)(cid:18)(cid:13)(cid:13)(cid:13) u (1) l (cid:13)(cid:13)(cid:13) X (cid:19) l ∈ Z ⋆ (cid:16)(cid:13)(cid:13) u (2) m (cid:13)(cid:13) X (cid:17) m ∈ Z ⋆ (cid:16)(cid:13)(cid:13) u (3) n (cid:13)(cid:13) X (cid:17) n ∈ Z (cid:13)(cid:13)(cid:13)(cid:13) ℓ ( Z ) ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:18)(cid:13)(cid:13)(cid:13) u (1) l (cid:13)(cid:13)(cid:13) X (cid:19) l ∈ Z (cid:13)(cid:13)(cid:13)(cid:13) ℓ ( Z ) (cid:13)(cid:13)(cid:13)(cid:16)(cid:13)(cid:13) u (2) m (cid:13)(cid:13) X (cid:17) m ∈ Z (cid:13)(cid:13)(cid:13) ℓ ( Z ) (cid:13)(cid:13)(cid:13)(cid:16)(cid:13)(cid:13) u (3) n (cid:13)(cid:13) X (cid:17) n ∈ Z (cid:13)(cid:13)(cid:13) ℓ ( Z ) = (cid:13)(cid:13) u (1) (cid:13)(cid:13) X (cid:13)(cid:13) u (2) (cid:13)(cid:13) X (cid:13)(cid:13) u (3) (cid:13)(cid:13) X , where finally Young’s inequality for convolutions in ℓ ( Z ) has been applied. Since the latterterm is finite, we infer u (1) ⋆ u (2) ⋆ u (3) ∈ X . (cid:3) Proof of Proposition 3.
Step Decay estimates
The proof of Proposition 3 requires convergence properties in order to handle the infiniteseries in the definition of U ( t, x ), which we first provide in the following two lemmas. Lemma 2.
The convolution operators R τµ : X → X satisfy for τ ∈ (0 , π ) and µ > ∀ f ∈ X (cid:13)(cid:13) R τµ [ f ] (cid:13)(cid:13) X ≤ C sin( τ ) (cid:18) √ µ (cid:19) · k f k X , (cid:13)(cid:13) R τµ [ f ] (cid:13)(cid:13) L ( R ) ≤ C √ µ sin( τ ) · k f k L ( R ) . The fact that a power of µ appears in the denominator is crucial since it will finally providethe convergence and regularity of the polychromatic sums where µ = µ k = ω k − m for k ∈ Z .The proof of Lemma 2 relies, via rescaling, on the respective estimates for µ = 1. Thesecan be found in [19], pp. 1038–1039 for the X - X estimate and in [10], Theorem 2.1 for the L / - L estimate. Lemma 3.
Let Γ ∈ L ∞ rad ( R ) ∩ C ( R ) and assume u = ( u k ) k ∈ Z ∈ X is a sequence of C functions which satisfy the following system of convolution equations: u k = R τ k µ k [Γ ( u ⋆ u ⋆ u ) k ] for all k ∈ Z with | k | > s where µ k = ω k − m and τ k ∈ ( δ, π − δ ) for some ω > m, δ ∈ (cid:0) , π (cid:1) . Then there holds:(i) For every α ≥ , there exists a constant C α ≥ with k u k k L ( R ) + k Γ ( u ⋆ u ⋆ u ) k k L ( R ) ≤ C α · ( k + 1) − α ( k ∈ Z ) . REATHER SOLUTIONS OF THE CUBIC KLEIN-GORDON EQUATION 13 (ii) For every ball B = B R (0) ⊆ R and α ≥ there exists a constant D α ( B ) ≥ with | u k ( x ) | + |∇ u k ( x ) | + | D u k ( x ) | ≤ D α ( B ) · ( k + 1) − α ( k ∈ Z , x ∈ B ) . (iii) For every α ≥ , there exists a constant E α ≥ with k u k k X ≤ E α · ( k + 1) − α ( k ∈ Z ) . The proof of Lemma 3 can also be found in detail in the author’s PhD thesis [21], Lemma 4.13.We present here the most important step and summarize the remainder briefly, since it ismainly based on the application of (standard) elliptic regularity estimates.As u k ∈ X ∩ C ( R ) for all k ∈ Z by assumption, it is straightforward to find constants asin the lemma for a finite number of elements u − s , ..., u s . Hence it is sufficient to study those k ∈ Z with | k | > s ; for these, we have µ k = k ω − m ≥ c s ( k + 1) for some positive c s > ω and m . The decay estimates of arbitrary order in k we aim toprove essentially go back to the L / - L scaling property stated in Lemma 2 above. Indeed,due to δ < τ k < π − δ , it provides C = C ( k Γ k ∞ , δ, ω, m, s ) ≥ k u k k L ( R ) ≤ C ( k + 1) k ( u ⋆ u ⋆ u ) k k L ( R ) for all k ∈ Z . (14)With that, assuming P k ∈ Z ( k + 1) α k u k k L ( R ) < ∞ for some α ≥ α = 0 since u ∈ X ), one can iterate as follows X k ∈ Z ( k + 1) α +1 / k u k k L ( R ) (14) ≤ C X k ∈ Z ( k + 1) α k ( u ⋆ u ⋆ u ) k k L ( R ) ≤ C X k ∈ Z X l + m + n = k (( l + m + n ) + 1) α k u l k L ( R ) k u m k L ( R ) k u n k L ( R ) ≤ α C X k ∈ Z X l + m + n = k h ( l + 1) α k u l k L ( R ) ( m + 1) α k u m k L ( R ) ( n + 1) α k u n k L ( R ) i = 2 α C X k ∈ Z ( k + 1) α k u k k L ( R ) ! < ∞ . This shows the first part of the estimate in (i), and the second part follows by combining theformer with the interpolation estimate k u l u m u n k L ( R ) ≤ k u l k L ( R ) k u m k L ( R ) k u n k L ( R ) ≤ h k u l k L ( R ) k u m k L ( R ) k u n k L ( R ) i h k u l k ∞ k u m k ∞ k u n k ∞ i ≤ h k u l k L ( R ) k u m k L ( R ) k u n k L ( R ) i k u k X . The local estimate in (ii) can be derived from the global L bounds in (i) using ellipticregularity, which first provides estimates in W , ( R ) and then is suitable H¨older spaces. The estimate (iii) in the X norm essentially uses the explicit representations (given f ∈ X ) R τµ [ f ]( x ) = Z R sin( | x − y |√ µ k + τ k )4 π | x − y | sin( τ k ) · f ( y ) d y = sin( | x |√ µ k + τ k ) | x | sin( τ k ) Z | x | sin( r √ µ k ) r √ µ k f ( r ) r d r + sin( | x |√ µ k ) | x | sin( τ k ) Z ∞| x | sin( r √ µ k + τ k ) r √ µ k f ( r ) r d r. Starting here, H¨older’s inequality and (i) yield (iii); again, for details, cf. [21].
Step Mapping properties of F resp. G . For λ ∈ R and v ∈ X , we set u := w + v and recall the defining equations (12a) and (12b): F ( v , λ ) k := v k − P µ [Γ ( u ⋆ u ⋆ u ) − Γ w ] k = 0 , R π/ µ s (cid:2) Γ ( u ⋆ u ⋆ u ) ± s (cid:3) +(cot( τ ± s ) − λ ) ˜Ψ µ s ∗ (cid:2) Γ ( u ⋆ u ⋆ u ) ± s (cid:3) k = ± s, R τ k µ k [Γ ( u ⋆ u ⋆ u ) k ] else; G ( v , λ ) k := v k − P µ [Γ ( u ⋆ u ⋆ u ) − Γ w ] k = 0 , R π/ µ s (cid:2) Γ ( u ⋆ u ⋆ u ) ± s (cid:3) +(1 − λ ) (cid:0) α ( µ s ) ( v ± s ) + β ( µ s ) ( v ± s ) (cid:1) ˜Ψ µ s k = ± s, R τ k µ k [Γ ( u ⋆ u ⋆ u ) k ] else . Our main concern will be convergence of the infinite sums related to the space X = ℓ ( Z , X ). Noticing that F and G only differ in the ± s -th component, and that the scalarparameter λ only appears as a multiplicative factor, we solely discuss smoothness of the map F ( · , λ ) : X → X with λ ∈ R fixed.The main tool is the following uniform norm estimate for the operators appearing in thecomponents of F . Recalling that τ k ∈ { π , π } for k = 0 , ± s by (10), Lemma 2 above (for k = 0 , ± s ) as well as the continuity properties stated in Lemmas 4 and 7 (for k = ± s and k = 0, respectively) provide a constant C = C ( λ, τ s , ω, m ) > (cid:13)(cid:13) R τ k µ k (cid:13)(cid:13) L ( X ,X ) ≤ C ( k ∈ Z \ {± s } ) , (cid:13)(cid:13) R π/ µ s (cid:13)(cid:13) L ( X ,X ) ≤ C , (cid:13)(cid:13)(cid:13) (cot( τ ± s ) − λ ) ˜Ψ µ s ∗ (cid:13)(cid:13)(cid:13) L ( X ,X ) ≤ C , kP µ k L ( X ,X ) ≤ C . (15)We now let v ∈ X and define u = v + w . Since Γ is assumed to be continuous andbounded, Proposition 1 implies that Γ ( u ⋆ u ⋆ u ) ∈ X . Thus every component F ( v , λ ) k isa well-defined element of X , and we estimate k F ( v , λ ) k X = X k ∈ Z k F ( v , λ ) k k X (15) ≤ k v k X + C (cid:13)(cid:13) Γ w (cid:13)(cid:13) X + C X k ∈ Z k Γ ( u ⋆ u ⋆ u ) k k X REATHER SOLUTIONS OF THE CUBIC KLEIN-GORDON EQUATION 15
Prop. ≤ k v k X + C k Γ k ∞ k w k X + C k Γ k ∞ k u k X . This is finite, hence F ( v , λ ) ∈ X as asserted. Since F ( · , λ ) is a combination of continuouslinear operators and polynomials in the convolution algebra, essentially the same estimatescan be used to show differentiability (to arbitrary order); one thus obtains in particular (13a). Step Solution properties of u k ( x ) . First of all, recalling that w = ( ..., , w , , ... ) and hence ( w ⋆ w ⋆ w ) k = δ k, w for k ∈ Z ,one can immediately see that F ( , λ ) = G ( , λ ) = for all λ ∈ R . Let us now assume that F ( v , λ ) = 0 resp. G ( v , λ ) = 0 for some v ∈ X and λ ∈ R . Again, we define u := v + w , andsummarize u − w = v = P µ (cid:2) Γ ( u ⋆ u ⋆ u ) − Γ w (cid:3) ,u ± s = v ± s = R π/ µ s (cid:2) Γ ( u ⋆ u ⋆ u ) ± s (cid:3) + ( (cot( τ s ) − λ ) ˜Ψ µ s ∗ (cid:2) Γ ( u ⋆ u ⋆ u ) ± s (cid:3) , (1 − λ ) (cid:0) α ( µ s ) ( v ± s ) + β ( µ s ) ( v ± s ) (cid:1) ˜Ψ µ s ,u k = v k = R τ k µ k [Γ ( u ⋆ u ⋆ u ) k ] ( k ∈ Z \ { , ± s } ) . By choice of τ k in equation (10), we observe in particular that the requirements of Lemma 3are satisfied with any δ < π , which we will rely on throughout the subsequent steps. Butfirst, according to Lemmas 4 and 7, v k , u k ∈ X ∩ C ( R ) satisfy the differential equations − ∆ v k − µ k v k = Γ( x ) (cid:2) ( u ⋆ u ⋆ u ) k − δ k, w (cid:3) on R or equivalently, in view of w = ( ..., , w , , ... ), of (4) and of µ k = ω k − m , − ∆ u k − ( ω k − m ) u k = Γ( x ) ( u ⋆ u ⋆ u ) k on R . (16)We now define formally for t ∈ R , x ∈ R U ( t, x ) := w ( x ) + v ( x ) + ∞ X k =1 ωkt ) v k ( x ) = X k ∈ Z e i ωkt u k ( x ) . (17)Since by assumption u = v + w ∈ ℓ ( Z , X ), the Weierstrass M-test asserts that the sumin (17) converges in X uniformly with respect to t ∈ R , and hence the map t U ( t, · ) iscontinuous as a map from R to X . We next show stronger regularity properties of U ( t, x ). Step Differentiability of U ( t, x ) . We prove that the map t U ( t, · ), when interpreted as a map from R to X , possesses twocontinuous time derivatives given by ∂ t U ( t, · ) = X k ∈ Z i ωk e i ωkt u k , ∂ t U ( t, · ) = X k ∈ Z − ω k e i ωkt u k . Indeed, term-by-term differentiation is justified since the sums above as well as in (17) con-verge in X uniformly with respect to time. This is a consequence of the Weierstraß M-testand the decay estimate in Lemma 3 (iii). Hence, as asserted, the map t U ( t, · ) is twicecontinuously differentiable as a map from R to X - the same strategy yields in fact C ∞ regularity in time. Similarly, the local regularity estimate in Lemma 3 (ii) implies U ∈ C ( R × B ) for everygiven ball B = B R (0) ⊆ R again via term-by-term differentiation. Since the radius of theball B is arbitrary, we conclude for t ∈ R and all x ∈ R (cid:2) ∂ t − ∆ + m (cid:3) U ( t, x ) = X k ∈ Z e i ωkt (cid:2) − ω k − ∆ + m (cid:3) u k ( x ) (16) = X k ∈ Z e i ωkt Γ( x ) X l + m + n = k u l ( x ) u m ( x ) u n ( x )= Γ( x ) X l ∈ Z e i ωlt u l ( x ) ! X m ∈ Z e i ωmt u m ( x ) ! X n ∈ Z e i ωnt u n ( x ) ! = Γ( x ) U ( t, x ) where the re-ordering of the summation is justified by absolute convergence of the sums.Thus U is shown to be a classical solution of the Klein-Gordon equation (1). (cid:3) Proof of Proposition 4.
We prove the statement for the map F and then comment on the aspects that differ in case of G . Using formula (13a), we find for k ∈ Z and q ∈ X , recalling that w k = 0 for k ∈ Z \ { } and that R τ s µ s = R π/ µ s + cot( τ s ) ˜Ψ µ s ∗ , DF ( , q ] k = q k − R τ k µ k [Γ ( q ⋆ w ⋆ w ) k ] = q k − R τ k µ k (cid:2) Γ w · q k (cid:3) ,DF ( , q ] = q − P µ [Γ ( q ⋆ w ⋆ w ) ] = q − P µ (cid:2) Γ w · q (cid:3) . For q ∈ ker DF ( , τ k in (10), the nondegeneracy properties (11)imply q k ≡ k ∈ Z , k = ± s . Since τ ± s = σ s in (10), Proposition 2 guarantees the existenceof a nontrivial solution q s ∈ X of q s = 3 R τ s µ s (cid:2) Γ w · q s (cid:3) (18)which is unique up to a multiplicative factor. Hence ker DF ( ,
0) has the asserted form. (Werecall here that we consider the subspace of symmetric sequences, whence q − s = q s .) Further,by Lemmas 4 and 7 in the final Section 4, the operators X → X , ( q k q k − R τ k µ k [Γ w · q k ] ( k = 0) q q − P µ [Γ w · q ]are linear compact perturbations of the identity and so ker DF ( ,
0) is 1-1-Fredholm. Inorder to verify transversality, we compute for k ∈ Z and q ∈ ker DF ( , \ { } ∂ λ DF ( , q ] k = ( µ s ∗ [Γ w q s ] , k = ± s, , else . Assuming for contradiction that ∂ λ DF ( , q ] = DF ( , p ] for some p ∈ X , we infer inparticular that the component p s satisfies the convolution identity p s − R τ s µ s (cid:2) Γ w · p s (cid:3) = 3 ˜Ψ µ s ∗ [Γ w · q s ](19) REATHER SOLUTIONS OF THE CUBIC KLEIN-GORDON EQUATION 17 and hence, following Lemmas 4, 5 − ∆ p s − µ s p s = 3 Γ( x ) w ( x ) p s on R , which is also nontrivially solved by q s as a consequence of (18). Due to the uniquenessstatement in Proposition 2, this implies that p s = c · q s for some c ∈ R . But then, applying (18)to (19), we obtain ˜Ψ µ s ∗ [Γ w · q s ] = 0. Hence by the asymptotic expansion in Lemma 4 \ Γ w q s ( √ µ s ) = 0and therefore, due to q s = 3 R τ s µ s [Γ w q s ] and Lemma 5, q s ( x ) = O (cid:18) | x | (cid:19) as | x | → ∞ . This contradicts Proposition 2 stating that the leading-order term as | x | → ∞ of a nontrivialsolution q s of − ∆ q s − µ s q s = 3 Γ( x ) w ( x ) q s cannot vanish.In the case τ s = 0, we see as above that q ∈ ker DG ( ,
0) if and only if q k = 0 for k = ± s ,and that q s = q − s can be chosen to be the (nontrivial) solution of q s = 3 R π/ µ s (cid:2) Γ w · q s (cid:3) + α ( µ s ) ( q s ) ˜Ψ µ s with β ( µ s ) ( q s ) = 0 . (20)Similarly, ker DG ( ,
0) is 1-1-Fredholm. We again assume for contradiction that there is p ∈ X with ∂ λ DG ( , q ] = DG ( , p ], which implies in particular(21) p s − R π/ µ s (cid:2) Γ w · p s (cid:3) − (cid:0) α ( µ s ) ( p s ) + β ( µ s ) ( p s ) (cid:1) ˜Ψ µ s = α ( µ s ) ( q s ) ˜Ψ µ s with β ( µ s ) ( q s ) = 0. Thus, according to Lemma 4, p s solves the differential equation − ∆ p s − µ s p s = 3 Γ( x ) w ( x ) p s on R , which is also solved by q s , see equation (20). As before, the uniqueness property in Proposi-tion 2 implies p s = c · q s for some c ∈ R , and inserting this into the identity (21), comparisonwith (20) yields α ( µ s ) ( q s ) = 0. Since also β ( µ s ) ( q s ) = 0, we infer from the definition ofthe functionals α ( µ s ) , β ( µ s ) preceding Lemma 6 that, again, q s ( x ) = O (1 / | x | ), contradictingProposition 2. (cid:3) Appendix: Stationary Linear Helmholtz and Schr¨odinger Equations
Given µ >
0, we study aspects of the solution theory of the linear equations − ∆ u ± µu = f on R . (22)In the case of a “+”, equation (22) is said to be a Schr¨odinger equation. Given any right-handside f ∈ L ( R ), a unique solution u ∈ H ( R ) can be obtained by applying the resolvent( − ∆ + µ ) − , which can be calculated explicitly by applying the Fourier transform u = ( − ∆ + µ ) − f = Z R ˆ f ( ξ ) | ξ | + µ e i h · ,ξ i d ξ (2 π ) / . In the case of a Helmholtz equation, i.e. of a “ − ” sign in (22), this is not possible since µ > − ∆ on R . A well-established strategy to find solutionsin spaces other than L ( R ) is known as Limiting Absorption Principle(s). The idea is to replace µ by µ + i ε , apply an L -resolvent, and pass to the limit ε → u = “ lim ε ց ” ( − ∆ − ( µ + i ε )) − f = “ lim ε ց ” Z R ˆ f ( ξ ) | ξ | − ( µ + i ε ) e i h · ,ξ i d ξ (2 π ) / . Using tools from harmonic analysis, such a construction of solutions of linear inhomogeneousHelmholtz equations has been successfully done by Agmon [2] in weighted L spaces, andby Kenig, Ruiz and Sogge [14] as well as Guti´errez [11] in certain pairs of L p spaces. Theresolvent-type operator is, then, for sufficiently nice f , given by a convolution u = e i | · |√ µ π | · | ∗ f. Such studies are completed by characterizations of the so-called Herglotz waves, i.e. thesolutions of the homogeneous equation − ∆ u − µu = 0 on the respective spaces, see e.g. [3].We study the case of (real-valued, radial) functions f ∈ X , u ∈ X with the Banach spaces X q := n v ∈ C rad ( R ) (cid:12)(cid:12) k v k X q := (cid:13)(cid:13)(cid:13) (1 + | · | ) q v (cid:13)(cid:13)(cid:13) ∞ < ∞ o , q ∈ { , } . These have been successfully applied in solving systems of cubic Helmholtz equations in [19].Let us again point out that the decay rate prescribed in X is the natural one for solutionsof Helmholtz equations on the full space R . Such solutions of the Helmholtz equation − ∆ u − µu = f on R (23)can be obtained using convolution operators with kernels Ψ µ , ˜Ψ µ given byΨ µ ( x ) = cos( | x |√ µ )4 π | x | , ˜Ψ µ ( x ) = sin( | x |√ µ )4 π | x | ( x ∈ R \ { } ) . Here Ψ µ , ˜Ψ µ are radial solutions of the homogeneous Helmholtz equation on R \ { } . Wenotice that ˜Ψ µ extends to a smooth solution of − ∆ u − µu = 0 in X and it is, up to constantmultiples, the only one. Moreover, the following holds: Lemma 4 ( [19], Proposition 4) . The convolution operators f Ψ µ ∗ f , f ˜Ψ µ ∗ f arewell-defined, linear and compact as operators from X to X . Moreover, given f ∈ X , thefunctions w := Ψ µ ∗ f and ˜ w := ˜Ψ µ ∗ f belong to X ∩ C ( R ) and satisfy − ∆ w − µw = f on R , w ( x ) = 4 π r π f ( √ µ ) Ψ µ ( x ) + O (cid:18) | x | (cid:19) ; − ∆ ˜ w − µ ˜ w = 0 on R , ˜ w ( x ) = 4 π r π f ( √ µ ) ˜Ψ µ ( x ) . Here ˆ f ( √ µ ) refers to the profile of the Fourier transform on R . Working in a radial settingwith strongly decaying inhomogeneities f ∈ X , the properties in the previous Lemma (andin the following ones) can be verified immediately by explicit calculations and need notbe derived from suitable Limiting Absorption Principles; for details, we refer to the earlierarticle [19]. REATHER SOLUTIONS OF THE CUBIC KLEIN-GORDON EQUATION 19
The study of conditions guaranteeing uniqueness of solutions of (23) in X involves thecharacterization of Herglotz waves in X , which are all multiples of ˜Ψ µ . As in [19], inspiredby the analysis of the so-called far field of solutions of Helmholtz equations in scatteringtheory, we impose asymptotic conditions governing the leading-order contribution of u ( x ) as | x | → ∞ . For τ ∈ (0 , π ), we introduce R τµ [ f ] = Ψ µ ∗ f + cot( τ ) ˜Ψ µ ∗ f = sin( | · |√ µ + τ )4 π sin( τ ) | · | ∗ f. Then, using the above Lemma 4, one obtains:
Lemma 5 ( [19], Corollary 5) . Let τ ∈ (0 , π ) and µ > . Then the operator R τµ : X → X is well-defined, linear and compact. Moreover, given f ∈ X , we have u = R τµ [ f ] if and onlyif u ∈ C with − ∆ u − µu = f on R , u ( x ) = c · sin( | x |√ µ + τ ) | x | + O (cid:18) | x | (cid:19) as | x | → ∞ for some c ∈ R , and in this case c = τ ) p π ˆ f ( √ µ ) . Handling the case of far field conditions with τ = 0 is somewhat more delicate since theexistence of the solution ˜Ψ µ (which satisfies exactly this condition) excludes an analogousuniqueness statement. For proving Theorem 1, the following setting is suitable. First, bythe Hahn-Banach Theorem, we define continuous linear functionals α ( µ ) , β ( µ ) ∈ X ′ with theproperty that, for u ∈ X with u ( x ) = α u · ˜Ψ µ ( x ) + β u · Ψ µ ( x ) + O (cid:18) | x | (cid:19) as | x | → ∞ , we have α ( µ ) ( u ) = α u and β ( µ ) ( u ) = β u , cf. [19], equation (13) and the following explanations.Then, the following analogue of Lemma 5 holds. Lemma 6.
Given f ∈ X , we have u = R π/ µ [ f ] + ( α ( µ ) ( u ) + β ( µ ) ( u )) · ˜Ψ µ if and only if u ∈ C with − ∆ u − µu = f on R , u ( x ) = c · sin( | x |√ µ ) | x | + O (cid:18) | x | (cid:19) as | x | → ∞ for some c ∈ R . In this case, β ( µ ) ( u ) = 0 . These results will allow to handle the nonlinear Helmholtz equations in (3b); for the proofs,we refer to the corresponding parts of [19]. Nonlinear Schr¨odinger equations such as − ∆ u + µu = f on R (24)for some µ > Lemma 7.
Let µ > . Then the operator P µ : X → X , f e −| · |√ µ π | · | ∗ f is well-defined, linear and compact. Moreover, given f ∈ X , we have u := P µ [ f ] ∈ X ∩ C ( R ) , and u is a solution in X of − ∆ u + µu = f on R . For details on the proof, which is similar to that of Lemma 4 but with less difficulties due tothe strongly localized kernel, cf. [21], Lemma 4.10.Let us remark that, in the Schr¨odinger case, we do not obtain a family of possible “resolvent-type” operators as R τ = R + cot( τ ) ˜ R , 0 < τ < π , in the Helmholtz case. This is due to thefact that the homogeneous Schr¨odinger equation − ∆ u + µu = 0 has no smooth and localizednontrivial solution in X . In particular, a major consequence in our study of Klein-Gordonbreathers is that we have to impose nondegeneracy of w as an assumption rather than, asin the Helmholtz case, generate it by choosing an appropriate resolvent R τ , as will be donein (10), (11a) below. Acknowledgements
Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) –project-id 258734477 – SFB 1173.The construction of breather solutions of a similar wave-type equation is a major result ofthe author’s dissertation thesis and can be found partly verbatim in [21, Chapter 4]. Specialthanks goes in particular to my PhD advisor Dr. Rainer Mandel who encouraged me to workon this topic and provided advice whenever asked.
References [1] M. J. Ablowitz, D. J. Kaup, A. C. Newell, and H. Segur. Method for Solving the Sine-Gordon Equation.
Phys. Rev. Lett. , 30 : 1262–1264, 1973.[2] S. Agmon. Spectral properties of Schr¨odinger operators and scattering theory.
Annali della Scuola Nor-male Superiore di Pisa - Classe di Scienze , Ser. 4, 2 (2): 151–218, 1975.[3] S. Agmon.
A Representation Theorem for Solutions of the Helmholtz Equation and Resolvent Estimatesfor The Laplacian . Academic Press, 1990.[4] T. Bartsch and M. Willem. Infinitely Many Nonradial Solutions of a Euclidean Scalar Field Equation.
Journal of Functional Analysis , 117 (2): 447–460, 1993.[5] T. Bartsch and M. Willem. Infinitely many radial solutions of a semilinear elliptic problem on R n . Archivefor Rational Mechanics and Analysis , 124 (3): 261–276, 1993.[6] P. W. Bates and J. Shi. Existence and instability of spike layer solutions to singular perturbation prob-lems.
Journal of Functional Analysis , 196 (2): 211–264, 2002.[7] B. Birnir, H. McKean, and A. Weinstein. The rigidity of sine-gordon breathers.
Communications onPure and Applied Mathematics , 47 (8): 1043–1051, 1994.[8] C. Blank, M. Chirilus-Bruckner, V. Lescarret, and G. Schneider. Breather Solutions in Periodic Media.
Communications in Mathematical Physics , (3): 815–841, 2011.[9] J.-M. Delort. Existence globale et comportement asymptotique pour l’´equation de Klein–Gordon quasilin´eaire `a donn´ees petites en dimension 1.
Annales Scientifiques de l’ ´Ecole Normale Sup´erieure , 34 (1):1–61, 2001.[10] G. Ev´equoz and T. Weth. Dual variational methods and nonvanishing for the nonlinear Helmholtzequation.
Advances in Mathematics , 280 : 690–728, 2015.[11] S. Guti´errez. Non trivial L q solutions to the Ginzburg-Landau equation. Mathematische Annalen , 328 (1):1–25, 2004.
REATHER SOLUTIONS OF THE CUBIC KLEIN-GORDON EQUATION 21 [12] N. Hayashi and P. I. Naumkin. The initial value problem for the cubic nonlinear Klein-Gordon equation.
Zeitschrift f¨ur angewandte Mathematik und Physik , 59 (6): 1002–1028, 2008.[13] A. Hirsch and W. Reichel. Real-valued, time-periodic localized weak solutions for a semilinear waveequation with periodic potentials.
Nonlinearity , 32 (4): 1408–1439, 2019.[14] C. E. Kenig, A. Ruiz, and C. D. Sogge. Uniform Sobolev inequalities and unique continuation for secondorder constant coefficient differential operators.
Duke Math. J. , 55 (2): 329–347, 1987.[15] S. Klainerman. Long-time behavior of solutions to nonlinear evolution equations.
Archive for RationalMechanics and Analysis , 78 (1): 73–98, 1982.[16] S. Klainerman. Global existence of small amplitude solutions to nonlinear Klein-Gordon equations infour space-time dimensions.
Communications on Pure and Applied Mathematics , 38 (5): 631–641, 1985.[17] M. K. Kwong and L. Q. Zhang. Uniqueness of the positive solution of ∆ u + f ( u ) = 0 in an annulus. Differential and Integral Equations , 4 (3): 583–599, 1991.[18] P.-L. Lions. The concentration-compactness principle in the calculus of variations. The locally compactcase, part 2.
Annales de l’I.H.P. Analyse non lin´eaire , 1 (4): 223–283, 1984.[19] R. Mandel and D. Scheider. Bifurcations of nontrivial solutions of a cubic Helmholtz system.
Advancesin Nonlinear Analysis , 9 (1): 1026–1045, 2019.[20] K. Moriyama. Normal forms and global existence of solutions to a class of cubic nonlinear Klein-Gordonequations in one space dimension.
Differential Integral Equations , 10 (3): 499–520, 1997.[21] D. Scheider.
On a Nonlinear Helmholtz System . PhD thesis, Karlsruher Institut f¨ur Technologie (KIT),2019.[22] J. Shatah. Global existence of small solutions to nonlinear evolution equations.
Journal of DifferentialEquations , 46 (3): 409–425, 1982.[23] J. Shatah. Normal forms and quadratic nonlinear Klein-Gordon equations.
Communications on Pureand Applied Mathematics , 38 (5): 685–696, 1985.[24] W. A. Strauss. Nonlinear invariant wave equations. In G. Velo and A. S. Wightman, editors,
InvariantWave Equations , pages 197–249. Springer Berlin Heidelberg, 1978.[25] J. Wei. On the Construction of Single-Peaked Solutions to a Singularly Perturbed Semilinear DirichletProblem.
Journal of Differential Equations , 129 (2): 315–333, 1996.
D. ScheiderKarlsruhe Institute of TechnologyInstitute for AnalysisEnglerstraße 2D-76131 Karlsruhe, Germany
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