aa r X i v : . [ m a t h . A P ] F e b A UNIQUENESS RESULT FOR THE SINE-GORDON BREATHER
RAINER MANDEL
Abstract.
In this note we prove that the sine-Gordon breather is the only quasimonochro-matic breather in the context of nonlinear wave equations in R N . Introduction
Breathers are time-periodic and spatially localized patterns that describe the propagation ofwaves. The most impressive solution of this kind is the so-called sine-Gordon breather forthe 1D sine-Gordon equation ∂ tt u − ∂ xx u + sin( u ) = 0 in R × R . It is given by the explicit formula u ∗ ( x, t ) = 4 arctan (cid:18) m sin( ωt ) ω cosh( mx ) (cid:19) for ( x, t ) ∈ R × R , (1)where the parameters m, ω > m + ω = 1. It is natural to ask if other real-valuedbreather solutions exist. We shall address this question in the broader context of more generalnonlinear wave equations of the form(2) ∂ tt u − ∆ u = g ( u ) in R N × R , where the space dimension N ∈ N and the nonlinearity g : R → R are arbitrary.The existence of radially symmetric breather solutions for the cubic Klein-Gordon equation g ( z ) = − m z + z , m > u ( · , t ) ∈ L q ( R N ) forsome q ∈ (2 , ∞ ) but u ( · , t ) / ∈ L ( R N ). In [10] infinitely many weakly localized breathers werefound for nonlinearities Q ( x ) | u | p − u where Q lies in a suitable Lebesgue space and p > Q as well as the space dimension N ≥
2. Up to now, nothingis known about the existence of strongly localized breathers of (2) satisfying u ( · , t ) ∈ L ( R N )for almost all t ∈ R and N ≥
2, see however [11] for a an existence result for semilinear curl-curl equations for N = 3. In the case N = 1 strongly localized breather solutions differentfrom the sine-Gordon breather have been found for nonlinear wave equations of the form s ( x ) ∂ tt u − u xx + q ( x ) u = f ( x, u ) ( x ∈ R )where the coefficient functions s, q are discontinuous and periodic, see [7, Theorem 1.3] and [1,Theorem 1.1]. Given the discontinuity of s, q it must be expected that these breathers arenot twice continuously differentiable. To sum up, the existence of smooth and strongly Date : February 24, 2021. localized breather solutions of (2) different from the sine-Gordon breather is not known. Stillfor N = 1 there are nonexistence results by Denzler [4] and Kowalczyk, Martel, Mu˜noz [9]dealing with small perturbations of the sine-Gordon equation respectively small odd breathers(not covering the even sine-Gordon breather). We are not aware of any other mathematicallyrigorous existence or nonexistence results for (2).One of the main obstructions for the construction of localized breathers is polychromaticity.Indeed, plugging in an ansatz of the form u ( x, t ) = P k ∈ Z u k ( x ) e ikt with u k = u − k one endsup with infinitely many equations of nonlinear Helmholtz type that typically do not possessstrongly localized solutions, see for instance [8, Theorem 1a]. For this reason the solutionsobtained in [10, 13] are only weakly localized. On the other hand, a purely monochromaticansatz like u ( x, t ) = sin( ωt ) p ( x ) cannot be successful either provided that g is not a linearfunction. In view of the formula (1) for the sine-Gordon breather we investigate whetherquasimonochromatic breathers exist. Definition 1.
We call the function u : R N × R → R a quasimonochromatic breather if u ( x, t ) = F (sin( ωt ) p ( x )) ( x ∈ R N , t ∈ R ) for some ω ∈ R \ { } and nontrivial functions F ∈ C ( R ) , p ∈ C ( R N ) such that F (0) = 0 and p ( x ) → as | x | → ∞ . We show that in one spatial dimension the sine-Gordon breather is, up to translation anddilation, the only one for (2) and that no such breathers exist in higher dimensions as longas g does not act like a linear function. In fact, to rule out L ∞ -small solutions of linear waveequations, we assume that g : R → R is not a linear function near zero, i.e., that there is anontrivial interval I ⊂ R containing 0 with the property that there is no β ∈ R such that g ( z ) = βz for all z ∈ I . Theorem 1.
Assume N ∈ N and that g : R → R is not a linear function near zero.(i) In the case N ≥ there is no quasimonochromatic breather solution of (2) .(ii) In the case N = 1 each quasimonochromatic breather solution of (2) is of the form u ( x, t ) = κu ∗ ( x − x , t ) for x ∈ R , m, ω, κ ∈ R \{ } and u ∗ as in (1) . The nonlinearitythen satisfies g ( z ) = − ( m + ω ) κ sin( κ − z ) whenever | z | < π | κ | . We stress that our result holds regardless of any smoothness assumption on g nor any kindof growth condition at 0 or infinity. Moreover, our considerations are not limited to smallperturbations of u ∗ or small breathers in whatever sense. Following the proof of Theorem 1one also finds that quasimonochromatic breathers of wave equations on any open set Ω ( R N with homogeneous Dirichlet conditions(3) ∂ tt u − ∆ u = g ( u ) in Ω × R , u = 0 on ∂ Ω × R with profile functions p ∈ C (Ω) do not exist either (even if N = 1) provided that g is nota linear function near zero. We will comment on this fact at the end of this paper. Asa consequence, we find that Rabinowitz’ C ([0 , × R )-solutions of the 1D wave equationfrom [12, Theorem 1.6] are not of quasimonochromatic type. This might be true as well for UNIQUENESS RESULT FOR THE SINE-GORDON BREATHER 3 the solutions from [2, 3], but here our argument does not apply in a direct way since thesolutions are not known to be twice continuously differentiable up to the boundary.For completeness we briefly comment on the linear case g ( z ) = βz , β ∈ R . Then the profilefunction p of any given quasimonochromatic breather of (2) satisfies the linear elliptic PDE − ∆ p − ( ω + β ) p = 0 in R N . For β < − ω there are positive, radially symmetric andexponentially decaying solutions p , see [5, Theorem 2]. In the case β > − ω , N ≥ | p ( x ) | + |∇ p ( x ) | & | x | − N in a suitableintegrated sense, see [14, Theorem 1] respectively [8, Theorem 1a]. For β > − ω , N = 1 allsolutions are linear combinations of sin and cos so that breather solutions do not exist. Sowe see that the picture is already quite complete in the case of linear wave equations.2. Proof of Theorem 1
In the following let u ( x, t ) = F (sin( ωt ) p ( x )) be a solution of (2) as in (1) with g as in theTheorem. Plugging in this ansatz we get for all x ∈ R N such that p ( x ) = 0, ∂ tt u ( x, t ) = − ω sin( ωt ) p ( x ) F ′ (sin( ωt ) p ( x )) + ω cos( ωt ) p ( x ) F ′′ (sin( ωt ) p ( x ))= − ω zF ′ ( z ) + ω ( p ( x ) − z ) F ′′ ( z ) , ∆ u ( x, t ) = sin( ωt )∆ p ( x ) F ′ (sin( ωt ) p ( x )) + sin( ωt ) |∇ p ( x ) | F ′′ (sin( ωt ) p ( x ))= ∆ p ( x ) p ( x ) zF ′ ( z ) + |∇ p ( x ) | p ( x ) z F ′′ ( z ) , where z = sin( ωt ) p ( x ) ∈ [ −k p k ∞ , + k p k ∞ ]. This and (2) imply for x ∈ R N , z ∈ R such that p ( x ) = 0 , z ∈ [ −k p k ∞ , + k p k ∞ ] g ( F ( z )) + ω zF ′ ( z ) + ω z F ′′ ( z ) = p ( x ) ω F ′′ ( z ) − ∆ p ( x ) p ( x ) zF ′ ( z ) − |∇ p ( x ) | p ( x ) z F ′′ ( z ) . (4)If F was linear on [ −k p k ∞ , + k p k ∞ ], then g would have to be linear on the nontrivial interval I := { F ( z ) : | z | ≤ k p k ∞ } as well. Since the latter is not the case by assumption, we knowthat z z F ′′ ( z ) does not vanish identically on that interval. Multiplying (4) with p ( x ) andchoosing z according to z F ′′ ( z ) = 0 we find that p does not change sign. Indeed, if p ( x ∗ ) = 0and R > p has a fixed sign in the open ball B R ( x ∗ ), thenHopf’s Lemma [6, Lemma 3.4] implies |∇ p | > ∂B R ( x ∗ ). But then (4) implies that ∆ p isunbounded on ∂B R ( x ∗ ), which contradicts p ∈ C ( R N ). Hence, p does not change sign andwe will without loss of generality assume that p is positive. So (4) holds for all x ∈ R N andall z ∈ [ −k p k ∞ , k p k ∞ ] and standard elliptic regularity theory gives p ∈ C ∞ ( R N ).Differentiating (4) with respect to x i we get(5) ∂ i ( p ( x ) ) ω F ′′ ( z ) − ∂ i (cid:18) ∆ p ( x ) p ( x ) (cid:19) zF ′ ( z ) − ∂ i (cid:18) |∇ p ( x ) | p ( x ) (cid:19) z F ′′ ( z ) = 0 . RAINER MANDEL
Since p is non-constant, we infer that F satisfies an ODE of the form(6) F ′′ ( z ) = − µ zω + µ z F ′ ( z ) ( | z | ≤ k p k ∞ , µ ∈ R , µ ∈ R \ { } ) . Here, µ = 0 is due to the fact that F is not a linear function. Each nontrivial solution ofsuch an ODE satisfies F ′ ( z ) = 0 for almost all z ∈ [ −k p k ∞ , k p k ∞ ]. Combining (5) and (6)we thus infer − ∂ i ( p ( x ) ) µ ω zω + µ z − ∂ i (cid:18) ∆ p ( x ) p ( x ) (cid:19) z + ∂ i (cid:18) |∇ p ( x ) | p ( x ) (cid:19) µ z ω + µ z = 0 . Since (6) holds for all i ∈ { , . . . , N } and z ∈ [ −k p k ∞ , k p k ∞ ], we get − µ ∂ i (cid:18) ∆ p ( x ) p ( x ) (cid:19) + µ ∂ i (cid:18) |∇ p ( x ) | p ( x ) (cid:19) = 0 , − µ ∂ i ( p ( x ) ) − ∂ i (cid:18) ∆ p ( x ) p ( x ) (cid:19) = 0 . Since µ = 0 we can find λ , λ ∈ R such that − µ ∆ pp + µ |∇ p | p = − λ µ + λ µ , − µ p − ∆ pp = − λ . This implies |∇ p | = λ p − µ p , − ∆ p + λ p = µ p . (7)We now use (7) and the positivity of p to show that p is radially symmetric about itsmaximum point x ∈ R N . We concentrate on the case N ≥ N = 1follows from the fact that x u ( x + x ) and x u ( x − x ) solve the same initial valueproblem. Since p vanishes at infinity, we must have λ ≥ p does not change sign, λ ≥
0, see [14, Theorem 1]. Moreover, p attains its maximum at some point x ∈ R N with p ( x ) > , |∇ p ( x ) | = 0 , ∆ p ( x ) ≤
0. This and (7) implies λ , µ > µ ≥
0. Sowe know that (7) holds for λ , µ > , λ , µ ≥ . In the case λ > x , so we are leftwith the case λ = 0.So let use assume λ = 0. Liouville’s Theorem implies that µ = 0 is impossible, so we have µ > α := 1 − µ µ ∈ ( −∞ , α ∈ (0 ,
1) the function ψ ( x ) := p ( x ) α satisfies − ∆ ψ = − α (∆ p ) p α − − α ( α − |∇ p | p α − = α (1 − α ) λ ψ. In view of α (1 − α ) λ > ψ has infinitely many nodaldomains, which contradicts the positivity of ψ . So this case cannot occur. In the case α ∈ ( −∞ ,
0) radial symmetry about x follows once more from [5, Theorem 2], so it remainsto discuss the case α = 0, i.e., µ = µ . Then ψ ( x ) := log( p ( x )) satisfies − ∆ ψ = − (∆ p ) p − + |∇ p | p − = λ ψ UNIQUENESS RESULT FOR THE SINE-GORDON BREATHER 5 and we find as above that ψ has to change sign, which is a contradiction. So we have shownthat p is radially symmetric about x also in the case λ = 0.So we have p ( x ) = p ( | x − x | ) where p ′ ( r ) = λ p ( r ) − µ p ( r ) , p ′ (0) = 0 . Solving this ODE gives p ( r ) = A cosh( mr ) where λ = m , µ = m A − for some A > , m = 0. So − ∆ p + λ p = µ p can only hold for N = 1 as well as λ = m , µ = 2 m A − . Plugging these values into (6) and solving the ODE we get from F (0) = 0 , F F ( z ) = 4 κ arctan (cid:16) mzAω (cid:17) for some κ ∈ R \ { } . This implies that the breather solution is given by u ( x, t ) = F (sin( ωt ) p ( x )) = F (sin( ωt ) p ( | x − x | )) = κu ∗ ( x − x , t )for u ∗ as in (1). So have proved the nonexistence of such breathers for N ≥ g , we combine (6) and (7) toget p ( x ) ω F ′′ ( z ) − ∆ p ( x ) p ( x ) zF ′ ( z ) − |∇ p ( x ) | p ( x ) z F ′′ ( z ) = m ( m z − A ω ) m z + A ω F ′ ( z ) z. So (4) implies g ( F ( z )) = − ω zF ′ ( z ) − ω z F ′′ ( z ) + m ( m z − A ω ) m z + A ω F ′ ( z ) z = ( m + ω )( m z − A ω ) m z + A ω zF ′ ( z )= 4 Amκω ( m + ω )( m z − A ω ) z ( m z + A ω ) . Plugging in z = Aωm tan( y κ ) for | y | < π | κ | we get F ( z ) = y and hence g ( y ) = 4 A ω κ ( m + ω )( A ω tan( y κ ) − A ω ) tan( y κ )( A ω tan( y κ ) + A ω ) = 4 κ ( m + ω )(tan( y κ ) −
1) tan( y κ )(tan( y κ ) + 1) = 4 κ ( m + ω ) (cid:16) sin( y κ ) − cos( y κ ) (cid:17) sin( y κ ) cos( y κ )= − κ ( m + ω ) cos( y κ ) sin( y κ )= − κ ( m + ω ) sin( yκ ) . RAINER MANDEL (cid:3)
Remark 1. (i) We explain why nonlinear quasimonochromatic breathers of (3) with profile functions p ∈ C (Ω) do not exist on open sets Ω ( R N . The arguments presented above revealthat any such breather is given by functions F, p as in Definition 1 such that for all x ∈ Ω , p ( x ) = 0 , | z | ≤ k p k ∞ we have as in (4) g ( F ( z )) + ω zF ′ ( z ) + ω z F ′′ ( z ) = p ( x ) ω F ′′ ( z ) − ∆ p ( x ) p ( x ) zF ′ ( z ) − |∇ p ( x ) | p ( x ) z F ′′ ( z ) . Now fix z ∈ ( −k p k ∞ , k p k ∞ ) such that z F ′′ ( z ) = 0 and choose x ∗ ∈ Ω such that p ( x ∗ ) = 0 . Let R > be largest possible such that | p | is positive in the open ball B R ( x ∗ ) ⊂ Ω . By the homogeneous Dirichlet boundary condition, we know R ≤ dist( x ∗ , ∂ Ω) < ∞ and that p vanishes on ∂B R ( x ∗ ) . So the same argument as inthe above proof (Hopf ’s Lemma) shows that | ∆ p | is unbounded on B R ( x ∗ ) , a contra-diction. As a consequence, such a profile function cannot exist and we obtain thenonexistence of quasimonochromatic breathers for (3) .(ii) In our proof we did not use the assumption p ( x ) → as | x | → ∞ when we provedthat | p | is positive. As a consequence, each profile function p of a solution u ( x, t ) = F (sin( ωt ) p ( x )) of (2) has a fixed sign regardless of its behaviour at infinity. Sim-ilarly, (7) holds without this hypothesis. So we conclude that any profile function p ∈ C ( R N ) of a quasimonochromatic breather is a positive solution of (7) providedthat the nonlinearity g is not a linear function on the interval { F ( z ) : | z | ≤ k p k ∞ } .Notice also that the assumption F (0) = 0 is not used either.(iii) Our notion of a quasimonochromatic breather does not allow for the solutions u ( x, t ) = u ∗ ( x , t ) ( x ∈ R N ), which are localized only with respect to one spatial direction.Accordingly, our nonexistence result for N ≥ is false under the weaker requirement (8) sup x ′ ∈ R N − | p ( x , x ′ ) | → as x → ∞ . One may conjecture that the solutions u ( x, t ) = u ∗ ( x · θ, t ) for θ ∈ S N − ⊂ R N are theonly quasimonochromatic breathers that are localized in some spatial direction. Thisopen problem bears some similarity to the Gibbon’s Conjecture or de Giorgi Conjectureabout the classification of monotone solutions of the Allen-Cahn equation ∆ u + u = u in R N that we recast in our setting below. Conjecture 1.
Let N ∈ N , N ≥ and let p ∈ C ( R N ) be a solution of (7) for some λ , λ , µ , µ ∈ R that satisfies (8) . Then there are γ, m, z ∈ R such that p ( x ) = γ cosh( m ( x − z )) . Conjecture 2.
Let N ∈ N , N ≥ and let p ∈ C ( R N ) be a solution of (7) for some λ , λ , µ , µ ∈ R that satisfies ∂ p ( x ) x < for all x ∈ R N such that x = 0 . Then there are UNIQUENESS RESULT FOR THE SINE-GORDON BREATHER 7 γ, m > such that p ( x ) = γ cosh( mx ) . Acknowledgements
Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) -Project-ID 258734477 - SFB 1173.
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R. MandelKarlsruhe Institute of TechnologyInstitute for AnalysisEnglerstraße 2D-76131 Karlsruhe, Germany
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