Linear instability and uniqueness of the peaked periodic wave in the reduced Ostrovsky equation
aa r X i v : . [ m a t h . A P ] J a n LINEAR INSTABILITY AND UNIQUENESS OF THE PEAKED PERIODICWAVE IN THE REDUCED OSTROVSKY EQUATION
ANNA GEYER AND DMITRY E. PELINOVSKY
Abstract.
Stability of the peaked periodic wave in the reduced Ostrovsky equation has re-mained an open problem for a long time. In order to solve this problem we obtain sharp boundson the exponential growth of the L norm of co-periodic perturbations to the peaked periodicwave, from which it follows that the peaked periodic wave is linearly unstable. We also provethat the peaked periodic wave with parabolic profile is the unique peaked wave in the space ofperiodic L functions with zero mean and a single minimum per period. Introduction
We address solutions of the Cauchy problem for the reduced Ostrovsky equation [33] writtenin the form(1.1) (cid:26) u t + uu x = ∂ − x u, t > ,u | t =0 = u , where u is a 2 π -periodic function with zero mean defined in the Sobolev space H s per ( − π, π ) forsome s ≥
0, which we simply write as H s per . We denote the subspace of 2 π -periodic functionswith zero mean in H s per by ˙ H s per . The operator ∂ − x : ˙ H s per → ˙ H s +1per denotes the anti-derivativewith zero mean, which can be defined using Fourier series.The reduced Ostrovsky equation is also known under the names of Ostrovsky–Hunter andOstrovsky–Vakhnenko equation, due to contributions of Hunter [26] and Vakhnenko [43].Local solutions to the Cauchy problem (1.1) with u ∈ ˙ H s per exist for s > [39], and werefer to [32] for a discussion on how the well-posedness in H s ( R ) is extended to ˙ H s per . Forsufficiently large initial data, the local solutions break in finite time, similar to the inviscidBurgers equation [32]. However, if the initial data u is suitably small, then the local solutionsfor s = 3 are continued for all times [19, 20]. Weak bounded solutions with shock discontinuitieswere constructed in [7, 8]. Weak solutions of the Cauchy problem (1.1) as the limiting solutionof the Cauchy problem for the regularized Ostrovsky equation were considered in [6].The reduced Ostrovsky equation with smooth solutions is completely integrable as it can bereduced to the integrable Tzizeica equation by a coordinate transformation [29]. This propertyenables a construction of a bi-infinite set of conserved quantities in the time evolution [5] andthe inverse scattering transform with the Riemann–Hilbert approach [1]. Two integrable semi-discretizations of the reduced Ostrovsky equation have been obtained by using bilinear forms[16]. Date : January 3, 2019.
Key words and phrases.
Peaked periodic wave, reduced Ostrovsky equation, characteristics, semigroup,instability.
Stability of smooth and peaked periodic waves in the reduced Ostrovsky equation has been re-cently addressed in a number of publications [14, 18, 21, 22, 38]. By using higher-order conservedquantities the smooth small-amplitude periodic waves were shown in [14] to be unconstrainedminimizers of a higher-order energy function. This result holds for subharmonic perturbations,that is, perturbations whose period is a multiple of the period of the smooth periodic waves.Since the higher-order conserved quantities are well-defined in the space ˙ H , where globalwell-posedness has been proven [20], it follows from the minimization properties that smoothsmall-amplitude periodic waves are both spectrally and orbitally stable. The minimization prop-erties were confirmed numerically for smooth periodic waves of large amplitude all the way up tothe limiting peaked wave of parabolic profile with maximal amplitude, for which the numericalresults were inconclusive [14].Spectral stability of smooth periodic waves with respect to co-periodic perturbations, that is,perturbations with the same period as the period of the periodic wave, was shown in [18] by usingthe standard variational formulation of the periodic waves as critical points of energy subjectto fixed momentum. This result holds also for the generalized reduced Ostrovsky equation withpower nonlinearity. Independently, spectral stability of smooth periodic waves in the reducedOstrovsky equation was shown in [22] by using a coordinate transformation of the spectralstability problem to an eigenvalue problem studied earlier in [38].Regarding the peaked periodic waves, some conflicting results were recently obtained. In [22],the peaked wave with the parabolic profile was addressed and claimed to be “unstable in theabsence of periodic boundary conditions”. A formal proof of this statement was obtained byconstructing explicit solutions of the spectral stability problem for a positive (unstable) eigen-value. However, this construction violates the periodic boundary conditions on the perturbationand hence does not provide an answer to the spectral stability question. In contrast, familiesof peaked periodic waves of small amplitude, which were previously unknown in the contextof the reduced Ostrovsky equation, were constructed in [21] and these families were shown tobe spectrally stable with respect to co-periodic perturbations by using the same coordinatetransformation as in [38].In this paper we give a simple and definite conclusion about existence, uniqueness and stabilityof peaked periodic waves in the reduced Ostrovsky equation. This is the first time, to the best ofour knowledge, that linear instability of peaked periodic waves is proven by means of semigrouptheory and energy estimates.The following theorem presents a summary of the main results of this paper. See Definitions1, 3 and Lemmas 2, 7 for precise statements. Theorem 1. (1)
Uniqueness:
The peaked periodic wave U ∗ with parabolic profile is the unique (up tospatial translations) peaked travelling wave solution of the reduced Ostrovsky equationin ˙ L having a single minimum per period. The solution is Lipschitz continuous andexists in ˙ H s per with s < / . Moreover, the reduced Ostrovsky equation does not admitany H¨older continuous solutions. (2) Instability:
The orbit generated by spatial translations of the peaked periodic wave U ∗ is linearly unstable with respect to perturbations in X , where (1.2) X := { v ∈ ˙ L : ( c ∗ − U ∗ ) v ∈ H } and c ∗ is the wave speed of the periodic wave U ∗ . NIQUENESS AND INSTABILITY OF THE PEAKED PERIODIC WAVE 3
Part (1) of Theorem 1 allows us to prove that the families of peaked periodic small-amplitudewaves constructed in [21] do not satisfy the reduced Ostrovsky equation, see Remark 6. Ouranalysis relies on Fourier theory and the existence of a first integral. Indeed, the reducedOstrovsky equation for smooth periodic waves can be rewritten as a second-order differentialequation with a conserved quantity. Although this equivalence can not be used when dealingwith peaked periodic waves, we can still use a first-order invariant of the second-order differentialequation to analyze the behavior of the smooth parts of the peaked periodic waves together withsharp estimates of the solution at the singularity, see Remark 4 and Lemma 2.Part (2) of Theorem 1 gives a definite conclusion on linear instability of the peaked periodicwave with parabolic profile with respect to co-periodic perturbations. We do not make anyclaims regarding the spectral stability problem related to the peaked periodic wave, see Remark19. Instead, we prove linear instability of the peaked periodic waves by obtaining sharp boundson the exponential growth of the L norm of the co-periodic perturbations in the linearizedtime-evolution problem in X , see Lemma 7. Note that ˙ H is continuously embedded into X but is not equivalent to X , see Remark 8.It is interesting to compare peaked periodic waves in the reduced Ostrovsky equation withpeaked waves in other related nonlinear dispersive equations such as the Whitham equation andthe Camassa–Holm equation. The existence of smooth periodic travelling waves in the Whithamequation has recently been established by [4, 10, 11, 12], where it was shown that the familyof smooth periodic waves terminates at the highest, peaked wave, similarly to what happensfor the reduced Ostrovsky equation. It was shown numerically in [36] that smooth periodicwaves of small amplitude are stable while smooth waves of large amplitude become unstable,even before reaching the highest wave. This is different from the reduced Ostrovsky equation,where all smooth periodic waves are stable even for large amplitudes up to the peaked wave, see[14], whereas the peaked periodic wave is unstable. For the Camassa-Holm equation, both thesmooth periodic waves of all amplitudes and the limiting peaked periodic wave are stable, see[30, 31] and the earlier result [9] on peakons. It is an open question to understand which precisemechanisms govern these surprisingly different stability behaviours.The paper is organized as follows. Section 2 contains the proof that the peaked wave withparabolic profile is unique up to spatial translations in the space of functions in ˙ L with asingle minimum per period. Section 3 gives the proof of linear instability of the peaked periodicwave with respect to co-periodic perturbations.2. Peaked periodic wave
The periodic travelling waves in the reduced Ostrovsky equation are given by u ( x, t ) = U ( x − ct ) , where c is the wave speed and U is a bounded 2 π -periodic wave profile with zero mean. Thewave profile U is to be found from the boundary-value problem(2.1) (cid:26) [ c − U ( z )] U ′ ( z ) + ( ∂ − z U )( z ) = 0 , for every z ∈ ( − π, π ) such that U ( z ) = c,U ( − π ) = U ( π ) , R π − π U ( z ) dz = 0 , where z = x − ct is the travelling wave coordinate. If U ∈ ˙ L , then ∂ − z U ∈ ˙ H . By Sobolev’sembedding, it follows that ∂ − z U ∈ C per so that the anti-derivative ∂ − z U with zero mean can be ANNA GEYER AND DMITRY E. PELINOVSKY expressed by the pointwise formula(2.2) ( ∂ − z U )( z ) = Z z U ( z ′ ) dz ′ − π Z π − π Z z U ( z ′ ) dz ′ dz, z ∈ [ − π, π ] . In what follows, we assume that U is at least continuous on [ − π, π ], that is, we assume that U ∈ C per . For α ∈ (0 , C α per be the space of α -H¨older 2 π -periodic continuous functionssuch that(2.3) | U ( x ) − U ( y ) | ≤ K | x − y | α , for all x, y ∈ [ − π, π ] , for some K ∈ R . We will adopt the following definition of single-lobe periodic waves. Definition 1.
We say that U ∈ C per is a single-lobe periodic wave if there exists z ∈ ( − π, π )such that U is non-increasing on [ − π, z ] and non-decreasing on [ z , π ]. Remark . Due to the condition U ( − π ) = U ( π ) and the symmetry of the equation( c − U ( z )) U ′ ( z ) + Z z U ( z ′ ) dz ′ − π Z π − π Z z U ( z ′ ) dz ′ dz = 0with respect to the reflection z
7→ − z , the single-lobe periodic waves in Definition 1 have evenprofile U with z = 0. In this case, ( ∂ − z U )( z ) = R z U ( z ′ ) dz ′ is odd.A family of smooth 2 π -periodic waves to the boundary-value problem (2.1) satisfying U ( z ) < c for every z ∈ [ − π, π ] was constructed in our previous work [18] in an open interval of the speedparameter c . By Theorem 1(a) and Lemma 3 in [18], we have the following result. Lemma 1.
There exists c ∗ > such that for every c ∈ (1 , c ∗ ) , the boundary-value problem (2.1)admits a unique smooth periodic wave in the sense of Definition 1 with the profile U ∈ ˙ H ∞ per satisfying U ( z ) < c for every z ∈ [ − π, π ] .Remark . For the smooth periodic waves U ∈ ˙ H ∞ per to the boundary-value problem (2.1), theperiodic boundary conditions are satisfied for all derivatives of U .At c = c ∗ , the periodic wave with parabolic profile has been known since the original workof Ostrovsky [33]. It is easy to check that the boundary-value problem (2.1) is satisfied by U ( z ) = ( z − c ) /
6, whereas the zero mean condition is satisfied if c = c ∗ := π /
9. This yieldsthe exact expression for the peaked periodic wave with zero mean(2.4) U ∗ ( z ) := 3 z − π , z ∈ [ − π, π ] , periodically continued beyond [ − π, π ]. Note that U ∗ ( ± π ) = π / c ∗ and ± U ′∗ ( ± π ) = π . Thepeaked periodic wave (2.4) can be represented by the Fourier cosine series U ∗ ( z ) = ∞ X n =1 − n n cos( nz ) , which is well defined in ˙ H s per for s < / Remark . The enclosed angle at the peak of the wave is steeper than the maximal 120 ◦ angle ofthe Stokes wave of greatest height, see [41, 42]. Indeed, for peaked periodic waves of the reducedOstrovsky equation the enclosed angle is ϕ = π − π/ ϕ = π − π/ NIQUENESS AND INSTABILITY OF THE PEAKED PERIODIC WAVE 5
Remark . The peaked periodic wave (2.4) belongs to solutions of the boundary-value problem(2.1) with profile U ∗ ∈ ˙ H satisfying U ∗ ( z ) < c for every z ∈ ( − π, π ) and U ∗ ( ± π ) = c . Thefirst derivative of U ∗ ∈ ˙ H has a finite jump singularity across the end points z = ± π . Moreprecisely, the profile U ∗ is Lipschitz continuous at ± π , that is, there exist constants 0 < c < c such that c | z − π | ≤ | U ∗ ( z ) − c ∗ | ≤ c | z − π | for | z − π | ≪ , which can be easily checked in view of the explicit expression (2.4).The next result states that the only single-lobe periodic wave with profile U ∈ C per satisfyingthe boundary-value problem (2.1) and having a singularity in the derivative at z = ± π is thepeaked periodic wave U ∗ given in (2.4). Lemma 2.
For every c ∈ R , the boundary-value problem (2.1) does not admit single-lobe periodicwaves in the sense of Definition 1 which are C α per with α ∈ [0 , . The only periodic wave witha singularity in the derivative at z = ± π is the peaked wave with parabolic profile (2.4), whichexists for c = c ∗ = π / and is Lipschitz at the peak.Proof. Let U ∈ ˙ L ∩ C per be a single-lobe periodic wave solution of the boundary-value problem(2.1). By Remark 1, U ∈ ˙ L is even, ∂ − z U ∈ ˙ H is odd, and ∂ − z U is represented by theFourier sine series which converges absolutely and uniformly, so that ( ∂ − z U )( ± π ) = 0. • Let us first consider the case where U ( z ) = c for every z ∈ ( − π, π ) and U ( ± π ) = c . Let α ∈ (0 , U of the boundary-valueproblem (2.1) with U ∈ C α per . If U ∈ C α per with α ∈ (0 , ∂ − z U ∈ C . Since U ( ± π ) = c and ( ∂ − z U )( ± π ) = 0 we find that c − U ( z ) ∼ ( π − z ) α and ( ∂ − z U )( z ) ∼ ( π − z ) at z = ± π .Since U satisfies the boundary value problem (2.1) we have that(2.5) U ′ ( z ) = − ( ∂ − z U )( z ) c − U ( z ) , z ∈ ( − π, π )which yields U ′ ( z ) ∼ ( π − z ) − α at z = ± π . Equation (2.5) also implies that U ′ ∈ C ( − π, π ) sowe find that U ′ ∈ C − α per . Since 1 − α ∈ (0 ,
1) we conclude that U ∈ C in contradiction to theassumption that U ∈ C α per with α ∈ (0 , α = 0, which refers to solutions U ∈ C per in view of Definition 1, can be proven in exactly the same way.We now show that the only peaked periodic solution with peak at U ( ± π ) = c is the solutionwith the parabolic profile (2.4). Since U ( z ) = c for every z ∈ ( − π, π ) and U ∈ C ( − π, π ), thefirst-order invariant E = 12 [ c − U ( z )] (cid:2) U ′ ( z ) (cid:3) + c U ( z ) − U ( z ) = 12 (cid:2) ( ∂ − z U )( z ) (cid:3) + c U ( z ) − U ( z ) , (2.6)holds for z ∈ ( − π, π ). Since ( ∂ − z U )( z ) is continuous in z = ± π with ( ∂ − z U )( ± π ) = 0, E iscontinuous and constant up to the boundary at z = ± π and we have E | z = ± π = c / E c . For c >
0, the level with E = E c (see the bold curve in Figure 1) gives rise to a peaked periodicwave solution with parabolic profile U ( z ) < c . We claim that this peaked wave is exactly thesolution (2.4) with speed c = c ∗ . Indeed, from the level set with E = E c we have that ( c − U ) ( U ′ ( z )) = c − c U + U = ( c − U ) ( c + 2 U ) ANNA GEYER AND DMITRY E. PELINOVSKY
UU’ −2 −1 0 1 2 3 4−4−2024
Figure 1.
Phase plane portrait obtained from level curves of the first-orderinvariant (2.6) for some c >
0. The dashed black line indicates the singularityline U = c . The solid black curve to the left of the singular line corresponds tothe parabolic profile (2.4).and hence U ′ ( z ) = √ √ c + 2 U > z ∈ (0 , π )subject to the boundary condition U ′ (0) = 0. By separation of variables we can solve thisequation uniquely to find that U ( z ) = ( z − c ). In view of the condition U ( π ) = c , thisimplies that c = π = c ∗ which proves the claim. • Let us now analyze the situation where there exists z ∈ (0 , π ) such that U ( ± z ) = c andequation (2.5) holds separately for z ∈ (0 , z ) and for z ∈ ( z , π ). Let us assume that U ( z ) < c for z ∈ (0 , z ). If U ( z ) > c for z ∈ (0 , z ), the proof is analogous. There are two possibilities,either ( ∂ − z U )( ± z ) = 0 or ( ∂ − z U )( ± z ) = 0.If ( ∂ − z U )( ± z ) = 0, then by the same argument as above the first-order invariant E iscontinuous and constant on [ − z , z ] with E | z = ± z = E z .For z ∈ ( z , π ] the solution corresponding to the level set E = E z can either be continueduniquely from the region with U ( z ) < c into the region with U ( z ) > c , or z = z representsa turning point (a local maximum for U ) and the solution can be continued uniquely into theregion with U ( z ) < c for z & z . However, U ( z ) > c > U ( z ) < c < z ∈ ( z , π ]. Therefore, the first variant implies that ( ∂ − z U )( z ) = R z U ( z ′ ) dz ′ = 0 forevery z ∈ ( z , π ] which contradicts ( ∂ − z U )( π ) = 0. The second continuation is possible but doesnot belong to the class of single-lobe periodic waves, see Remark 5. The notation z & z means that 0 < z − z < ε for some small ε >
0, and equivalently for the reverseinequality.
NIQUENESS AND INSTABILITY OF THE PEAKED PERIODIC WAVE 7
If ( ∂ − z U )( ± z ) = 0, then the contradiction arises from the fact that, since ( ∂ − z U )( z ) iscontinuous at z and equation (2.5) holds separately for z ∈ (0 , z ) and for z ∈ ( z , π ), thechange of the sign of U ′ ( z ) across z is determined by the change of the sign of c − U ( z ) across z . Indeed, if U ( z ) < c both for z . z and z & z , then it follows from (2.5) that the signof U ′ ( z ) remains the same for z ∈ (0 , z ) and z ∈ ( z , π ). But this is impossible since U ′ ( z )must change sign for z . z and z & z if U ( z ) < c on both sides of z . If on the other hand U ( z ) < c for z . z and U ( z ) > c for z & z , then it follows again from (2.5) that the sign of U ′ ( z ) changes, in contradiction with the monotone increase of U ( z ) for all z ∈ (0 , π ). Hence,both possibilities with ( ∂ − z U )( ± z ) = 0 yield a contradiction.Combing all these arguments we find that the only single-lobe peaked periodic wave hasparabolic profile (2.4), which is Lipschitz at the peak U ( ± π ) = c . (cid:3) Remark . There is a simple way to obtain other peaked periodic waves in the boundary-valueproblem (2.1). One can flip the periodic wave with parabolic profile at a point z ∈ (0 , π ) andpack two such waves over one period. This possibility is allowed in the proof of Lemma 2, butnot in the class of single-lobe periodic waves. Similarly, one can pack three and more periods ofthe peaked wave with parabolic profiles. Definition 1 eliminates this type of non-uniqueness ofthe peaked periodic wave in the boundary-value problem (2.1). Remark . With the following formal transformation(2.7) U ( z ) = u ( ζ ) , z = Z ζ ( c − u ( ζ ′ )) dζ ′ , smooth periodic waves with profile U satisfying the quasilinear second-order equation (2.8) ddz [ c − U ( z )] dUdz + U = 0are related to smooth periodic waves with profile u satisfying the semi-linear second-order equa-tion(2.9) d u dζ + ( c − u ) u = 0 . Although all periodic solutions of (2.9) are smooth, the coordinate transformation (2.7) fails tobe invertible if u ( ζ ) = c for some ζ . Such points generate singularities in the periodic solutionsof the quasi-linear equation (2.8) since U ′ ( z ) = u ′ ( ζ ) c − u ( ζ ) . In [21], small-amplitude peaked periodic waves of (2.8) with c < U for such peaked periodic waves has a square root singularity of the form(2.10) U ( z ) = c + O ( p π − z ) as z → ± π. The quasi-linear equation (2.8) is a derivative of the first equation in the boundary-value problem (2.1) in z ,which is justified for smooth periodic waves in Lemma 1. ANNA GEYER AND DMITRY E. PELINOVSKY
Our analysis in the proof of Lemma 2 shows that such solutions cannot exist, since the expansion(2.10) implies that U ∈ C / . We conclude that the small-amplitude peaked periodic wavesconstructed in [21] are artefacts of the construction method and do not satisfy the boundary-value problem (2.1). Remark . In [40] several different types of travelling wave solutions to the reduced Ostrovskyequation were constructed by means of phase plane analysis. Three types of solitary waves (seeFig. 9 in [40]) were found for c <
0. One of them is a loop soliton, given by a multi-valuedfunction, which is studied in many publications [15, 43, 44]. The other two solutions havepoints of infinite slope (cusps), either at the maximum or at the inflection points. The cuspedsolitary waves were also constructed in [38] by using the transformation (2.7). By using similararguments as in the proof of Lemma 2, the existence of the cusped waves as weak solutions tothe reduced Ostrovsky equation can be disproved.3.
Linear instability of the peaked periodic wave
We add a co-periodic perturbation v to the travelling wave U , that is, a perturbation withthe same period 2 π . Truncating the quadratic terms and moving with the reference frame ofthe travelling wave yields the linearized evolution problem in the form(3.1) (cid:26) v t + ∂ z [( U ( z ) − c ) v ] = ∂ − z v, t > ,v | t =0 = v . The linearized evolution equation can be formulated in the form v t = ∂ z Lv defined by theself-adjoint operator(3.2) L = P (cid:0) ∂ − z + c − U ( z ) (cid:1) P : ˙ L → ˙ L , where P : L → ˙ L is the projection operator that removes the mean value of 2 π -periodicfunctions. The form v t = ∂ z Lv is related to the formulation of the reduced Ostrovsky equationin the travelling wave coordinate z = x − ct as a Hamiltonian system defined by the symplecticoperator ∂ z and the conserved energy function H c ( u ) = H ( u ) + cQ ( u ), where(3.3) H ( u ) = Z π − π (cid:20) − ( ∂ − z u ) − u (cid:21) dz, Q ( u ) = Z π − π u dz are the conserved energy and momentum functionals for the reduced Ostrovsky equation (1.1).The periodic wave u = U is a critical point of H c ( u ) and the self-adjoint operator L is theHessian operator of the energy function H c ( u ) at the periodic wave u = U .Thanks to the translational invariance of the boundary-value problem (2.1), L∂ z U = 0, where ∂ z U ∈ ˙ L , holds for both the smooth periodic waves of Lemma 1 and the peaked periodic wave(2.4) in Lemma 2. Associated to the translational eigenvector is the symplectic orthogonalityconstraint h U, v i = 0. This constraint is used to study both the evolution of the Cauchy problem(3.1) and the spectrum of the linearized operator(3.4) ∂ z L : X ⊂ ˙ L → ˙ L , Solutions of [21] have nonzero mean value, hence Lemma 2 does not apply directly. However, the argumentsin the proof lead to the same conclusion also for solutions with nonzero mean. Indeed, if U has a non-zero mean, ∂ − z U ( z ) may not be zero at z = ± π . However, if we translate the solution by half a period so that the singularityis placed at z = 0, then ∂ − z U (0) = 0 by oddness of ∂ − z U and we can use the same contradiction as the oneobtained from (2.5). NIQUENESS AND INSTABILITY OF THE PEAKED PERIODIC WAVE 9 where X = { v ∈ ˙ L : ( c − U ) v ∈ H } is the maximal domain of ∂ z L . See [3, 24, 34]. Remark . For smooth periodic waves we have c − U ( z ) > z ∈ [ − π, π ] so that X ≡ ˙ H . For the peaked periodic wave U ∗ with speed c ∗ , the space ˙ H is continuouslyembedded into X since U ∗ is bounded, but ˙ H is not equivalent to X . Indeed, if aperturbation v to U ∗ is piecewise C with a finite jump-discontinuity at z = ± π , then v / ∈ ˙ H per but v ∈ X since ( c ∗ − U ∗ ) v ∈ ˙ H in view of the fact that U ∗ ( ± π ) = c ∗ .In what follows, h· , ·i and k v k L denote the inner product and the L norm with integrationover [ − π, π ], respectively. In the case of the peaked periodic wave U ∗ with the speed c ∗ , weequip X with the norm(3.5) k v k X := k v k L + k ∂ z [( c ∗ − U ∗ ) v ] k L . We distinguish two concepts of stability of 2 π -periodic waves with respect to linearization. Definition 2.
The travelling wave U is said to be spectrally stable if σ ( ∂ z L ) ⊂ i R in ˙ L .Otherwise, it is said to be spectrally unstable. Definition 3.
The travelling wave U is said to be linearly stable if for every v ∈ X satisfying h U, v i = 0, there exists C > v ∈ C ( R , X ) to the Cauchyproblem (3.1) such that(3.6) k v ( t ) k X ≤ C k v k X , t > . Otherwise, it is said to be linearly unstable.In [18], we have proved that the smooth periodic waves of Lemma 1 are spectrally stable inthe sense of Definition 2. Here we intend to show that the peaked periodic wave U ∗ of Lemma2 given in (2.4) is linearly unstable in the sense of Definition 3. The linear instability is due tothe sharp exponential growth of the unique global solution to the Cauchy problem (3.1) with U = U ∗ :(3.7) C k v k L e πt/ ≤ k v ( t ) k L ≤ k v k L e πt/ , t > , for some C ∈ (0 , ∂ − z v and obtain the sharp bounds (3.7) for all initial conditions v ∈ X satisfying the constraint(3.8) Z π − π zv ( z ) dz = 0 . In the second step, carried out in Section 3.2, we will show that the bounds (3.7) remain truein the full linearized equation (3.1) for a subset of initial conditions v ∈ X satisfying theconstraint (3.8) and the additional constraint(3.9) Z π − π z v ( z ) dz = 0 , which arises due to the orthogonality condition h U, v i = 0 in Definition 3 and the zero-meancondition on v . Regarding spectral stability or instability of the peaked periodic wave (2.4),we will show in Section 3.3 that σ ( L ) in ˙ L is given by a continuous spectrum on [0 , π / which includes the embedded eigenvalue λ = 0 with the eigenvector ∂ z U , and a simple negativeeigenvalue λ <
0. As a result, no spectral gap appears between λ = 0 and the continuousspectrum, hence it is impossible to solve the spectral stability problem by applying the standardmethods from [3, 24, 34].3.1. Linear instability of truncated evolution.
For the peaked periodic wave (2.4), weobtain the simple expression(3.10) U ∗ ( z ) − c ∗ = 16 ( z − π ) , z ∈ [ − π, π ] . By removing the term ∂ − z v from the linearized evolution problem (3.1) and using the explicitexpression (3.10), we can write the truncated evolution problem in the form(3.11) (cid:26) v t + ∂ z (cid:2) ( z − π ) v (cid:3) = 0 , t > ,v | t =0 = v , where the initial data v is taken in X . The evolution problem can be solved by the methodof characteristics along the family of characteristic curves z = Z ( s, t ), where s ∈ [ − π, π ] is aparameter for the initial data and t ≥ (cid:26) ddt Z ( s, t ) = (cid:2) Z ( s, t ) − π (cid:3) , t > ,Z ( s,
0) = s, and setting V ( s, t ) := v ( Z ( s, t ) , t ) yields the evolution problem in the form(3.13) (cid:26) ddt V ( s, t ) = − Z ( s, t ) V ( s, t ) , t > ,V ( s,
0) = v ( s ) . The family of characteristic curves is obtained by integrating the differential equation (3.12)with the parameter s ∈ [ − π, π ]. Because Z = ± π are critical points of the differential equation(3.12), the family of characteristic curves remain inside the invariant region [ − π, π ] for every t ≥
0. The family of characteristic curves can be obtained in the explicit form(3.14) Z ( s, t ) = π s cosh( πt/ − π sinh( πt/ π cosh( πt/ − s sinh( πt/ , s ∈ [ − π, π ] , t ∈ R . For later use of the chain rule we compute(3.15) e R t Z ( s,t ′ ) dt ′ = ∂∂s Z ( s, t ) = π [ π cosh( πt/ − s sinh( πt/ , s ∈ [ − π, π ] , t ∈ R . The explicit solution for V in characteristic variables is obtained by integrating the differentialequation (3.13) with respect to the parameter s ∈ [ − π, π ]: V ( s, t ) = v ( s ) e − R t Z ( s,t ′ ) dt ′ . In view of (3.15), the explicit solution is given by(3.16) V ( s, t ) = 1 π [ π cosh( πt/ − s sinh( πt/ v ( s ) , s ∈ [ − π, π ] , t ∈ R . Remark . Since Z ( ± π, t ) = ± π for every t ∈ R , we have V ( ± π, t ) = e ∓ πt/ v ( ± π, t ), hence V ( − π, t ) = V ( π, t ) for t = 0 if and only if v ( ± π ) = 0, in which case V ( ± π, t ) = 0 for every t ∈ R . Therefore, V ( · , t ) / ∈ ˙ H for t = 0 if v ∈ ˙ H with v ( ± π ) = 0. NIQUENESS AND INSTABILITY OF THE PEAKED PERIODIC WAVE 11
By using the explicit solutions (3.14) and (3.16), we are able to state and prove the followinglinear instability result for the truncated evolution problem (3.11).
Lemma 3.
For every v ∈ X , there exists a unique global solution v ∈ C ( R , X ) to theCauchy problem (3.11) satisfying the upper bound (3.17) k v ( t ) k L ≤ k v k L e πt/ , t > . If R π − π sv ( s ) ds = 0 , then the global solution satisfies the lower bound (3.18) 12 k v k L e πt/ ≤ k v ( t ) k L , t > . Proof.
Existence of a global solution in the explicit form (3.14) and (3.16) is obtained from themethod of characteristics. By using the chain rule and (3.15), we verify that the mean-zeroconstraint is preserved by the time evolution: Z π − π v ( z, t ) dz = Z π − π V ( s, t ) ∂Z∂s ds = Z π − π v ( s ) ds = 0 . The explicit expression (3.16) implies that V ( · , t ) ∈ X if v ∈ X and t ∈ R . On the otherhand, the explicit expression (3.14) implies that for every τ >
0, there exists C τ > ∂∂s Z ( s, t ) ≥ C τ , s ∈ [ − π, π ] , t ∈ [ − τ, τ ] . Hence, the chain rule implies that v ( · , t ) ∈ X if v ∈ X and t ∈ R . Uniqueness of suchglobal solutions follows by standard theory (see Theorem 3.1 in [2]).It remains to prove the sharp exponential growth in the bounds (3.17) and (3.18). By thechain rule, we obtain Z π − π v ( z, t ) dz = Z π − π V ( s, t ) ∂Z∂s ds = 1 π Z π − π [ π cosh( πt/ − s sinh( πt/ v ( s ) ds. From here, we have the upper bound k v ( t ) k L ≤ e πt/ k v k L and the lower bound under the additional condition R π − π sv ( s ) ds = 0: k v ( t ) k L = cosh( πt/ k v k L + 1 π sinh( πt/ k sv k L ≥ e πt/ k v k L . Taking the square root of these bounds yields (3.17) and (3.18). (cid:3)
Remark . By the chain rule, we also have Z π − π (cid:2) ∂ z ( π − z ) v ( z, t ) (cid:3) dz = 1 π Z π − π [ π cosh( πt/ − s sinh( πt/ (cid:2) ∂ s ( π − s ) v ( s ) (cid:3) ds, from which the sharp exponential growth with the same growth rate as in the bounds (3.17)and (3.18) can be established for the second term in the X norm given by (3.5). Remark . The global solution in Lemma 3 remains bounded in L . This follows from thechain rule: Z π − π | v ( z, t ) | dz = Z π − π | V ( s, t ) | ∂Z∂s ds = Z π − π | v ( s ) | ds. Since(3.19) k v k L ≤ (2 π ) / k v k L , hence v ∈ ˙ L implies v ∈ L . Extending this bound to the time-dependent solution,(3.20) k v ( t ) k L ≤ (2 π ) / k v ( t ) k L , t > , shows that the L norm of the global solution v ( t ) may remain bounded even if the L norm ofthis solution grows exponentially. Remark . Truncating a quadratic form associated with the self-adjoint operator L in (3.2)and using the chain rule yield the energy conservation for the truncated evolution (3.11): Z π − π ( π − z ) v ( z, t ) dz = Z π − π (cid:2) π − Z ( s, t ) (cid:3) V ( s, t ) ∂Z∂s ds = Z π − π ( π − s ) v ( s ) ds. The energy conservation shows that the truncated evolution leads to the exponential growth of k v ( t ) k L and k zv ( t ) k L but the difference between the two squared norms remains bounded. Remark . For the smooth periodic waves of Lemma 1 satisfying U ( z ) < c for every z ∈ [ − π, π ],the truncated energy R π − π ( c − U ) v dz is coercive in the L norm, hence the energy conservation Z π − π [ c − U ( z )] v ( z, t ) dz = Z π − π [ c − U ( z )] v ( s ) ds implies a global time-independent bound on k v ( t ) k L , where v ( t ) is a solution of the truncationof the linear evolution equation (3.1) without the ∂ − z v term. Remark . For the smooth periodic waves of Lemma 1, the characteristic curves reach theboundaries z = ± π in finite time because z = ± π are not critical points of the differentialequations for the characteristic curves. On the other hand, for the peaked periodic wave (2.4),the characteristic curves reach the boundaries z = ± π in infinite time. The latter propertyinduces exponential growth of the global solutions to the Cauchy problem (3.11) as is shown inLemma 3.3.2. Linear instability of full evolution.
Here we consider the full linearized evolution prob-lem (3.1) with (3.10) and rewrite the evolution problem in the form(3.21) (cid:26) v t + ∂ z (cid:2) ( z − π ) v (cid:3) = ∂ − z v, t > ,v | t =0 = v , where the initial data v is taken in X . Lemma 4.
For every v ∈ X there exists a unique global solution v ∈ C ( R , X ) of theCauchy problem (3.21) .Proof. By Lemma 3, the Cauchy problem (3.11) with v ∈ X has a unique global solution v ∈ C ( R , X ). In the framework of semigroup theory, the evolution equation (3.11) can bewritten in the form v t = A v , where A := 16 ∂ z ( π − z ) v. NIQUENESS AND INSTABILITY OF THE PEAKED PERIODIC WAVE 13
Existence of a unique global solution v ∈ C ( R , X ) implies that the operator A with domain D ( A ) = X is the infinitesimal generator of a strongly continuous semigroup ( S ( t )) t ≥ on˙ L . Since ∂ − z : ˙ L → ˙ L is a bounded operator, the Bounded Perturbation Theorem (seeTheorem III,1.3 on p. 158 in [13]) implies that the operator A := A + ∂ − z with the same domain D ( A ) = D ( A ) = X also generates a strongly continuous semigroup( S ( t )) t ≥ on ˙ L . Therefore, the evolution equation in the Cauchy problem (3.21) can be viewedas a bounded perturbation of the evolution equation in the Cauchy problem (3.11). The assertionof the Lemma then follows by Proposition 6.2 on p. 145 in [13]. (cid:3) In what follows, we obtain bounds on the global solution v ∈ C ( R , X ) to the Cauchyproblem (3.21). First, we note the following upper bound on the growth of the global solution. Lemma 5.
A global solution v ∈ C ( R , X ) to the Cauchy problem (3.21) in Lemma 4 satisfiesthe upper bound (3.22) k v ( t ) k L ≤ k v k L e πt/ , t > . Proof.
Note the following integration yields Z π − π v ( ∂ − z v ) dz = 12 ( ∂ − z v ) | z = πz = − π = 0 , since ∂ − z v ∈ H and hence ∂ − z v ∈ C per by Sobolev’s embedding. Integrating by parts yieldsthe following balance equation ddt k v ( t ) k L = 16 Z π − π v∂ z (cid:2) ( π − z ) v (cid:3) dz = − Z π − π ( π − z ) v∂ z vdz = − Z π − π zv dz. Hence ddt k v ( t ) k L ≤ π k v ( t ) k L and Gronwall’s inequality yields the desired bound (3.22). (cid:3) In order to obtain the lower bound on the L norm of the global solution to the Cauchyproblem (3.21), we use the generalized method of characteristics and treat ∂ − z v ( z, t ) as a sourceterm in (3.11). This term satisfies the following useful bound (also proven in [32]). Lemma 6. If g := ∂ − z v ∈ ˙ H , then (3.23) k g k L ∞ per ≤ k v k L . Proof.
By Sobolev embedding of H into C per , g is a continuous 2 π -periodic function with zeromean. Therefore, there exists ζ ∈ [ − π, π ] such that g ( ζ ) = 0. For every z ∈ [ − π, π ], we canwrite g ( z ) = Z zζ v ( z ′ ) dz ′ , from which bound (3.23) follows. Note that L is continuously embedded into L because of thebound (3.19). (cid:3) By using the family of characteristic curves z = Z ( s, t ) with s ∈ [ − π, π ] and t ≥
0, where Z is defined by the same initial-value problem (3.12), and setting V ( s, t ) := v ( Z ( s, t ) , t ) and G ( s, t ) := g ( Z ( s, t ) , t ), we obtain the evolution problem in the form(3.24) (cid:26) ddt V ( s, t ) = − Z ( s, t ) V ( s, t ) + G ( s, t ) , t > ,V ( s,
0) = v ( s ) . The family of characteristic curves Z is still given by the same explicit form (3.14). Integratingthe differential equation (3.24) with an integrating factor yields the explicit solution for V inthe form(3.25) V ( s, t ) = (cid:20) v ( s ) + Z t G ( s, t ′ ) e R t ′ Z ( s,t ′′ ) dt ′′ dt ′ (cid:21) e − R t Z ( s,t ′ ) dt ′ By using the explicit solution (3.25), we are able to prove the linear instability result for theCauchy problem (3.21).
Lemma 7.
There exists v ∈ X and C > such that the unique global solution v ∈ C ( R , X ) to the Cauchy problem (3.21) in Lemma 4 satisfies the lower bound (3.26) k v ( t ) k L ≥ C k v k L e πt/ , t > . Proof.
By the chain rule, the explicit expression (3.25) with the help of (3.15) yields the followingequation: Z π − π v ( z, t ) dz = Z π − π V ( s, t ) ∂Z∂s ds = 1 π Z π − π [ π cosh( πt/ − s sinh( πt/ × (cid:20) v ( s ) + Z t π G ( s, t ′ )[ π cosh( πt ′ / − s sinh( πt ′ / dt ′ (cid:21) ds. Let us assume the same constraint R π − π sv ( s ) ds = 0 as in Lemma 3. Neglecting positive termsin the lower bound, we obtain k v ( t ) k L ≥ e πt/ k v k L (3.27) − Z π − π Z t | v ( s ) || G ( s, t ′ ) | [ π cosh( πt/ − s sinh( πt/ [ π cosh( πt ′ / − s sinh( πt ′ / dt ′ ds. Let us define for any t > K ( t, t ′ , s ) := π cosh( πt/ − s sinh( πt/ π cosh( πt ′ / − s sinh( πt ′ / , t ′ ∈ [0 , t ] , s ∈ [ − π, π ] . We prove that for every 0 ≤ t ′ ≤ t ,(3.28) sup s ∈ [ − π,π ] K ( t, t ′ , s ) = e π ( t − t ′ ) / . Indeed, K ( t, t ′ , s ) = e π ( t − t ′ ) / M ( t, t ′ , s ), where M ( t, t ′ , s ) := ( π − s ) + ( π + s ) e − πt/ ( π − s ) + ( π + s ) e − πt ′ / , NIQUENESS AND INSTABILITY OF THE PEAKED PERIODIC WAVE 15 and M is monotonically decreasing since ∂ s M ( t, t ′ , s ) ≤ t ′ ∈ [0 , t ] and s ∈ [ − π, π ].Therefore, M has a maximum at s = − π , where M ( t, t ′ , − π ) = 1.By using (3.27) and (3.28), we obtain k v ( t ) k L ≥ e πt/ k v k L − k v k L Z t k g ( t ′ ) k L ∞ per e π ( t − t ′ ) / dt ′ ≥ e πt/ k v k L − k v k L Z t k v ( t ′ ) k L e π ( t − t ′ ) / dt ′ ≥ e πt/ k v k L − √ π k v k L k v k L e πt/ Z t e − πt ′ / dt ′ , where (3.20), (3.22), and (3.23) have been used in the last two inequalities. Hence, k v ( t ) k L e − πt/ ≥ k v k L k v k L − √ √ π k v k L ! and since k v k L can be much larger than k v k L by the bound (3.19), there exist v ∈ X and C ∈ (0 , /
4) such that(3.29) k v k L ≤ √ π (1 − C )48 √ k v k L , and hence k v ( t ) k L e − πt/ ≥ C k v k L . (3.30)This yields the desired bound (3.26). (cid:3) Remark . Let us show that there exist functions v ∈ X satisfying the constraints (3.8),(3.9), and (3.29). Indeed, if v is odd, then v is even, hence the two constraints (3.8) and (3.9)are satisfied simultaneously. From the class of odd initial data we need to pick functions in X that satisfy the inequality (3.29) for a fixed C ∈ (0 , / H ⊂ X (3.31) v ( x ) = x ( π − x )1 + a x , x ∈ [ − π, π ] , where a > k v k L = (cid:18) π a + 1 a (cid:19) log(1 + π a ) − π a and k v k L = 1 a (cid:20)(cid:18) π + 6 π a + 5 a (cid:19) arctan( πa ) − π (15 + 13 π a )3 a (cid:21) . Since k v k L = O (log( a ) a − ) decays to zero as a → ∞ faster than k v k L = O ( a − / ),inequality (3.29) can be satisfied for sufficiently large a . Remark . If v ( ± π ) = 0 like in the example (3.31), then v ∈ ˙ H and the truncated linearizedevolution (3.11) preserves the constraint v ( ± π, t ) = 0 for every t ∈ R , see Remark 9. However,the integral term ∂ − z v in the full linearized evolution (3.21) does not generally preserve thesame constraint because it is uniquely defined from the condition that ∂ − z v has zero mean. As a result, the full linearized equation does not generally admit a solution v ∈ C ( R , ˙ H ) even if v ∈ ˙ H . Remark . In the presence of the source term G , we are not able to show that k v ( t ) k L remains bounded as t → ∞ , see Remark 11. By using the integral Z π − π π [ π cosh( πt ′ / − s sinh( πt ′ / ds = 2 π, t ′ ∈ [0 , t ] , we obtain the bound k v ( t ) k L ≤ k v k L + 2 π Z t k g ( t ′ ) k L ∞ per dt ′ , in view of (3.15) and (3.25). Thanks to the bound (3.23), the inequality is closed as follows: k v ( t ) k L ≤ k v k L + 2 π Z t k v ( t ′ ) k L dt ′ . By Gronwall’s inequality, this bound gives the fast exponential growth k v ( t ) k L ≤ k v k L e πt , which cannot be sharp because k v ( t ) k L is bounded by a slowly growing exponential functionthat follows from the bounds (3.20) and (3.22). Remark . There exists a conserved energy for the Cauchy problem (3.21), see Remark 12,which is given by(3.32) h Lv ( t ) , v ( t ) i = h Lv , v i , where the self-adjoint operator L is defined by (3.2). However, the conserved quantity (3.32)does not prevent k v ( t ) k L from growing exponentially fast as t → ∞ because the boundedoperator L is not coercive under the constraint (3.9), see Lemma 8.3.3. Spectrum of the linear self-adjoint operator L . Here we consider the spectrum σ ( L )of the linear self-adjoint operator L defined by (3.2). We will prove that σ ( L ) consists of thecontinuous spectrum on [0 , π / λ = 0 with theeigenvector ∂ z U , and a simple negative eigenvalue λ <
0. No spectral gap appears between λ = 0 and the continuous spectrum. The following lemma gives the corresponding result. Lemma 8.
The spectrum of the self-adjoint operator L given by (3.2) is (3.33) σ ( L ) = { λ } ∪ (cid:20) , π (cid:21) , where λ < is the unique zero of the transcendental equation (3.34) ( π + 3 λ ) log √ π − λ + π √ π − λ − π − π p π − λ = 0 , λ < . Proof.
By the spectral theorem (see, e.g., Definition 8.39, Theorem 8.70, and Theorem 8.71 in[35]), the spectrum of the self-adjoint operator L in ˙ L denoted by σ ( L ) may consist of onlytwo disjoint sets on the real line: the point spectrum of eigenvalues with eigenvectors in ˙ L denoted by σ p ( L ) and the continuous spectrum denoted by σ c ( L ), where the resolvent operatorexists but is unbounded. NIQUENESS AND INSTABILITY OF THE PEAKED PERIODIC WAVE 17
The self-adjoint operator L in (3.2) is given by the sum of a bounded operator L and acompact operator K given by(3.35) L := 16 P (cid:0) π − z (cid:1) P : ˙ L → ˙ L and(3.36) K := P ∂ − z P : ˙ L → ˙ L . Moreover, the compact operator is in the trace class since P ∞ n =1 n − < ∞ . By Kato’s Theorem[27] (see Theorem 4.4 on p. 540 in [28]), σ c ( L ) = σ c ( L ). We show that [0 , π / ⊆ σ c ( L ) byconsidering the odd functions in ˙ L , which can be represented by the Fourier sine series. Letus denote the space of odd functions in ˙ L by L , odd . Then, L f = 16 ( π − z ) f, ∀ f ∈ L , odd . Then, σ c ( L ) in L , odd coincides with the range of the multiplicative function h ( z ) = ( π − z )for z ∈ [ − π, π ], which is [0 , π / , π / ⊆ σ c ( L ) in ˙ L .Let us show that [0 , π / ≡ σ c ( L ) by working with the resolvent equation ( L − λI ) f = g for given g ∈ ˙ L and λ / ∈ [0 , π / z ∈ [ − π, π ]:16 ( π − λ − z ) f ( z ) − k ( f ) = g ( z ) , k ( f ) := 112 π Z π − π ( π − z ) f ( z ) dz, where f ∈ ˙ L is supposed to satisfy the zero-mean constraint R π − π f ( z ) dz = 0. Computing thesolution explicitly, f ( z ) = 6 π − λ − z [ g ( z ) + k ( f )] , and using the zero mean constraint, we can define k ( f ) in terms of g : k ( f ) = R π − π g ( z ) π − λ − z dz R π − π π − λ − z dz . For every λ / ∈ [0 , π / C λ , C ′ λ > z ∈ [ − π,π ] | π − λ − z | ≤ C λ , (cid:12)(cid:12)(cid:12)(cid:12)Z π − π π − λ − z dz (cid:12)(cid:12)(cid:12)(cid:12) ≥ C ′ λ . As a result, we obtain the bound k f k L ≤ C λ h k g k L + | k ( f ) |√ π i ≤ C λ (cid:2) π ( C ′ λ ) − C λ (cid:3) k g k L . Therefore, the resolvent operator ( L − λI ) − : ˙ L → ˙ L is bounded for every λ / ∈ [0 , π / σ c ( L ) = [0 , π / σ p ( L ) ∈ R \ [0 , π / L withthe spectral parameter λ / ∈ [0 , π / P (cid:0) π − z (cid:1) w + P ∂ − z w = λw, w ∈ ˙ L . Since ∂ − z w ∈ H , bootstrapping arguments show that w ∈ H on any compact subset in( − π, π ). Iterations of bootstrapping arguments yield w ∈ H ∞ loc . Therefore, the spectral problem(3.37) can be differentiated twice on a compact subset in ( − π, π ), after which it is rewritten asthe second-order differential equation(3.38) (cid:0) π − z − λ (cid:1) d wdz − z dwdz + 4 w ( z ) = 0 , w ∈ H ∞ loc , with the two linearly independent solutions for λ ∈ R \ [0 , π / w ( z ) = z and w ( z ) = ( − z π − z − λ ) + z √ π − λ log √ π − λ + z √ π − λ − z , λ < , − z π − z − λ ) − z √ λ − π arctan z √ λ − π , λ > π . The first solution corresponds to the eigenvector ∂ z U of the spectral problem (3.37) for theeigenvalue λ = 0, which is embedded into σ c ( L ) = [0 , π / w of thespectral problem (3.37) such that h w, w i = 0. Therefore, we take w = w and extend it from H to ˙ L . This extension is achieved if and only if w has zero mean, that is,(3.39) 0 = 12 π Z π − π w ( z ) dz = ( − + π +3 λ π √ π − λ log √ π − λ + π √ π − λ − π λ < , − − π +3 λ π √ λ − π arctan π √ λ − π λ > π . The piecewise graph of the right-hand side of the zero-mean constraint (3.39) on ( −∞ ,
0) and( π / , ∞ ) is shown on Figure 2. The first line of the zero-mean constraint (3.39) is equivalentto the transcendental equation (3.34) and it has only one simple zero at λ ≈ − . λ < σ p ( L ). (cid:3) Remark . For the smooth periodic waves of Lemma 1, we proved in [18] that σ ( L ) in ˙ L includes a simple negative eigenvalue, a simple zero eigenvalue with the eigenvector ∂ z U , and therest of the spectrum is positive and bounded away from zero. Hence, the spectral gap is presentin the case of smooth periodic waves. This enabled us in [18] to use Hamilton-Krein index theoryto deduce that σ ( ∂ z L ) ⊂ i R and hence to deduce spectral stability of the smooth periodic wavesaccording to Definition 2. By the standard analysis involving the conserved quantity (3.32),see [23], this spectral stability result transfers to linear stability of the smooth periodic wavesaccording to Definition 3. In the spectral problem for the peaked periodic wave, however, thisspectral gap is not present. Therefore, we are not able to deduce spectral instability of thepeaked periodic wave from the spectrum of L .4. Discussion
We have studied peaked periodic traveling wave solutions of the reduced Ostrovsky equation(1.1). We found that the peaked periodic wave with parabolic shape U ∗ is the unique periodictraveling wave with a single minimum per period and that the boundary-value problem (2.1)does not admit H¨older continuous solutions, see Lemma 2. As a consequence, existence of cusped Note that h w , w i = 0 because w is odd and w is even. NIQUENESS AND INSTABILITY OF THE PEAKED PERIODIC WAVE 19 −6 −4 −2 0 2 4 6 8−60−40−20020406080
Figure 2.
The graph of the right-hand side of the zero-mean constraint (3.39)on ( −∞ ,
0) and ( π / , ∞ ) as a function of the spectral parameter λ . Only onesimple zero λ < U ∗ is linearly unstable with respectto co-periodic perturbations in the space X , which is the maximal domain of the linearizedoperator ∂ z L , see Lemma 7. This result was obtained using sharp exponential bounds on the L norm of perturbations v of U ∗ in X satisfying the Cauchy problem (3.1) with the peakedperiodic wave U ∗ and for the wave speed c ∗ .Passing from linear to nonlinear instability is often a delicate issue. Several authors haveshown that linear instability directly implies nonlinear instability if a part of the spectrum ofthe linearized operator is located in the right half of the complex plane, see for instance [17, 37]and Theorem 5.1.5 in [25]. However, these approaches do not work for the reduced Ostrovskyequation since the linearized evolution is defined in the space X whereas the nonlinear evo-lution is defined in ˙ H s per with s > /
2. It is not clear if the local well-posedness results can beextended to the space X . It is also unclear how the peaks of the peaked periodic wave moveunder the flow of the reduced Ostrovsky equation (1.1). For these reasons, nonlinear instabilityof the peaked periodic wave in the reduced Ostrovsky equation remains an open problem fornow. Acknowledgements.
This project was initiated during the research program on NonlinearWater Waves at Isaac Newton Institute at Cambridge in August 2017. Both authors thankMats Ehrnstr¨om for useful discussions during the program and afterward. Computations in theproof of Lemma 8 were performed back in 2015 in collaboration with Ted Johnson (UCL). DEPacknowledges a financial support from the State task program in the sphere of scientific activityof Ministry of Education and Science of the Russian Federation (Task No. 5.5176.2017/8.9) and from the grant of President of Russian Federation for the leading scientific schools (NSH-2685.2018.5). Finally, the authors thank the referees for useful remarks which helped to improvethe manuscript.
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E-mail address : [email protected] (D. Pelinovsky)
Department of Mathematics and Statistics, McMaster University, Hamilton, On-tario, Canada, L8S 4K1
E-mail address : [email protected] (D. Pelinovsky)(D. Pelinovsky)