STABILITY OF PERIODIC WAVES FOR THE GENERALIZED BBM EQUATION

on his 70th birthday

STABILITY OF PERIODIC WAVES FOR THE GENERALIZED BBM EQUATION

MARIANA H€R€GU“

We consider the generalized Benjamin-Bona-Mahony (gBBM) equation with poly- nomial nonlinearity of the form(u^p+1)x,p≥1. The aim of this paper is to in- vestigate the stability of the periodic travelling waves with speedsc > 1 which are small perturbations of the constant stateu= (c−1)^1/p. For general bounded perturbations, we show that these waves are spectrally stable for all speedsc >1 when1≤p≤2, and that forp≥3, there exists a critical speedcp,1< cp<_p−3^p , such that the waves are stable forc∈(cp,_p−3^p ), and unstable otherwise.

AMS 2000 Subject Classication: 35Q99.

Key words: generalized Benjamin-Bona-Mahony equation, periodic waves, spec- tral stability.

1. INTRODUCTION

We consider the generalized Benjamin-Bona-Mahony (gBBM) equation (1.1) ut(x, t)−uxxt(x, t) +ux(x, t) + (u^p+1(x, t))x = 0,

wherex∈R,t∈R,u(x, t)∈R, andp≥1 is an integer. The gBBM equation possesses periodic travelling-wave solutions of the form

(1.2) u(x, t) =q(x−ct), x∈R, t∈R,

where the wave speed c > 1 and the wave prole q is a periodic function of its argument. The aim of the present paper is to investigate the stability properties of these particular solutions, when the wave prole q stays close to the constant state u= (c−1)^1/p.

The casep= 1, which corresponds to the Benjamin-Bona-Mahony (BBM) equation, or the regularized long-wave equation, has been derived as a model for the description of gravity water waves in the long-wave regime [4, 18, 19].

In this context, it arises as an alternative model to the well-known Korteweg-de Vries (KdV) equation, exhibiting several of the striking phenomena encoun- tered in the dynamics of the water-wave problem. Among these, questions

REV. ROUMAINE MATH. PURES APPL., 53 (2008), 56, 445463

concerning solitary-wave dynamics have received particular interest, for the BBM equation, but also the gBBM equation. The gBBM equation possesses travelling solitary waves of the form (1.2) in which the wave speed c >1, just as for the periodic travelling waves, and the wave prole q is exponentially decaying to zero asx−ct→ ±∞. These solitary waves are orbitally stable for all speeds c >1when 1≤p≤4, and there exists a critical speed cp>1when p≥5, such that the solitary waves are orbitally stable forc > c_p, and unstable for c ∈ (1, cp) (e.g., see [6, 24, 26]). In addition, for the BBM equation, the solitary waves are asymptotically stable [9, 17].

In contrast to travelling solitary waves, for which the stability question is well understood, the stability of periodic travelling waves for dispersive equa- tions, and more precisely their nonlinear stability with respect to bounded per- turbations, is an open problem. So far, there are two types of stability results for periodic waves: orbital stability with respect to periodic perturbations, and spectral stability with respect to bounded or localized perturbations.

Orbital stability results rely upon a now classical energetic approach which goes back to Benjamin [3] and which has been extensively used for solitary waves (e.g., see [5, 12, 13, 25]). While for solitary waves this method gives a rather complete answer to the stability question, in the case of periodic waves it allows to prove orbital stability only under the restriction that pertur- bations have the same periodicity properties as the original wave. Such results are now available for dierent dispersive models, and in particular for the KdV equation and the nonlinear Schrödinger (NLS) equation (see [1, 2, 11] and the references therein), but also for a generalized BBM equation [14]. However, these results do not completely answer the stability question. In particular, the periodic analytical set-up used in these approaches does not allow to detect instabilities due to non periodic perturbations, such as side-band instabilities, and which have been detected in dierent models (e.g. the focusing NLS equa- tion [10], or the generalized KdV equation [15]).

For non periodic perturbations, stability results for periodic waves in dispersive models concern their spectral stability, that is, the location of the spectrum of the linearization about the wave. In contrast to the orbital sta- bility which gives necessary conditions for stability, the spectral stability pro- vides sucient stability conditions. For periodic waves, the spectral analysis relies upon the so-called Bloch-wave decomposition, which is well-known for Schrödinger operators with periodic potentials [20] and have been extensively used in dissipative problems [16, 21, 22, 23]. As for dispersive problems, we refer to [7, 8, 11, 15], and the references therein, for a number of spectral stability results for periodic waves in NLS- and KdV-type equations.

In this paper we study the spectral stability of the periodic travelling waves (1.2) of the gBBM equation (1.1). A periodic wave is spectrally stable if

the spectrum of the linearization at the wave entirely lies in the closed left half complex plane, and spectrally unstable otherwise. The spectrum of the linear operator is considered in either L²(R), for spatially localized perturbations, or C_b(R), for bounded uniformly continuous perturbations. We restrict to the periodic waves which are close to the constant state u= (c−1)^1/p. The main result shows that the periodic waves are spectrally stable, in both L²(R) and C_b(R), for all speeds c > 1 when 1 ≤ p ≤2, and that for p ≥ 3 there exists a critical speed cp, 1 < cp < _p−3^p , such that such that the periodic waves are spectrally stable for c∈(cp,_p−3^p ), and unstable for c∈(1, cp)∪(_p−3^p ,∞).¹

The instability is of side-band type, so that it cannot be detected in a periodic set-up, as for example, the one used in the analysis of orbital stability.

We point out that all these waves are spectrally stable in L²_per(0, T), that is, with respect to perturbations which have the same period T as the wave.

The proof is based on the Bloch-wave decomposition mentioned above, which reduces the spectral study of the linearized operator in the two spaces above to the study of the spectra of a family of linear operators in a space of periodic functions. The advantage of such a decomposition is that the re- sulting operators have compact resolvent, and therefore only point spectra.

In contrast to previous approaches to spectral stability of periodic waves, in which the linear operators were dierential operators with periodic coecients, in the present case the linearization at the periodic wave is a nonlocal oper- ator. However, using the Floquet theory, we show that the same result holds for the Bloch-wave decomposition. The next step in the analysis consists in locating the point spectra of the operators in the family resulting from the Bloch-wave decomposition. We rely on perturbation arguments for linear op- erators in which we regard these operators as small perturbations of operators with constant coecients. The latter ones are obtained from the linearization of (1.1) at the constant state u = (c−1)^1/p, and Fourier analysis allows to compute their spectra explicitly. The restriction to periodic waves which are close to this constant state is essential in this perturbation argument, and we do not know whether spectral stability holds for the other waves.

The remainder of the paper is organized as follows. In Section 2, we describe the set of periodic travelling waves of (1.1), and introduce an an- alytic parametrization of the periodic waves close to the constant solution u = (c−1)^1/p which will be used in the stability analysis. We discuss the spectral stability problem and give the result on the Bloch-wave decomposi- tion in Section 3. In Section 4, we determine the spectra of the unperturbed operators, and then in Section 5 we use the perturbation argument mentioned

1 Here, and in the rest of the paper p−3^p =∞, whenp= 3. In particular, forp= 3, the waves are spectrally stable forc∈(cp,∞)and unstable forc∈(1, cp).

above and locate the spectra of the operators resulting from the Bloch-wave decomposition. We conclude with an Appendix in which we give the proof of the result on the Bloch-wave decomposition.

2. PERIODIC WAVES

In this section we describe the set of periodic travelling waves of equa- tion (1.1), and give an analytic parametrization of the periodic waves close to the constant solution u= (c−1)^1/p.

The travelling-wave solutions (1.2) of equation (1.1) satisfy the ODE (2.1) cq⁰⁰= (c−1)q−q^p+1+B,

for some real constant B, where ⁰ denotes the derivative with respect to y = x−ct. We restrict here to speeds c > 1. This is the situation in which the ODE possesses homoclinic solutions to zero, for B = 0, which correspond to the solitary travelling waves of (1.1).

In the (q, q⁰) phase plane, equation (2.1) with B = 0 has equilibria at (q, q⁰) = (0,0) and (q, q⁰) = ((c−1)^1/p,0). The equilibrium ((c−1)^1/p,0) is a center and is enclosed in a one-parameter family of periodic orbits (q_α, q_α⁰), parameterized by 0 < α < 1, say. As α → 0, one nds the equilibrium ((c−1)^1/p,0), and as α → 1 these periodic orbits tend to an orbit (q₁, q⁰₁) which is homoclinic to the saddle point(0,0). The periodT_α of these periodic orbits depends smoothly upon α, and satises

α→0limTα= 2π√ c

pp(c−1), lim

α→1Tα=∞.

Up to spatial translations, we then have a one-parameter family of periodic travelling waves qα to (1.1). Notice that qα → (c−1)^1/p as α → 0, and that in the limit α → 1 one nds the solitary travelling wave of the gBBM equation (1.1).

Remark 2.1. (i) Ifp ≥1 is odd, the orbits described above are the only bounded orbits of (2.1) when B = 0. Ifp≥1 is even, the ODE withB = 0is equivariant under the reection q → −q, so that the set of its bounded orbits also contains the reected equilibrium (q, q⁰) = (−(c−1)^1/p,0), the reected periodic orbits(−q_α,−q⁰_α), and the reected homoclinic orbit(−q₁,−q⁰₁). This reection also is a symmetry for the gBBM equation (1.1), so that these re- ected orbits have the same dynamic properties as the corresponding non re- ected ones.

(ii) For speeds c ∈ (0,1), the ODE (2.1) with B = 0 also possesses a family of periodic orbits, now surrounding the origin (0,0)in the phase plane (q, q⁰). In contrast, for speeds c < 0 periodic orbits exist only when p ≥ 1

is odd. The stability of all these periodic orbits can be studied in the same manner as that of the periodic orbits for c >1discussed in the present paper.

For small values of B 6= 0, the phase portrait of the ODE (2.1) does not qualitatively change, so that we nd a family (q_α,B, q⁰_α,B) of periodic orbits.

Up to spatial translations, we then have a two-parameter family of periodic travelling waves q_α,B to (1.1). Notice that due to the invariance of (2.1) under the reection y → −y, the periodic travelling waves in the familyqα,B can be chosen to be even in y.

We now introduce an analytic parametrization of this family of periodic waves, for small α and B. First, it is convenient to replace the parameter B =b(c−1)^1+1/p, and then ODE (2.1) has the equilibriumqb with expansion

q_b = (c−1)^1/p

1 +1

pb−p+ 1

2p² b²+O(|b|³)

,

for small b. Next, for the periodic solutions we write q_α,b(y) = P_α,b(k_α,by), with k_α,b= 2π/T_α,b,T_α,b being the period of the wave, and replaceα by

a= 1

2π(c−1)^1/p Z 2π

0

Pα,b(z)e^−izdz,

so that α is proportional to the rst nonzero Fourier coecient of the 2π- periodic function P_α,b. Then

q_a,b(y) =P_a,b(k_a,by), wherePa,b is a2π-periodic even solution of the ODE

ck_a,b² v⁰⁰ = (c−1)v−v^p+1+b(c−1)^1+1/p, such that

P0,b=qb, k_0,b² = p+ 1

c q_b^p−c−1 c . A straightforward calculation gives the expansions Pa,b(z) =qb+(c−1)^1/p

cos(z)a−p+1

4 a²+ p+1

12 cos(2z)a²+O(|a|(a²+b²))

,

k²_a,b= c−1 c

p+ (p+ 1)b−p(p+ 1)(p+ 4)

12 a²− p+ 1

p b²+O(|a|³+|b|³)

. We point out that with this parametrization, replacingaby−agives the same solution up to a spatial translation,

(2.2) P−a,b(z) =P_a,b(z+π), k−a,b=k_a,b.

Together with the spatial translation invariance this family of solutions gives the gBBM equation (1.1), a three-parameter family of periodic travelling waves qa,b(·+d).

3. SPECTRAL STABILITY

AND BLOCH-WAVE DECOMPOSITION

In this section, we obtain the spectral stability problem and state the main result. Then we describe the Bloch-wave decomposition, and discuss some symmetry properties of the linear operators, which are important for the analysis in Section 5.

We consider solutions of (1.1) of the form

u(x, t) =U(z, t), z=k_a,b(x−ct), whereU satises

(1−k_a,b² ∂zz)(Ut−cka,bUz) +ka,bUz+ka,b(U^p+1)z= 0.

Then the 2π-periodic prole P_a,b, corresponding to the travelling waveq_a,b, is a stationary solution of this equation. Linearizing about P_a,b we nd

(1−k_a,b² ∂_zz)(V_t−ck_a,bV_z) +k_a,bV_z+ (p+ 1)k_a,b(P_a,b^p V)_z = 0, or equivalently,

Vt=cka,bVz−ka,b(1−k_a,b² ∂zz)⁻¹ Vz+ (p+ 1)(P_a,b^p V)z

.

The spectral stability is concerned with the spectrum of the linear oper- ator in the right hand side of this equation,

(3.1) A_a,bV =cka,bVz−ka,b(1−k_a,b² ∂zz)⁻¹ Vz+ (p+ 1)(P_a,b^p V)z

, that we consider in either the Hilbert space X = L²(R) (localized perturba- tions) or the Banach space X = Cb(R) (bounded uniformly continuous per- turbations). In both cases, this operator is closed with dense domain H¹(R) when X =L²(R), and C_b¹(R) whenX =Cb(R). The travelling periodic wave q_a,b is spectrally stable with respect to perturbations in X if the spectrum of the linear operator A_a,b inX is contained closed left half complex plane, and it is spectrally unstable otherwise. Our main result is as follows.

Theorem 1 (Spectral stability). LetX =L²(R) or X =C_b(R).

(i) Assume that 1≤p≤2. Then for any c >1, there exists εc >0 such that for any (a, b) with k(a, b)k ≤ ε_c, the spectrum of the linear operator A_a,b in X entirely lies on the imaginary axis. Consequently, the periodic wave q_a,b is spectrally stable in X.

(ii) Assume that p ≥ 3. There exists a critical speed c_p, 1 < c_p < _p−3^p , and for any c >1there existsε_c >0, such that for any(a, b)with k(a, b)k ≤ε_c, the following properties hold. If c∈ (c_p,_p−3^p ), then the spectrum of the linear operator A_a,b in X entirely lies on the imaginary axis, whereas if c∈(1, c_p)∪ (_p−3^p ,∞), then the spectrum of the linear operatorA_a,binXcontains a complex

number λ with Reλ > 0. Consequently, the periodic wave qa,b is spectrally stable in X if c∈(c_p,_p−3^p ), and spectrally unstable ifc∈(1, c_p)∪(_p−3^p ,∞).

The spectral analysis ofA_a,brelies upon the so-called Bloch-wave decom- position for dierential operators with periodic coecients. For a dierential operator A(z, ∂_z)with2π-periodic coecients, this decomposition shows that the spectrum is exactly the same in both spaces L²(R) and C_b(R), and that it is given by the union of the spectra of the family of operatorsA(z, ∂_z+ iγ), parameterized by the Bloch wavenumber γ ∈ (−¹₂,¹₂], acting in the space of 2π-periodic functions L²_per(0,2π). The fact that the operators A(z, ∂_z + iγ) act inL²_per(0,2π), implies that these operators have compact resolvent, so that their spectra are purely point spectra, consisting of eigenvalues, only. This is a major advantage, since it replaces the problem of locating the spectra of A(z, ∂_z) in L²(R) and C_b(R), which are typically continuous, by that of locating the eigenvalues of the operators A(z, ∂_z+ iγ).

Here, the operator A_a,b in (3.1) is nonlocal due to the presence of the inverse (1−k²_a,b∂_zz)⁻¹. Nevertheless, we show in the Appendix that a similar result holds for this operator, as well. More precisely, we prove

Lemma 3.1. LetX =L²(R) or X =C_b(R). Consider the operator A_a,b in (3.1) acting in X, and the Bloch operators

A_a,b,γ=ck_a,b(∂_z+ iγ)V−k_a,b(1−k²_a,b(∂_z+ iγ)²)⁻¹(∂_z+ iγ)(V+(p+1)P_a,b^p V), acting in L²_per(0,2π) with domainH_per¹ (0,2π). Then

(3.2) σ(A_a,b) = [

γ∈(−¹

2,¹₂]

σ(A_a,b,γ).

(Here σ(A) denotes the spectrum of an operator A.)

Using this lemma, Theorem 1 will follow from the next result.

Proposition 3.2. Let c_p and ε_c be as in Theorem 1.

(i) If 1≤p ≤2, then the spectrum of the linear operator A_a,b,γ consists of purely imaginary eigenvalues for any c > 1, any (a, b) with k(a, b)k ≤ ε_c, and any γ ∈(−¹₂,¹₂].

(ii) If p ≥ 3, then the spectrum of the linear operator A_a,b,γ consists of purely imaginary eigenvalues for anyc∈(cp,_p−3^p ), any(a, b)withk(a, b)k ≤εc, and anyγ ∈(−¹₂,¹₂]. Forc∈(1, c_p)∪(_p−3^p ,∞)and any(a, b)withk(a, b)k ≤ε_c, the linear operatorA_a,b,γ possesses an unstable eigenvalue λwithReλ >0, for suciently small γ=o(|a|).

We prove this result in Section 5. The proof is done for the operators B_a,b,γ= 1

ka,b

A_a,b,γ=c(∂_z+iγ)V−(1−k²_a,b(∂_z+iγ)²)⁻¹(∂_z+iγ)(V+(p+1)P_a,b^p V).

Clearly, it is enough to show that Proposition 3.2 holds forB_a,b,γ.

Remark 3.3 (Periodic perturbations). For periodic perturbations which have the same period as the wave, the spectral stability problem consists in considering the operatorA_a,bin the space of2π-periodic functionsL²_per(0,2π), or equivalently in considering the operator A_a,b,γ withγ = 0. An immediate consequence of the results in Section 5, is that the spectrum of A_a,b,0 consists of purely imaginary eigenvalues only, so that the periodic waves are spectrally stable in L²_per(0,2π) for all p ≥1 and any speed c >1. The instability found for p≥3and c∈(1, cp)∪(_p−3^p ,∞), is of side-band type, due to perturbations with periods close to the period of the wave (small nonzero values of γ in the Bloch-wave decomposition).

We end this section with a brief discussion of some symmetry properties of the spectra of the operatorsA_a,bandA_a,b,γ which will be used in the following.

The same properties also hold for operators B_a,b= _k¹

a,bA_a,b and B_a,b,γ.

First, since the operator A_a,b has real coecients, its spectrum is sym- metric with respect to the real axis. For the Bloch operator A_a,b,γ, the cor- responding property is σ(A_a,b,γ) = σ(A_a,b,−γ). Next, it is straightforward to check that A_a,b has a reversibility symmetry, i.e., it anti-commutes with the isometry S dened by

(3.3) SV(z) =V(−z).

Thus SA_a,b =−A_a,bS, which implies that the spectrum of A_a,b is symmetric with respect to the origin in the complex plane. The corresponding property for the Bloch operators is SA_a,b,γ = −A_a,b,−γS, which implies that σ(A_a,b,γ) =

−σ(A_a,b,−γ). Finally, from relations (2.2), we nd that σ(A_a,b) = σ(A_−a,b) and σ(A_a,b,γ) = σ(A_−a,b,γ). Summarizing, the spectra of the Bloch operators A_a,b,γ satisfy

σ(A_a,b,γ) =σ(A_a,b,−γ) =−σ(A_a,b,−γ) =−σ(A_a,b,γ), (3.4)

σ(A_a,b,γ) =σ(A_−a,b,γ).

In particular, the spectrum ofA_a,b,γis symmetric with respect to the imaginary axis, and we can restrict ourselves to positive values γ ∈ [0,¹₂] without loss of generality.

4. UNPERTURBED OPERATORS AND LINEAR DISPERSION The spectral analysis for the operators B_a,b,γ is based on perturbation arguments in which we regard B_a,b,γ as a perturbation of the operator with constant coecientsB_0,0,γ.

We consider the dierence

B_a,b,γ¹ =B_a,b,γ− B_0,0,γ = (∂_z+ iγ)(1−k²₀(∂_z+ iγ)²)⁻¹(V + (p+ 1)P₀^pV)

−(∂_z+ iγ)(1−k_a,b² (∂_z+ iγ)²)⁻¹(V + (p+ 1)P_a,b^p V),

where we write k0 andP0 instead of k0,0 andP0,0, respectively. This operator is compact in L²_per(0,2π), with normkB¹_a,b,γk=O(|a|+|b|), as(a, b)→(0,0). Notice that this estimate is uniform for γ ∈ [0,¹₂]. A standard perturbation argument then shows that the spectra of B_a,b,γ and B_0,0,γ stay close for small (a, b). More precisely, we have the following result.

Lemma 4.1. Let p≥1 and c >1. For any δ >0 there exists ε >0 such that, for any γ ∈[0,¹₂]and any (a, b) ∈R² with k(a, b)k ≤ε, the spectrum of B_a,b,γ satises

σ(B_a,b,γ)⊂ {λ∈C; dist(λ, σ(B_0,0,γ))< δ}.

In order to locate the spectrum ofB_a,b,γ, we need to determine the spec- trum of B_0,0,γ. SinceB_0,0,γ is a dierential operator with constant coecients, a straightforward Fourier analysis allows to compute its spectrum explicitly as (4.1) σ(B_0,0,γ) ={iω_n,γ;n∈Z} ⊂iR,

whereω_n,γ are determined from the linear dispersion relation

(4.2) ω(k) =ck²₀ k³−k

1 +k²₀k²,

through ωn,γ = ω(n+γ). Any λ∈ σ(B_0,0,γ) is a semi-simple eigenvalue with multiplicity equal to the number of distinct n ∈Z satisfying λ = iω_n,γ while the associated eigenvectors are en= e^inz.

It is now important to determine the more precise location of these eigen- values and, in particular, their multiplicity. (As we shall see, simple eigenvalues remain purely imaginary for small nonzero(a, b), but multiple eigenvalues may leave the imaginary axis and induce instabilities.)

The location of the eigenvalues ofB_0,0,γ depends upon the value of k₀²= c−1

c p.

We distinguish upon k²₀ ≤3, which corresponds to any c >1 when1≤p≤3, and to c∈ (1,_p−3^p ]when p ≥4, and k₀² >3, which corresponds top ≥4 and c > _p−3^p .

First, atγ = 0 we nd that

ω−1,0 =ω0,0 =ω1,0 = 0, and

· · ·< ω−3,0< ω−2,0<0< ω2,0< ω3,0<· · ·,

for all values of k₀². Next, for γ ∈(0,¹₂]we have

· · ·< ω−3,γ < ω−2,γ < ω_0,γ <0< ω−1,γ < ω_1,γ < ω_2,γ < ω_3,γ <· · · , if k₀² ≤ 3 whereas for k²₀ > 3 there are pairs of eigenvalues which collide at certain values of γ. However, it turns out that these collisions do not play a role for our stability result, so we omit the detailed calculations here. We point out that when k²₀ >3 we have

· · ·< ω−3,γ < ω−2,γ <0< ω1,γ < ω2,γ <· · ·

and ω_0,γ <0< ω−1,γ.Then, fork²₀ ∈(3,4]the positive eigenvaluesiω−1,γ and iω1,γ collide at some γ ∈ (0,¹₂], and all other eigenvalues are distinct. As for the negative eigenvalues, we nd collisions betweeniω_0,γ andiω−n,γ, forn≥2, when k₀² ≥4. More precisely, there is a sequence

κ₂ = 4, κ_n= (2n²−5) + 2p

(n²−1)(n²−4), n≥3,

such that the eigenvalues iω0,γ and iω−n,γ collide at some γ ∈ (0,¹₂], when k²₀ ≥κn.

We summarize in the following lemma the properties of the eigenvalues iωn,γ which are needed to locate the spectra ofB_a,b,γ in the next section.

Lemma 4.2. Assume k₀² >0.

(i) The spectrum of B_0,0,γ consists of semi-simple eigenvalues λ= iω_n,γ, n∈Z, with associated eigenfunctions e_n= e^inz.

(ii) There exist γ0 ∈ (0,¹₂) and C0 > 0 such that for any γ ∈ [0, γ0] we have

|ω_n,γ| ≤ ck²₀

1 + 4k₀², n= 0,±1, |ω_n,γ| ≥ 4ck²₀

1 + 4k²₀, n6= 0,±1, and

|ω_n,γ−ωp,γ| ≥C0, ∀p, n∈Z\ {0,±1}.

(iii) Assume k²₀ ≤ 3. Then for any γ∗ ∈ (0,¹₂) there exists a positive constant C∗ > 0 such that for any γ ∈ [γ∗,¹₂] we have |ω_n,γ −ω_p,γ| ≥ C∗,

∀p, n∈Z.

Together with Lemma 4.1, the result on the location of the eigenvalues of B_0,0,γ in Lemma 4.2 is a key ingredient in the perturbation arguments used to locate the spectrum ofB_a,b,γ in the next section.

Remark 4.3 (Hamiltonian structure). As is well-known, the gBBM equa- tion possesses a Hamiltonian structure. A consequence of this property is that the linear operator B_a,b decomposes asB_a,b=J L_a,b,where

J =∂x, L_a,b=c−(1−k²_a,b∂zz)⁻¹(1 + (p+ 1)P_a,b^p ).

Similarly, for the Bloch operators we have B_a,b,γ =J_γL_a,b,γ,with

J_γ=∂x+ iγ, L_a,b,γ =c−(1−k_a,b² (∂z+ iγ)²)⁻¹(1 + (p+ 1)P_a,b^p ).

The operatorsJ andJ_γare skew-adjoint, whereas the operatorsL_a,bandL_a,b,γ are self-adjoint. We point out that the symmetry with respect to the imaginary axis of the spectrum of B_a,b,γ can be deduced from this decomposition, as well (e.g., see [15, Proposition 2.5]).

A general result from the theory of Hamiltonian systems shows that for linear operators of this type, upon changing parameters, colliding purely imagi- nary eigenvalues do not leave the imaginary axis when they have the same Krein signatures. In other words, when perturbing an operator with purely imagi- nary spectrum, such asB_0,0,γ in our case, eigenvalues may leave the imaginary axis, and so become unstable, only through a collision of purely imaginary eigenvalues with opposite Krein signatures. For the operator B_0,0,γ here, the Krein signature of an eigenvalueiωn,γ is given by the sign of the scalar product (L_0,0,γe^inz,e^inz),and a direct calculation shows that the eigenvaluesiω0,γ and iω−1,γ have negative Krein signature while all other eigenvalues have positive Krein signature. In the discussion on the location of iωn,γ above, we have detected three types of collisions: of iω_0,γ, iω_1,γ and iω−1,γ, at γ = 0, of iω_1,γ and iω−1,γ, and of iω_0,γ and iω−n,γ, n ≥2, at some γ 6= 0, when k₀² >3. In all these cases, we nd eigenvalues with opposite Krein signature, which could then leave the imaginary axis under small perturbations. Since this is only a necessary condition, an additional analysis is required in order to detect the eigenvalues which indeed leave the imaginary axis and become unstable (as, for example, the one given in the proof of Proposition 5.1 for the eigenvalues iω0,γ,iω1,γ and iω−1,γ which collide whenγ = 0).

A rigorous proof of this result on colliding eigenvalues, for a general class of Hamiltonian linear operators, including operators with periodic coecients, has been recently given in [15]. Though this result does not directly apply to the operators B_a,b,γ, the method of proof can be adapted to the present situation and used for the proof of Proposition 3.2. However, we rely here on the more direct approach in [10], which does not make explicit use of the Hamiltonian structure of the problem.

5. LOCATION OF THE BLOCH SPECTRA

In this section, we locate the spectra of the operators B_a,b,γ and prove Proposition 3.2. First, we determine the location of the three eigenvalues which are the continuation of iω0,γ and iω±1,γ for small (a, b), when γ is suciently small (Proposition 5.1). In particular, this will prove the instability result in Proposition 3.2 (ii), for p≥3 and c∈(1, cp)∪(_p−3^p ,∞). Next, we restrict to

0 < k²₀ < 3, i.e., speeds c > 1 for 1 ≤ p ≤ 2, and 1 < c < _p−3^p for p ≥ 3.

We locate the remaining eigenvalues for γ suciently small, and the entire spectrum of B_a,b,γ for γ ∈[γ∗,¹₂], for some γ∗ ∈(0,¹₂) (Proposition 5.2). This will complete the proof of Proposition 3.2.

We start by considering γ ≈ 0. The result in Lemma 4.2 (ii), together with the perturbation result in Lemma 4.1, provides us with a spectral splitting

σ(B_a,b,γ) =σ1(B_a,b,γ)∪σ2(B_a,b,γ), for B_a,b,γ, with

σ1(B_a,b,γ)⊂B

0; 2ck²₀ 1 + 4k₀²

, σ2(B_a,b,γ)∩B

0; 3ck₀² 1 + 4k²₀

=∅.

The set σ₁(B_a,b,γ) consists of the three eigenvalues which are the continua- tion, for small (a, b), of the eigenvalues iω0,γ, iω±1,γ of B_0,0,γ, whereas the set σ₂(B_a,b,γ) is the continuation of the eigenvalues iω_n,γ, n ∈ Z\ {0,±1}. We determine the location of the three eigenvalues in σ₁(B_a,b,γ) in the next proposition.

Proposition 5.1. Assume p ≥ 1 and c > 1. There exist cp > 1 and positive constants γ1 and ε1, such that for any γ ∈ [0, γ1], and (a, b) with k(a, b)k ≤ε₁, the following holds.

(i) If 1≤p≤2 then the set σ1(B_a,b,γ) consists of three purely imaginary eigenvalues.

(ii) If p ≥ 3 and c ∈ (cp,_p−3^p ), then the set σ1(B_a,b,γ) consists of three purely imaginary eigenvalues.

(iii) If p≥ 3 and c ∈(1, cp)∪(_p−3^p ,∞), then the set σ1(B_a,b,γ) contains one unstable eigenvalue λwith Reλ >0, for suciently small γ =o(|a|).

Proof. We have to locate the three eigenvalues inside σ₁(B_a,b,γ), which are the continuation for small (a, b) of the eigenvalues iω0,γ, iω±1,γ of B_0,0,γ. We construct a suitable basis for the associated spectral subspace and compute the 3×3 matrix M_a,b,γ representing the action ofB_a,b,γ on this space. Then we determine the eigenvalues of this matrix, with the help of the characteristic polynomial.

First, ata=b= 0we have the operatorB_0,0,γ, and a basis for the spectral subspace is in this case formed by the eigenvectors 1and e^±iz associated with the eigenvalues iω_0,γ and iω±1,γ, respectively. For our purpose, it is more convenient to work with the real basis

ξ0 = cos(z), ξ1= sin(z), ξ2 = 1,

in which we then nd the matrix

M_0,0,γ =







i

2(ω1,γ+ω−1,γ) ¹₂(ω1,γ −ω−1,γ) 0

−¹₂(ω1,γ−ω−1,γ) ₂ⁱ (ω1,γ +ω−1,γ) 0

0 0 iω_0,γ





.

Next, atγ = 0we claim that zero is a triple eigenvalue ofB_a,b,0, with geometric multiplicity 2. The origin of this eigenvalue is the fact that together with spatial translations we have a three parameter family of periodic waves, Pa,b(·+d).

Indeed, due to the spatial translation invariance of the gBBM equation we have B_a,b,0(∂_zP_a,b) = 0. Furthermore, dierentiating the equation satised by P_a,b with respect toa we nd

B_a,b,0(∂aP_a,b) =c∂a(k²_a,b) (1−k_a,b² ∂zz)⁻¹∂_z³P_a,b,

and a similar equation is obtained by dierentiating with respect to b. Conse- quently,

B_a,b,0(∂_b(k_a,b² )∂_aP_a,b−∂_a(k_a,b² )∂_bP_a,b) = 0,

which gives a second vector in the kernel of B_a,b,0. Finally, we have B_a,b,0(P_a,b) =p(1−k_a,b² ∂_zz)⁻¹∂_zP_a,b−pc ∂_zP_a,b, and then a straightforward calculation shows that

B_a,b,0(c∂_b(k_a,b² )P_a,b−p k²_a,b∂_bP_a,b) =pc(1−c)∂_b(k²_a,b)∂zP_a,b.

This gives a principal vector in the generalized kernel of B_a,b,0 and, together with the two vectors above, proves the claim and provides us with a basis for the spectral subspace for γ = 0. Using the expansions of these three vectors for small(a, b), we modify this basis in order to obtain one which is compatible with the basis found at a=b= 0. We take

ξ₀ = c

(p+ 1)(c−1)^1+1/p ∂_b(k_a,b² )∂_aP_a,b−∂_a(k_a,b² )∂_bP_a,b

= cos(z)−2p−1

6 a+ p+ 1

6 cos(2z)a−2

pcos(z)b+O(a²+b²), ξ1 =− 1

(c−1)^1/pa∂zP_a,b= sin(z) +p+ 1

6 sin(2z)a+O(a²+b²),

ξ2 = 1

(1 +k₀²)(c−1)^1+1/p c∂_b(k_a,b² )P_a,b−p k_a,b² ∂_bP_a,b

= 1 + p+ 1

1 +k₀²cos(z)a− p+ 1

p(1 +k²₀)b+O(a²+b²).

In this basis we obtain

M_a,b,0 =





0 0 0

0 0 m_a,b

0 0 0



, where

m_a,b= pc

1 +k₀²∂_b(k²_a,b)a= (c−1)(p+ 1)

1 +k²₀ (pa−2ab) +O(|a|(a²+b²)).

The two bases found at a=b= 0 and γ = 0 above, can be extended to a basis for smalla,b, and γ, in which we then have the expansion

M_a,b,γ =M_0,0,γ+M_a,b,0+aγM₁+bγM₂+O(|γ|(a²+b²) +γ²(|a|+|b|)).

For our purposes it is enough to compute the matrix M₁ for which, after a lengthy calculation, we nd

M1 =















0 0 2ic(p+ 1)k₀²

1 +k₀²

0 0 0

−i 6

ck₀²(p(k₀²−3) + 4k₀²+ 6)

1 +k₀² 0 0













 .

In order to determine the location of the three eigenvalues ofM_a,b,γ we consider the characteristic polynomial

P(λ) =λ³+c2λ²+c1λ+c0,

which has coecients cj depending upon a, b, and γ. Since the spectrum of B_a,b,γ is symmetric with respect to the imaginary axis, it is not dicult to conclude that the coecientsc2 andc0are purely imaginary whereasc1is real.

Furthermore, from the last equality in (3.4) we conclude that the coecients c_j of the polynomial P are even functions in a, and from the rst equality in (3.4) we nd that c2 and c0 are odd in γ while c1 is even in γ. Together with the expansion of M_a,b,γ above, these properties imply that the eigenvalues of M_a,b,γ are of the formλ= iγX, withX root of the polynomial

Q(X) =X³+d₂X²+d₁X+d₀,

with real coecients d_j which are all even in a and γ. The location of the roots of this polynomial is determined by its discriminant,

∆a,b,γ = 18d2d1d0+d²₂d²₁−4d³₂d0−4d³₁−27d²₀, that we write as

(5.1) ∆a,b,γ = ∆0,b,γ+αa²+O(a²(a²+|b|+γ²)).

Now, notice that at a = 0 the operator B_a,0,γ has constant coecients, and it is then easy to show that it is skew-adjoint. Consequently, the three eigenvalues ofM_0,b,γ are purely imaginary, so that∆0,b,γ ≥0. The eigenvalues are obtained, just as for B_0,0,γ, from the dispersion relation

ω_b(k) =ck_0,b² k³−k 1 +k²_0,bk²,

by taking k=γ and k=±1 +γ. Then a direct calculation gives (5.2) ∆_0,b,γ = 4c⁶k_0,b¹²(3−k_0,b² )²(3 +k²_0,b)⁴

(1 +k_0,b² )⁸ γ²+O(γ²(γ²+|b|)).

Next, from the explicit expansion of M_a,0,γ, using Maple, we also compute, α= 2

3

(p+ 1)c⁶k_0,b¹²(3−k²₀)(3 +k²₀)³(k²₀(p+ 4) + 3(2−p))

(1 +k₀²)⁷ .

Analyzing the sign of α it is then straightforward to check that α > 0 for 1≤p≤2and for p≥3if c∈(cp,_p−3^p ), where

cp = p(p+ 4) p²+p+ 6 >1,

and that α < 0 for p ≥ 3 if c ∈(1, c_p)∪(_p−3^p ,∞). From (5.1) and (5.2), we conclude that the discriminant ∆_a,b,γ is positive for suciently smalla,b, and γ, when α > 0, and negative for suciently small γ = o(|a|), when α < 0. This completes the proof.

In order to complete the proof of Proposition 3.2, it remains to show that, for 0< k²₀ <3,

• the set σ2(B_a,b,γ) consists of purely imaginary eigenvalues for su- ciently smallγ and(a, b), and that,

• for any γ∗ ∈ (0,¹₂), the spectrum of B_a,b,γ is purely imaginary for all γ ∈[γ∗,¹₂], and suciently small (a, b).

(Recall that the set σ₂(B_a,b,γ)is the continuation of the eigenvaluesiω_n,γ,n∈ Z\ {0,±1} ofB_0,0,γ.) The key ingredient here is the result in Lemma 4.2 (ii) (iii), which shows that the distance between any pair of eigenvalues iω_n,γ, n∈Z\ {0,±1}, ofB_0,0,γ is strictly positive, uniformly for suciently smallγ, and that the same holds for any pair of eigenvalues of B_0,0,γ when γ ∈[γ∗,¹₂]. Together with the perturbation result in Lemma 4.1, this allows us to construct an innite sequence of mutually disjoint balls B(iωn,γ;δ∗),n∈Z\ {0,±1}, for suciently small γ (resp. n∈ Z, for γ ∈ [γ∗,¹₂]), with the property that the setσ2(B_a,b,γ)(resp. σ(B_a,b,γ)) is contained in their union, and that inside each ball the operator B_a,b,γ has precisely one simple eigenvalue. The symmetry of the spectrum of B_a,b,γ with respect to the imaginary axis then implies that

this simple eigenvalue is purely imaginary, so that the set σ2(B_a,b,γ) (resp.

σ(B_a,b,γ)) consists of purely imaginary eigenvalues. The proof of this result is the same as the proof of [10, Proposition 4.4], and we therefore only state the precise result here.

Proposition 5.2. Assume 0< k₀²<3.

(i) There exist positive constants γ2, δ2, and ε2 such that for any γ ∈ [0, γ₂]and (a, b) with k(a, b)k ≤ε₂, the spectrum σ₂(B_a,b,γ) satises

σ₂(B_a,b,γ) ⊂ [

n6=±1,0

B(iω_n,γ;δ₂),

where the closed balls B(iωn,γ;δ2) are mutually disjoints. Inside each ball B(iω_n,γ;δ₂), the operator B_a,b,γ has precisely one purely imaginary eigenvalue.

(ii) Fixγ∗∈(0,¹₂). Then there exist positive constantsδ∗ >0andε∗ >0 such that for any γ ∈[γ∗,¹₂]and any (a, b) with k(a, b)k ≤ε∗, the spectrum of B_a,b,γ satises

σ(B_a,b,γ)⊂ [

n∈Z

B(iωn,γ;δ∗)

and the closed balls B(iω_n,γ;δ∗) are mutually disjoints. Inside each ball B(iωn,γ;δ∗), the operator B_a,b,γ has precisely one eigenvalue, which is purely imaginary.

Appendix A. THE BLOCH-WAVE DECOMPOSITION The key observation in the proof of Lemma 3.1 is that the spectral prob- lem (λ− A_a,b)V =F is equivalent to

(A.1) (λD_a,b− M_a,b)V =G, G=D_a,bF, where

D_a,b= 1−k_a,b² ∂_zz, M_a,b=k_a,b∂_z −ck_a,b² ∂_zz + (c−1)−(p+ 1)P_a,b^p . Here,D_a,bandM_a,bare dierential operators with2π-periodic coecients, and we can use the Floquet theory in order to solve (A.1). We prove the following result, which implies Lemma 3.1.

Proposition A.1. LetX =L²(R) or X=C_b(R). Assume λ∈C. The following statements are equivalent:

(i) λbelongs to the spectrum of the closed operator A_a,b acting inX; (ii) the closed operatorλD_a,b− M_a,b acting inX is not invertible;

(iii) there exists a nonzero function V ∈ C_b³(R) of the form V(z) = W(z)e^iγz, with W being a 2π-periodic function, and some γ ∈ (−¹₂,¹₂] such that (λD_a,b− M_a,b)V = 0;

(iv) there exists γ ∈ (−¹₂,¹₂] such that λ belongs to the spectrum of the closed operator A_a,b,γ acting inL²_per(0,2π).

Proof. First, since the operatorD_a,bis invertible in bothL²(R)andC_b(R), we conclude that (i)⇔(ii). Next, a direct calculation and a bootstrapping argument show that (iii) holds if and only if the kernel of λ− A_a,b,γ is not trivial. The operator A_a,b,γ acting in L²_per(0,2π) has compactly embedded domain H_per¹ (0,2π)and, therefore, it has compact resolvent. Consequently, its spectrum consists of isolated eigenvalues, and in particular λ belongs to its spectrum if and only if the operator λ− A_a,b,γ has a nontrivial kernel. This shows (iii)⇔(iv).

Assume now that (iii) holds. Then the operatorλD_a,b− M_a,b= 0has a non trivial kernel inC_b(R), which implies (ii) whenX=C_b(R). ForX=L²(R) we use a truncation argument. Consider a smooth positive cut-o function χ:R→Rsuch thatχ(z) = 1 for|z| ≤1and χ(z) = 0 for|z| ≥2. Set

V_n(z) =χ(z/2πn)V(z) =χ(z/2πn)W(z)e^iγz.

Then a direct calculation shows that there exist positive constants C >0 and c >0 such that

kV_nk²_L2(R)= Z

R

χ z

2πn 2

|W(z)|²dz≥ Z

|z|≤2πn

|W(z)|²dz≥cn and

k(λD_a,b−M_a,b)V_nk²_L2(R)= Z

2πn≤|z|≤4πn

(λD_a,b−M_a,b) χ z

2πn

V(z)

2

dz≤C n. This implies that λD_a,b− M_a,b does not have a bounded inverse in L²(R), so that (ii) holds when X=L²(R), as well.

Finally, we prove (ii)⇒(iii). We start by rewriting equation (A.1) as a rst order system

(A.2) d

dzv=M(z)v+g,

where v = (V, V_z, V_zz), g = (0,0, G/ck_a,b³ ), and M(z) is a 3×3 matrix with 2π-periodic coecients, namely,

M(z) = 1 ck³_a,b









0 ck³_a,b 0

0 0 ck_a,b³

−λ−(p+1)k_a,b∂z(P_a,b^p ) ka,b(c−1−(p+1)P_a,b^p ) λk_a,b²







 .

(We do not explicitly indicate the dependence uponλ,a, andb, for notational simplicity.) According to the Floquet theory for dierential equations with

periodic coecients, any solution of the homogeneous dierential equation as- sociated with (A.2) is of the form

v(z) =Q(z) e^Czv(0),

whereQ(z)is a3×3matrix with2π-periodic coecients andCa3×3matrix with constant coecients. The eigenvaluesν of Care the Floquet exponents.

Assume that (iii) does not hold. Then the matrixChas no purely imag- inary Floquet exponents, and the matrix Q(z) is invertible for any z. Set v(z) =Q(z)w(z), so that system (A.2) becomes

d

dzw=Cw+Q(z)⁻¹g.

Using the variation of constant formula, one concludes that for anyg∈C_b(R) this equation has a unique bounded solution given by

(A.3) w(z) = Z z

−∞

e^C(z−ζ)P_sQ(ζ)⁻¹g(ζ)dζ− Z ∞

z

e^C(z−ζ)P_uQ(ζ)⁻¹g(ζ)dζ, where Pu and Ps are the spectral projectors associated with the eigenvalues of C which have positive and negative real parts, respectively. Going back to equation (A.1) it is now easy to check that it has a unique solution V ∈ C_b³(R) for G∈C_b(R), so that the operator λD_a,b− M_a,b :C_b³(R) →C_b⁰(R) is invertible. Furthermore, the same variation of constant formula allows to show that wgiven by (A.3) is the unique solution inH¹(R) of (A.2) forg∈L²(R), whence we conclude that the operator λD_a,b − M_a,b : H³(R) → L²(R) is invertible, as well. These prove that (ii)⇒(iii), which completes the proof.

Remark A.2. The proof of Proposition A.1 can be easily generalized to operators of the form A=D⁻¹M, where Dand M are dierential operators with periodic coecients, andD is invertible.

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Received 31 March 2008 Université de Franche-Comté Laboratoire de Mathématiques

16 Route de Gray 25030 Besançon, France mharagus@univ-fcomte.fr