Stability for the modified and fourth-order Benjamin-Bona-Mahony equations

In this work we establish new results about the existence of 
smooth, explicit families of periodic traveling waves for the 
modified and fourth-order Benjamin-Bona-Mahony equations. We also 
prove, under certain conditions, that these families are 
nonlinearly stable in the energy space. The techniques employed 
may be of further use in the study of the stability of periodic 
traveling-wave solutions of other nonlinear evolution equations.


1.
Introduction. Nonlinearity is a prevalent feature of mathematical models of natural phenomena. One active area where nonlinearity is centrally important is the theory of nonlinear dispersive evolution equations. In particular the well-known traveling-wave solutions (solitary wave, cnoidal waves) owe their existence to a balance between nonlinearity and dispersion. The study of existence and stability/instability of these special states of motion turn out to be very important to the understanding of phenomena observed in various scientific fields such as fluid mechanics, plasma physics and nonlinear optics. In the present essay, interest will focus upon periodic traveling-wave solutions of certain nonlinear, dispersive wave equations to be introduced presently. The first work about the existence and orbital stability of periodic traveling waves was that of Benjamin [9], a study of periodic waves of cnoidal type for the Korteweg-de Vries equation. Certain lacunae in the stability theory developed in [9] were recently addressed by Angulo, Bona and Scialom [8]. In the last years, several papers about periodic traveling-wave solutions have appeared in the literature dealing with existence, properties, nonlinear stability/instability and spectral stability of solutions. The spectral stability is quite an interesting aspect of this kind of solution, see for instance [2,3,4,6,7,8,17,18,19,23].
where p is a positive integer and u is a real-valued function. The case p = 1 corresponds to the Benjamin-Bona-Mahony (BBM) equation, which has been derived as a model to describe water waves in the long-wave regime, see [11,25,26]. When p = 2, equation (1) is known as the modified BBM equation, which describes wave propagation in a one-dimensional nonlinear lattice [29,30]. Thus the generalization considered here is not only of mathematical interest.
In the present work, we develop a stability theory for the periodic traveling-wave solutions of equation (1), in the cases p = 2 (mBBM) and p = 4 (4-BBM). The other possible cases, p = 3, 5, 6, 7, . . ., are not considered here because the methods used to obtain solutions present some difficulties. For instance, if odd p ≥ 3, we do not know the explicit Fourier transform of the solitary wave solution associated to the gBBM equation (see (4) below), which is needed in order to apply the Poisson Summation Theorem. Using the quadrature method, we believe that there are chances of getting periodic waves in the cases not studied here. We will present these results in a future work.
The traveling-wave solutions to be considered here will be of the general form u(x, t) = φ c (x − ct), where φ c : R → R is a smooth periodic function (of period L) and c > 1. Substituting this type of solution in (1) and integrating once, we obtain where d is a constant of integration. Now, since our theory of stability is based on the ideas of Benjamin [10] and Weinstein [31], we will consider the solutions φ c as critical points for the functional E + (c − 1)F, where E and F represent conserved quantities with respect to (1), given by Assuming that the function φ c is a critical point of E + (c − 1)F we conclude that the constant of integration d must be zero and thus φ c satisfies For c > 1, the gBBM equation considered in R has solitary-wave solutions of the form These solitary waves are orbitally stable for all speeds c > 1 if 1 ≤ p ≤ 4. For p ≥ 5, there exists a critical speed c p > 1 such that the solitary waves are nonlinearly stable if c > c p , and nonlinearly unstable when c ∈ (1, c p ), see [27,31]. In the particular case of the BBM equation it was proved [16,22] that the solitary waves are asymptotically stable. Zeng [33] proved the existence and stability of solitarywave solutions for a family of equations of BBM type using variational methods.
Recently, in the periodic setting, Hȃrȃguş [19] considered the spectral stability of periodic waves of the gBBM equations, which are small perturbations of the constant solution u = (c − 1) 1/p in L 2 (R) or C b (R). If 1 ≤ p ≤ 2, Hȃrȃguş obtained spectral stability for all c > 1. For p = 3 there exists c 3 > 1 such that the waves are stable for c > c 3 and unstable in (1, c 3 ), and in the case p > 3 there exists a critical speed c p such that the periodic waves are spectrally stable for c ∈ (c p , p p−3 ), and unstable for c ∈ (1, c p ) ∪ ( p p−3 , ∞).
Angulo, Banquet and Scialom in [5] considered the following general model where p is a positive integer and H is a differential or pseudo-differential operator.
In the context of periodic functions, they obtained sufficient conditions for the orbital stability of traveling waves associated with equation (5). In particular, they proved the existence and nonlinear stability of a family of periodic traveling-wave solutions of cnoidal type for the BBM equation (H = −∂ 2 x ) with minimal period L > 2π and wave speeds c > L 2 L 2 −4π 2 .
In this work we establish, for the mBBM equation, via the Poisson Summation Theorem, the existence of a smooth curve of periodic traveling-wave solutions of dnoidal type of (3) with minimal period L > √ 2π and c > where η and k are positive smooth functions depending on the wave speed c. For the 4-BBM equation, following the ideas in Angulo and Natali [7], we prove the existence of a explicit family of periodic traveling-wave solutions of (3), with minimal period L > π and c > L 2 L 2 −π 2 . The smooth curve of periodic solutions is given by where the parameters η 3 , g, α and k depend smoothly on the wave speed c. It is worth noting that all our stability results hold for any fixed period L > 0 such that pL 2 > 4π 2 and not only for large periods.
To obtain our result of stability for the mBBM and the 4-BBM equation we use the ideas developed by Benjamin [10], Bona [12], Weinstein [32] and Angulo and Natali [6] combined with the recent theory developed by Angulo, Banquet and Scialom in [5]. In [5], sufficient conditions were obtained that garantee certain spectral properties of the linear operator which, in turn, are required to prove stability. Here H is defined as Hu(n) = α(n) u(n), for all n ∈ Z. The symbol α is assumed to be a real, mensurable, locally bounded, even function on R, satisfying the conditions where 1 ≤ m 1 ≤ m 2 , |n| > n 0 , α(n) > b for all n ∈ Z, with b a real constant and A i > 0, for i = 1, 2. Actually, we confirmed that the following set of conditions which guarantee the stability of φ c in H 1 per ([0, L]) are valid in our case too.
(C 0 ) There is a nontrivial smooth curve of periodic solutions for (3) of the form c ∈ I ⊂ R → φ c ∈ H 1 per ([0, L]). (C 1 ) L p has a unique negative eigenvalue and it is simple. (C 2 ) The eigenvalue zero is simple.
Concerning the Cauchy problem associated to equation (1) in the periodic setting, the global well-posedness in the Sobolev space H s per ([0, L]) with s ≥ 1 is good enough for our purposes. This result follows in a straightforward fashion from the result of Albert [1], where it was obtained for the continuous case. It is worth to note that, in the non-periodic case of the BBM equation, there exists a better result given by Bona and Tzvetkov in [13], where they proved that the initial value problem associated to the BBM equation is globally well-posed in H s (R) if s ≥ 0. This result is sharp in the sense that the BBM equation cannot be solved by iteration of a bounded mapping in H s (R) for s < 0. Bona and Chen in [14] have additional results on the well-posedness of a family of equations which includes the gBBM equation.
In the last part of this paper, the stability theory of constant solutions for the gBBM is considered. We use the approach given in [8] to obtain the nonlinear stability of the family of solutions of (3) in the form φ 0 (ξ) = (c − 1) 1/p . Note that the restrictions on the period L and the speed c given by L > 2π √ p and c > pL 2 pL 2 −4π 2 , always come up when we prove stability in the case of non-constant periodic solutions for the BBM, mBBM and 4-BBM equations. If we impose the same restriction on L and assume 1 < c < pL 2 pL 2 −4π 2 , the constant solutions of the gBBM equation are stable, but if we suppose that 0 < L ≤ 2π √ p , we obtain orbital stability using only the assumption c > 1, which is a necessary condition for the existence of solutions.
The outline of this paper is as follows. We introduce notation to be used throughout the whole article and the well-posedness results in Section 2. Section 3 is devoted to showing the existence and stability of periodic traveling waves for the mBBM equation. In Section 4, we deal with the existence and nonlinear stability of periodic solutions for the 4-BBM equation. Finally, in Section 5, the stability of constant solutions is established.
2. Notation and preliminaries. The L 2 -based Sobolev spaces of periodic functions are defined as follows (for further details see Iorio and Iorio [20]). Let P = C ∞ per denote the collection of all functions f : R → C which are C ∞ and periodic with period L > 0. The collection P of all continuous linear functionals from P into C is the set of periodic distributions. If Ψ ∈ P then we denote the value of Ψ at ϕ by Ψ(ϕ) = Ψ, ϕ . Define the functions Θ k (x) = exp(2πikx/L), k ∈ Z, x ∈ R. The Fourier transform of Ψ is the function Ψ : Z → C defined by the formula Ψ(k) = 1 L Ψ, Θ −k , k ∈ Z. So, if Ψ is a periodic function with period L, we have

MODIFIED AND FOURTH BBM EQUATIONS 855
For s ∈ R, the periodic Sobolev space of order s,, with period [0, L] is simply denoted by H s per . It is the set of all f ∈ P such that (1 + |k| 2 ) We also note that H s per is a Hilbert space with respect to the inner product In the case s = 0, H 0 per is denoted by L 2 per , and its norm by · L 2 per . Of course H s per ⊂ L 2 per , for any s ≥ 0. Moreover, (H s per ) , the topological dual of H s per , is isometrically isomorphic to H −s per for all s ∈ R. The duality is defined by the pairing Thus, if f ∈ L 2 per and g ∈ H s per , with s ≥ 0, it follows that f, g s = (f, g).
For the sake of completeness we present the Poisson Summation Theorem. It will be used in Section 3 to find the periodic traveling-wave solutions for the mBBM equation.
where A > 0 and δ > 0 (so f and f can be assumed to be continuous functions). Then, for any The two series above converge absolutely.
Well-posedness result: Using a fixed point argument combined with the fact that H s per with s > 1/2 is a Banach algebra we obtain the next result.  (1) is globally well-posed in H 1 per . We finish this section presenting the definition of the class P F (2) discrete and one of its properties which will be used repeatedly.
Theorem 2.4. Let α and β be two even sequences in the class P F (2) discrete, then the convolution α * β is in P F (2) discrete (if the convolution makes sense).
3. Existence and stability of periodic traveling-wave solutions for the mBBM equation. In this section we establish the existence of a smooth curve of periodic traveling-wave solutions for the mBBM equation The equation which determines the periodic traveling-wave solution is The solitary wave solutions for the mBBM equation are given by with Fourier transform ϕ R w (ξ) = √ 2wπ sech πξ 2 w w−1 . Then, from Theorem 2.1 we obtain the following periodic function where w > 1 will be chosen later for ψ w to be a periodic traveling-wave solution of (10).
On the other hand, we have the Fourier expansion of the dnoidal Jacobi elliptic function of period L (see Oberhettinger [24]) where K = K(k) is the complete elliptic integral of the first kind and K (k) = K( √ 1 − k 2 ). Because of the shape of the series that determines ψ w , we consider φ c given in (6) with η > 0 and k ∈ (0, 1) fixed, a periodic solution (with period L) for the equation in (11). Then, substituting this type of solution in (11) and using the fact that the fundamental period of the dnoidal function is 2K, we obtain the following compatibility relations

MODIFIED AND FOURTH BBM EQUATIONS 857
Thus, for k ∈ (0, 1) we should have that η ∈ ( √ c − 1, 2(c − 1)) and c−1 c > 2π 2 L 2 . Combining the two equations given in (15) it is easy to see that and since c > 1 we obtain the a priori estimate L > √ 2π. The compatibility relations in (15) also imply that Using again the assumption c > 1 we should have that there exists k L ∈ (0, 1) such that For example, for L = 5, using Maple program we get k L = 0.9041841218 (see Figure  1). If we consider φ c given by (6) with period T φc , we obtain from (15) that the fundamental period of φ c is a function of η given by which is the solitary wave solution for the mBBM. If η → √ c − 1, we obtain the constant solution for the mBBM φ c (x) = √ c − 1.

Now, note that (18) is equivalent to
, therefore we have that k(c) ∈ (0, k L ) for all c ∈ I. The rest of the proof follows from the smoothness of the functions involved and using the fact that k 2 (c) = 2 − 2(c−1) η 2 (c) and (18) imply the compatibility relations given in (15).
In the next corollary we choose the speed w = w(c) in such way that ψ w in (13) becomes a periodic traveling wave with dnoidal profile.
where k = k(c) ∈ (0, k L ) and c > L 2 Proof. From the definition of w and (16) it is easy to see that w w−1 = 2K L K . Then, using (13) and (14) we obtain where in the second identity we used the relation η √ 2c = 2K L , which is obtained from (15). This finishes the proof of the corollary. Proof. We have that Γ(Λ(c), c) = L, for all c ∈ I(c 0 ). By the Implicit Function Theorem we get dΛ(c) dc = − Γc Γη . Since Γ η > 0, we just have to prove that Γ c < 0. In fact, since 20) and 2k 2 K(k)−(1+k 2 )E(k) < 0, for all k ∈ (0, 1), we obtain Γ c < 0, which finishes the proof of the corollary.
where w is given by (19). Then So, it follows that Using the inequality E > K , we get that (2 − k 2 )E (k) − k 2 K (k) > 0, for all k ∈ (0, 1). Then, to finish the proof we just have to show that dk dc > 0. It is enough to study the sign of 2 Corollary 3 implies dη dc = − Γc Γη . Note that From (20), we obtain Therefore 2(c − 1) dη dc − η = 2(c − 1)A > 0. Before proving the principal result of this section we define the type of stability we are considering. per : f = φ c (· + r) for some r ∈ R}, and, for γ > 0 With this terminology, we say that φ c is (orbitally) stable in H 1 per by the flow generated by (1) if the following conditions hold: (i) there exists s 0 such that H s0 per ⊂ H 1 per and the initial value problem associated to (1) is globally well-posed in H s0 per . (ii) for every > 0, there is δ( ) > 0 such that, for all u 0 ∈ U δ ∩ H s0 per , the solution u of (1) with u(0, x) = u 0 (x) satisfies u(t) ∈ U for all t > 0. Otherwise, we say that φ c is unstable in H 1 per . The proof of the next stability theorem is obtained following the ideas in Angulo, Banquet and Scialom [5], see also [10,12,32]. Theorem 3.3. Let φ c be a periodic traveling-wave solution of (3), and suppose that part (i) of the definition of stability holds. Suppose also that the operator L p = −c∂ 2 x + (c − 1) − (p + 1)φ p c has properties (C 1 ) and (C 2 ) in (9). Choose χ ∈ L 2 per such that L p χ = φ c − φ c , and define I = (χ, φ c − φ c ) L 2 per . If I < 0 then φ c is stable.
Next, we present the main result of this section, the stability theorem for the mBBM equation. Proof. First we prove that (C 1 ) and (C 2 ) given in (9) hold for the operator L 2 = −c∂ 2 x − 1 + c − 3φ 2 c . Since φ c is positive and even, from Theorem 8.1 in Angulo, Banquet and Scialom [5], we just have to show that φ c > 0 and φ 2 c belongs to P F (2) discrete. In fact, Corollary 2 implies φ c (n) = c w ψ w (n), for all n ∈ Z. By the Poisson Summation Theorem we obtain ψ w (n) = 1 L ϕ w R ( n L ), where ϕ w is the solitary wave given in (12). Therefore We conclude that φ c ∈ P F (2) discrete, because f (x) = Asech(Bx) ∈ P F (2) continuous with A > 0 and B ∈ R \ {0}. From Theorem 2.4, we also obtain φ 2 c = φ c * φ c ∈ P F (2) discrete. Since φ c > 0, it is easy to see that φ 2 c > 0. Therefore (C 1 ) and (C 2 ) hold for L.
Now, we prove that (C 3 ) given in (9) holds. Since the mapping c → φ c ∈ H n per ([0, L]) is smooth, we have that L − d dc φ c = φ c −φ c . Therefore, using Parseval identity we obtain Hence, where the constant C = C(c, L) > 0. Since the sequence n tanh πn where d = 0 as in (11). Therefore Multiplying (22) by φ c and integrating we arrive at where A φc is constant. Since we are interested in a positive solution (to apply the theory established in Angulo, Banquet and Scialom [5]) we may suppose that Let η 1 , η 2 and η 3 be the nonzero roots of the polynomial F (t) = −t 4 +3(c−1)t 2 +6At. From the last identity we obtain and
If we consider ψ c given in (26) with fundamental period T ψc , we get from (25) that is a strictly increasing function (we prove it later), then we obtain the following a priori estimate (ii). If we suppose that ψ c (0) = η 2 and ψ c (ξ 0 ) = η 3 for some ξ 0 ∈ (0, L), we obtain from formula 256.00 in [15] that This positive solution converges to zero when k → 1 − . In a future work the stability theory for this family of solutions will be addressed. (iii). Other possible choices for η 1 , η 2 and η 3 always produce a solution that is either negative or zero at some point . We do not have a theory for solutions which can be negative. Now, we prove the existence of a smooth curve of periodic solutions with fixed minimal period L > π for the equation (21). Since the square root is a smooth curve on (0, +∞) and the fundamental periods of φ c and ψ c are the same, we will study the curve
where ψ is given in (26), determine by η 3 has fundamental period L and satisfies (22). Furthermore Proof. The proof follows the ideas of Theorem 4.2 in Angulo and Natali [7]. For this reason we only present a sketch.

Now, note that (35) is equivalent to
, therefore we obtain k(c) ∈ (0, k L ), for all c ∈ I. The rest of the proof follows from the smoothness of the functions involved.
Proof. By Theorem 4.1, we have that Γ(Λ(c), c) = L for all c ∈ I(c 0 ). Then using the Implicit Function Theorem, we obtain We already showed that Γ η > 0, then we only have to prove that Γ c < 0. In fact, We consider two cases. If Since k 2 (2 + k 2 )K − 2E < 0, we have Γ c < 0, which finishes the proof of the corollary.
Fourier expansion of ψ c : Using a very similar analysis as in Angulo and Natali [7] we obtain the following Fourier expansion of the periodic traveling-wave solution given in (26) ψ c (ξ) where Λ 0 is the Heuman's Lambda function, see Byrd and Friedman [15].
Since it will be necessary to establish our result of stability for the 4-BBM equation we express η 1 , η 2 , and η 3 as functions of the modulus k ∈ (0, k L ). Replacing c given in (33) on the first equation of (31) we obtain Then, from the last identity and (29) we have that Next, we present our theorem of stability for the 4-BBM equation. It is worth to note that the next result differs from that one obtained by Angulo and Natali in [7] for periodic traveling-wave solutions of the critical KdV equation. Indeed, the authors in [7] proved the existence of a critical speed c L such that the periodic traveling-wave solutions ϕ c (of period L), associated to the critical KdV, are orbitally stable if c ∈ ( π 2 L 2 , c L ) and orbitally unstable if c > c L . In our case we do not have such a critical speed.

Note that we have
1+f (k) . If we define the function m (which does not depend on the period L) by we obtain that On the other hand, multiplying (22) by φ c and integrating from 0 to L we get Integrating (23) from 0 to L yields Combining (41) and (42), it is easy to see that From the last identity, (40), (25) and the expression given for η 1 , η 2 and η 3 , we have that Thus, from (33) we get that From Corollary 6 and the fact that for any L > π, the function P L is strictly increasing in (0, k L ) (see Figure 3 and 4), we obtain that d Finally, using a very similar analysis as in Angulo and Natali [7] we prove that φ 4 ∈ P F (2) and φ(n) > 0, for all n ∈ Z, which finishes the proof of the theorem. The proof of our result of stability is based on the conserved quantities given in (2). The next functional will be useful to obtain our result of stability for the constant solutions x − p(c − 1). Let us start studying the spectral properties of L 0 . √ p and c < pL 2 pL 2 −4π 2 , the operator L 0 has its first eigenvalue negative and the rest of the eigenvalues are double and positive.
(ii) If L > 2π √ p and c = pL 2 pL 2 −4π 2 , the operator L 0 has its first eigenvalue negative, zero is a double eigenvalue and the rest of the eigenvalues are double and positive. (iii) If L ≤ 2π √ p , we obtain the same result in (i) for all c > 1.
Next we present our theorem of stability for the constant solutions.
Proof. Define the functionals E, F : H 1 per ([0, L]) −→ R given by (2). E and F are well-defined in H 1 per and are continuous. Consider v := u − φ 0 ∈ H 1 per and let u 0 be the initial data associated to the periodic problem u t + u x + (p + 1)u p u x − u xxt = 0, t ≥ 0, x ∈ R u(x, 0) = u 0 (x).
Define B := E + (c − 1)F. Then using the immersion H 1 per → L q per , for all q ≥ 2, it is easy to see that Assume that F(u 0 ) = F(φ 0 ), then F(u(t)) = F(φ 0 ). Since v = u − φ 0 we obtain Define v ⊥ := v − v, where v = 1 L L 0 vdx. Note that L 0 v ⊥ dx = 0. So, it follows from the Poincaré's inequality that Using the last inequality, we have that Since 0 < c−1 c < 4π 2 L 2 p , we obtain β 1 = 4π 2 c L 2 − p(c − 1) > 0, and therefore On the other hand, using (44) we get that Therefore