Orbital stability of one-parameter periodic traveling waves for dispersive equations and applications
aa r X i v : . [ m a t h . A P ] A p r ORBITAL STABILITY OF ONE-PARAMETER PERIODIC TRAVELINGWAVES FOR DISPERSIVE EQUATIONS AND APPLICATIONSThiago Pinguello de Andrade
Departamento de Matem´atica - Universidade Tecnol´ogica Federal do Paran´aAv. Alberto Carazzai, 1640, CEP 86300-000, Corn´elio Proc´opio, PR, [email protected]
Ademir Pastor
IMECC-UNICAMPRua S´ergio Buarque de Holanda, 651, CEP 13083-859, Campinas, SP, [email protected]
Abstract.
This paper establishes sufficient conditions for the orbital stability of one-parameter spatially periodic traveling-wave solutions for one-dimensional dispersive equa-tions. Our method of proof combines known techniques with some new ideas. As a con-sequence of our result, we give several applications for well known dispersive equations.Extension of the theory to regularized equations is also established. Introduction
This paper sheds new contributions on the orbital stability theory of one-parameterperiodic traveling-wave solutions for nonlinear dispersive models which can be written inthe forms u t − M u x + ∂ x ( f ( u )) = 0 (1.1)and u t + M u t + ∂ x ( u + f ( u )) = 0 , (1.2)where f : R → R is a (at least) C -function, in general representing the nonlinearity, and M is a differential or pseudo-differential operator.Equations as in (1.1) and (1.2) appear in many physical situations. For instance, itdescribes long-crested, long-wavelength disturbances of small amplitude propagating pri-marily in one direction in a dispersive media (see [14]). In particular, when M = − ∂ x and f is a function having the form f ( u ) = u p +1 , where p ≥ u t + u xxx + ∂ x ( u p +1 ) = 0 , (1.3)whereas (1.2) reduces to the generalized regularized long-wave equation or generalizedBenjamin-Bona-Mahony (BBM) equation u t − u xxt + u x + ∂ x ( u p +1 ) = 0 . (1.4)We point out that we will be primarily interested in equations having the form (1.1),because the theory can be easily extended to (1.2) (see Section 4). Traveling-wave solutions(or, for short, traveling waves) for (1.1) are those solutions having the form u ( x, t ) = φ ( x − ct ) , (1.5)where φ : R → R is a suitable function representing the profile of the wave and c is a realconstant representing the wave speed. The traveling waves of main interest are classifiedinto two classes: solitary and periodic traveling waves. Solitary waves are those for which Mathematics Subject Classification.
Primary 35A01, 35Q53 ; Secondary 35Q35.
Key words and phrases.
Dispersive equations; Orbital stability; Periodic waves. φ , together with all its derivatives go to zero at infinity, whereas periodic traveling wavesare those for which φ is a periodic function of its argument with a fixed period L >
0. Thestudy of existence and stability (linear and nonlinear) of traveling waves has gained a lotof attention is recent decades. The interested reader will find a vast number of works inthe current literature, which we refrain from list them here. Instead, we will focus in theworks closed to ours.To obtain traveling waves, we usually replace the ansatz (1.5) into (1.1) and try todetermine φ and c which provide the desired solutions. Thus, by replacing this waveformin (1.1), we see that φ must be a solution of the differential or pseudo-differential equation( M + c ) φ − f ( φ ) + A = 0 , (1.6)where A appears as an integration constant. Thus, when we are interested in the studyof traveling waves, solving equation (1.6), which turns out to be a two-parameter equa-tion, is the first step. There are several manner of finding traveling waves and the mostpopular ones, to cite a fews, are the quadrature method, by using the implicit functiontheorem or the Lyapunov-Schmidt method, applications of the critical point theory, andconcentration-compactness techniques.Let us now describe the framework in order to present our main result. Operator M shall be formally defined through its Fourier transform by d M u ( m ) = α ( m ) b u ( m ) , m ∈ Z . (1.7)Here and in what follows b u denotes the Fourier transform of the periodic function u .The symbol α is assumed to be a measurable, locally bounded, even, and real functionsatisfying the following:(i) there are real constants s , s > s ≤ s , and A , A > A | m | s ≤ α ( m ) ≤ A (1 + | m | ) s , (1.8)for all m sufficiently large;(ii) there is a real constant γ such that inf m ∈ Z α ( m ) ≥ γ .In many applications the parameters c and A appearing in (1.6) are not independentthemselves and it turn out to depend on a third parameter, which we shall denote by k . This is the case, for instance, when M is a differential operator and f is a power-lawfunction: in several situations the solutions of (1.6) depend on the well-known Jacobianelliptic functions and, as a consequence, the parameters c and A depend on the ellipticmodulus k ∈ (0 , (H0) There are an interval J ⊂ R , C -functions k ∈ J c = c ( k ) and k ∈ J A = A ( k ), and a nontrivial smooth curve of L -periodic solutions for (1.6), say, k ∈ J φ k := φ ( c ( k ) ,A ( k )) ∈ H s per ([0 , L ]) with c = c ( k ) > − γ .Here and in what follows H sper ([0 , L ]) will denote the periodic Sobolev space of order s (see definition below). The condition c > − γ is necessary in order to make the operator M + c positive.After the linearization of (1.1) around a periodic solution φ = φ k , we are faced anunbounded, closed, self-adjoint operator L k : D ( L k ) ⊂ L per ([0 , L ]) → L per ([0 , L ]), definedon a dense subspace, by L k u := ( M + c ) u − f ′ ( φ k ) u. (1.9)Our assumptions concerning L k are the following. (H1) The linearized operator L k := L ( c ( k ) ,A ( k )) has a unique negative eigenvalue, whichis simple. TABILITY OF PERIODIC WAVES FOR DISPERSIVE EQUATIONS 3 (H2)
Zero is a simple eigenvalue of L k with associated eigenfunction φ ′ k .In order to introduce the remaining assumptions, let us recall that, at least formally,(1.1) conserves the quantities E ( u ) = 12 Z L ( u M u − F ( u )) dx, with F ( u ) = Z u f ( s ) ds, (1.10) Q ( u ) = 12 Z L u dx, (1.11)and V ( u ) = Z L u dx. (1.12)Note that (1.6) is nothing but the Euler-Lagrange equation associated with the functional F k := E + cQ + AV, (1.13)that is, solutions of (1.6) are critical points of F k . Thus, as is well understood, the natureof these points are crucial to determine their stability. In addition, L k is nothing but thesecond order Fr´echet derivative of F k at φ k , that is, L k = F ′′ k ( φ k ). Thus, it is expectedthat the spectrum of L k plays a crucial role in the stability analysis. We will show that (H1)-(H2) is sufficient to our purposes.Next, for a fixed k ∈ J , we introduce the functional M k ( u ) := ∂c∂k Q ( u ) + ∂A∂k V ( u ) , where ∂c∂k and ∂A∂k denote, respectively, the derivatives of the functions c ( k ) and A ( k ) at k .We assume the following. (H3) The quantity Φ defined by Φ := D L k (cid:16) ∂φ k ∂k (cid:17) , ∂φ k ∂k E is negative. (H4) It holds M k ( φ k ) = − ∂c∂k Q ( φ k ).Once property (H0) has been proved, our main goal is to show that (H1)-(H4) combineto establish the orbital stability of the traveling wave φ k , for each k ∈ J fixed. In orderto make clear the definition of orbital stability, let us observe that (1.8) implies that theoperator L k is well-defined on H s per ([0 , L ]) and the natural Sobolev space to consider theflow of (1.1) is the energy space H s / per ([0 , L ]). Definition 1.1.
Let φ k be an L -periodic solution of (1.6) . We say that φ k is orbitallystable (by the flow of (1.1) ) in H s / per ([0 , L ]) if, for any ε > , there exists δ > such thatif u ∈ H s / per ([0 , L ]) satisfies k u − φ k H s / per < δ, then the solution u ( t ) of (1.1) , with initial data u , exists globally and satisfies sup t ∈ R inf r ∈ R k u ( t ) − φ k ( · + r ) k H s / per < ε. Otherwise, we say that φ k is H s / per -unstable. Remark 1.2.
Note that, by definition, if the Cauchy problem associated with (1.1) is notglobally well-posed in H s / per ([0 , L ]) , at least for initial data in a small neighborhood of φ k ,then any traveling wave φ k is H s / per -unstable. Since the issue of well-posedness is out ofthe scope of this manuscript, in what follows we will assume that (1.1) always admit global STABILITY OF PERIODIC WAVES FOR DISPERSIVE EQUATIONS solutions for initial data in H s / per ([0 , L ]) , that is, for any u ∈ H s / per ([0 , L ]) , (1.1) has aunique solution u satisfying u (0) = u and u ∈ C ([ − T, T ]; H s / per ([0 , L ])) , for any T > . Our main theorem concerning orbital stability reads as follows.
Theorem 1.3 (Orbital stability) . Under assumptions (H0)-(H4) , for each k ∈ J , theperiodic traveling wave φ k is orbitally stable in H s / per ([0 , L ]) . The strategy to prove Theorem 1.3 follows the classical arguments in [16] and [24].Roughly speaking, if one restricts the augmented energy functional to a suitable manifold(see (2.2)) then the critical point in question is a minimum, which in turn implies theorbital stability.Before proceeding, a few words of explanation concerning assumptions (H3)-(H4) arein order. Assume for the moment that A = 0 and (1.6) has a family of solutions c φ c ,for c in an open interval. In this case, as is well-known in the current literature (see, forinstance, [16] and [24]), the Vakhitov-Kolokolov type condition ddc Z φ c dx > , (1.14)together with assumptions (H1)-(H2) (with L k replaced by L c = M + c − f ′ ( φ c )) issufficient to imply the stability of φ c . Taking the derivative with respect to c in (1.6) wededuce that L c ( ∂ c φ c ) = − φ c . Therefore,12 ddc Z φ c dx = Z φ c ∂ c φ c dx = − Z L c ( ∂ c φ c ) ∂ c φ c dx = −hL c ( ∂ c φ c ) , ∂ c φ c i , and (1.14) is equivalent to hL c ( ∂ c φ c ) , ∂ c φ c i < . (1.15)Thus, condition Φ < c = k ,we have that (H4) is fulfilled provide φ c = 0. Thus, (H4) appears as an extra assumptionin our context. It should be noted that (H4) is indeed used to prevent M k ( φ k ) from beinga critical value of a parabola defined in the proof of Theorem 1.3. We believe, however,that this assumption is not restrictive as is shown in our applications.Next, let us try to relate our work with the ones in the current literature (see alsoour applications below). The stability of spatially periodic traveling waves for dispersiveequations was initiated by T.B. Benjamin (see [12] and [13]), when studying the stabilityof cnoidal waves of the KdV equation. It should pointed out, however, that the orbitalstability of such waves was completed a couple of decade later (see [11]). Specially after[11], much effort has been expended on the stability theory of periodic traveling waves, andthe issue has been attracted the attention of a much broader community of mathematiciansand physicists.Most of the works in the literature do not assume that c and A depend on a thirdparameter and, instead, solutions of (1.6) are parametrized by c and A themselves (oreven more parameters). Specially in the case M = − ∂ x , the main results in this directionare provided by M. Johnson [27] and collaborators (see also [18], [28], and referencestherein). Equation (1.6) are then written as − φ ′′ + cφ − f ( φ ) + A = 0 . (1.16) TABILITY OF PERIODIC WAVES FOR DISPERSIVE EQUATIONS 5
Multiplying (1.16) by φ ′ and integrating once, we obtain −
12 ( φ ′ ) + c φ − F ( φ ) + Aφ = B, (1.17)where B in another integration constant, interpreted as the energy. If the effective poten-tial Γ( φ ) = − c φ + F ( φ ) − Aφ has a local minimum, then from the theory of differential equations, (1.17) has periodicsolutions which can be parametrized by the parameters ( c, A, B ). In [27], [18] the authorthen establishes some criteria to determine the orbital stability of such waves dependingon the sign of certain determinants, which encode some geometric information about theunderlying manifold of periodic solutions. This approach is much general. Indeed, it bringsmany important contributions to the theory of orbital stability for periodic traveling wavesand it has been successfully applied in several situations. For one hand, our approach ismore particular and it does not provide some criterion to study the orbital stability of all periodic solution of (1.6) (or (1.16)). On the other hand, when applicable, our methodyields some simplifications and even provides new results.Besides this introduction, the paper is organized as follows. In Section 2 we prove The-orem 1.3. In Section 3 we give the applications of our method to the Korteweg-de Vries,modified Korteweg-de Vries, Gardner, Intermediate Long Wave, and Schamel equations.Extension to regularized equations as in (1.2) will be given in Section 4. Applications tothe regularized Schamel and modified Benjamin-Bona-Mahony equations are also provided. Notation.
For s ∈ R , the Sobolev space H sper = H sper ([0 , L ]) is the set of all periodicdistributions such that || f || H sper := L P + ∞ k = −∞ (1 + | k | ) s | b f ( k ) | < ∞ , where b f is the (pe-riodic) Fourier transform of f . For s = 0, H sper ([0 , L ]) is isometric to L per ([0 , L ]). Thenorm and inner product in L per will be denoted by k · k and ( · , · ) L per , respectively. By h· , ·i we mean the duality pairing H sper - H − sper . The symbols SN( · , k ), DN( · , k ), and CN( · , k )represent the Jacobi elliptic functions of snoidal , dnoidal , and cnoidal type, respectively.Recall that SN( · , k ) is an odd function, while DN( · , k ), and CN( · , k ) are even functions.For k ∈ (0 , K ( k ) and E ( k ) will denote the complete elliptic integrals of the first andsecond type, respectively (see e.g., [20]).2. Proof of Theorem 1.3
Our goal in this section is to prove Theorem 1.3, that is, to show how Assumptions (H1)-(H4) implies the orbital stability of the periodic traveling wave φ k .To begin with, in H s / per ([0 , L ]), let us introduce the pseudo-metric ρ defined by ρ ( u, v ) := inf r ∈ R k u − v ( · + r ) k H s / per . (2.1)Given any real number ε > U ε ( φ k ) shall denote the ε -neighborhood of φ k with respectto ρ , that is, U ε ( φ k ) = { u ∈ H s / per ([0 , L ]); ρ ( u, φ k ) < ε } . Also, we introduce the manifold Σ k asΣ k := { u ∈ H s / per ([0 , L ]); M k ( u ) = M k ( φ k ) } . (2.2)Now, we state two classical lemmas. The proofs in our case are very close to the originalones. STABILITY OF PERIODIC WAVES FOR DISPERSIVE EQUATIONS
Lemma 2.1.
There exist ε > and a C map ω : U ε ( φ k ) → R , such that for all u ∈ U ε ( φ k ) , (cid:16) u ( · + ω ( u )) , φ ′ k (cid:17) L per = 0 . Proof.
The proof is based on an application of Implicit Function Theorem. See [16, Lemma4.1] or [6, Lemma 7.7] for details. (cid:3)
Lemma 2.2.
Let A = n ψ ∈ H s / per ([0 , L ]); ( ψ, M ′ k ( φ k )) L per = ( ψ, φ ′ k ) L per = 0 o . Under assumptions ( H0 ) - ( H3 ) there exists C > such that hL k ψ, ψ i ≥ C k ψ k H s / per , ψ ∈ A . Proof.
The proof is quite standard by now, so we omit the details. We refer the interestedreader to [24, Theorem 3.3] or [6, Lemma 7.8]. It should be noted that the assumption d ′′ ( c ) > < (cid:3) In the next lemma we prove that φ k is a local minimum of the functional F k restrictto the manifold Σ k . It worth mentioning that its proof relies on the classical ideas withsome changes in the spirit of [27, Lemma 4.6]. Lemma 2.3.
Under the above assumptions, there exist ε > and a constant C = C ( ε ) such that F k ( u ) − F k ( φ k ) ≥ Cρ ( u, φ k ) , (2.3) for all u ∈ U ε ( φ k ) satisfying M k ( u ) = M k ( φ k ) .Proof. Since F k is invariant by translations, we have F k ( u ) = F k ( u ( · + r )), for all r ∈ R .Thus, it suffices to prove that F k ( u ( · + ω ( u ))) − F k ( φ k ) ≥ Cρ ( u, φ k ) , where ω is given in Lemma 2.1.By fixing u ∈ U ε ( φ k ) ∩ Σ k (with ε as in Lemma 2.1) and making use of Lemma 2.1, itfollows that there exists C ∈ R such that v := u ( · + ω ( u )) − φ k = C M ′ k ( φ k ) + y, where y ∈ T k := { M ′ k ( φ k ) } ⊥ ∩ { φ ′ k } ⊥ . Since u belongs to U ε ( φ k ), up to a translation in φ k , we may assume that v = u ( · + ω ( u )) − φ k satisfies k v k H s / per < ε .Let us prove that C = O ( k v k ). In fact, using the invariance by translation of M k , aTaylor expansion gives M k ( u ) = M k ( u ( · + ω ( u ))) = M k ( φ k ) + h M ′ k ( φ k ) , v i + O ( k v k ) . (2.4)On the other hand, since y ∈ T k , we have h M ′ k ( φ k ) , y i = 0 and h M ′ k ( φ k ) , v i = h M ′ k ( φ k ) , C M ′ k ( φ k ) + y i = C h M ′ k ( φ k ) , M ′ k ( φ k ) i = C N, (2.5)where N is a constant depending only on k . Therefore, since M k ( u ) = M k ( φ k ), it followsfrom (2.4) and (2.5) that C = O ( k v k ) . (2.6)A Taylor expansion at u ( · + ω ( u )) = φ k + v now yields F k ( u ) = F k ( u ( · + ω ( u ))) = F k ( φ k ) + h F ′ k ( φ k ) , v i + 12 h F ′′ k ( φ k ) v, v i + o ( k v k ) . TABILITY OF PERIODIC WAVES FOR DISPERSIVE EQUATIONS 7
Since F ′ k ( φ k ) = 0 and F ′′ k ( φ k ) = L k , we have F k ( u ) − F k ( φ k ) = 12 hL k v, v i + o ( k v k ) . (2.7)Note that the equality v = C M ′ k ( φ k ) + y provides hL k v, v i = C hL k M ′ k ( φ k ) , M ′ k ( φ k ) i + 2 C hL k M ′ k ( φ k ) , y i + hL k y, y i . (2.8)In view of (2.6), we obtain positive constants C , C , and C , depending only on k , suchthat | C hL k M ′ k ( φ k ) , M ′ k ( φ k ) i| ≤ C k v k and | C hL k M ′ k ( φ k ) , y i| ≤ | C |kL k M ′ k ( φ k ) kk y k≤ | C |kL k M ′ k ( φ k ) k (cid:16) k y + C M ′ k ( φ k ) k + k C M ′ k ( φ k ) k (cid:17) ≤ C k v k + C k v k . This last two inequalities together with (2.8) imply hL k v, v i = hL k y, y i + o ( k v k ) . (2.9)Therefore, combining (2.7) with (2.9), we obtain F k ( u ) − F k ( φ k ) = 12 hL k y, y i + o ( k v k ) . By using that y ∈ T k , we have y ∈ A . Thus, Lemma 2.2 gives hL k y, y i ≥ C k y k H s / per , and, consequently, F k ( u ) − F k ( φ k ) ≥ C k y k H s / per + o ( k v k ) . (2.10)By using the definition of v and (2.6) it is easily seen that k y k H s / per ≥ k v k H s / per + o ( k v k H s / per ) , (2.11)provided v is small enough (if necessary we can take a smaller ε > F k ( u ) − F k ( φ k ) ≥ C k v k H s / per + o ( k v k H s / per ) , which, for ε > F k ( u ) − F k ( φ k ) ≥ C ( ε ) k v k H s / per ≥ C ( ε ) ρ ( u, φ k ) and completes the proof of the lemma. (cid:3) Finally, we are in a position to prove Theorem 1.3.
Proof of Theorem 1.3.
The proof follows classical arguments as the ones in [16] and [24].Assume by contradiction that φ k is H s / per -unstable. Then, we can choose ε > w n := u n (0) ∈ U n ( φ k ), n ∈ N , such that ρ ( w n , φ k ) → t ≥ ρ ( u n ( t ) , φ k ) ≥ ε, STABILITY OF PERIODIC WAVES FOR DISPERSIVE EQUATIONS where u n ( t ) is the solution of (1.1) with initial data w n . Here, by taking a smaller ε ifnecessary, we can assume that ε > t , we can take the first time t n > ρ ( u n ( t n ) , φ k ) = ε . (2.12)The strategy now is to obtain a contradiction with (2.12), for n sufficiently large. Let f n be the function defined as f n ( α ) = M k ( αu n ( t n ))= α ∂c∂k Z L | u n ( t n ) | dx + α ∂A∂k Z L u n ( t n ) dx. Since Q and V are conserved quantities, we then see that f n ( α ) = α ∂c∂k Z L | w n | dx + α ∂A∂k Z L w n dx = α Q k ( w n ) + αV k ( w n ) , (2.13)where we have denoted Q k ( u ) := ∂c∂k Q ( u ) and V k ( u ) := ∂A∂k V ( u ) . On the other hand, since ρ ( w n , φ k ) →
0, as n → ∞ , Q and V are invariant by translationsand continuous, we have Q k ( w n ) −→ Q k ( φ k ) =: a, V k ( w n ) −→ V k ( φ k ) =: b. (2.14)Hence, for any α ∈ R , we obtain from (2.13), f n ( α ) → f ( α ) , (2.15)where f ( α ) = α Q k ( φ k ) + αV k ( φ k ) = α a + αb. Before proceeding, we shall show that under assumption (H4) there exist real sequences( α n ) such that M k ( α n u n ( t n )) = M k ( φ k ). Lemma 2.4.
Assume that (H4) holds. We have the following. (i) If ∂c∂k = 0 then there exist two sequences ( α n ) and ( ˜ α n ) , and real numbers θ < θ such that, for all n sufficiently large, M k ( α n u n ( t n )) = M k ( ˜ α n u n ( t n )) = M k ( φ k ) (2.16) and ˜ α n ≤ θ < θ ≤ α n . (2.17) In addition, up to a subsequence, either ( α n ) or ( ˜ α n ) converges to 1. (ii) If ∂c∂k = 0 then there exists a sequence ( α n ) such that, for all n ∈ N , M k ( α n u n ( t n )) = M k ( φ k ) . In addition, ( α n ) converges to 1. TABILITY OF PERIODIC WAVES FOR DISPERSIVE EQUATIONS 9
Proof. (i) Without loss of generality, we shall assume that ∂c∂k >
0. The case, ∂c∂k < a = Q k ( φ k ) >
0. Thus the parabola f has a minimum at x = − b/ a with minimum value f ( x ) = − b / a . In addition, note that f (0) = f (cid:18) − ba (cid:19) = 0 , f (1) = f (cid:18) − b + aa (cid:19) = a + b, and a + b = M k ( φ k ) is not the minimum value of f , otherwise we would have a + b = − a ,contradicting assumption (H4) .Now, fix any real number β satisfying f ( x ) < β < a + b and let K ⊂ R be a compact setcontaining the interval [ x − , x + 1]. Since f n → f in R , we see that f n → f uniformlyin K . Thus, there is N ∈ N such that for any α ∈ K , n ≥ N ⇒ | f n ( α ) − f ( α ) | < a + b − β . Moreover, the continuity of f at x implies the existence of δ ∈ (0 ,
1) such that | α − x | < δ ⇒ | f ( α ) − f ( x ) | < a + b − β . Consequently, if | α − x | < δ and n ≥ N , we deduce f n ( α ) ≤ | f n ( α ) − f ( α ) | + | f ( α ) − f ( x ) | + f ( x ) < a + b + f ( x ) − β< a + b. This mean that a + b is not the minimum value of the parabola f n . Hence, for n ≥ N ,there are real numbers ˜ α n and α n satisfying˜ α n ≤ x − δ < x + δ ≤ α n , such that f n ( ˜ α n ) = f n ( α n ) = a + b . By taking θ = x − δ and θ = x + δ we obtain(2.16) and (2.17).It remains to show that either ( α n ) or ( ˜ α n ) admit a subsequence converging to 1. Since Q and V are conserved quantities, using (2.16), we have ̺ := | α n Q k ( w n ) + α n V k ( w n ) − ( Q k ( w n ) + V k ( w n )) | = | α n Q k ( u n ( t n )) + α n V k ( u n ( t n )) − ( Q k ( w n ) + V k ( w n )) | = | Q k ( α n u n ( t n )) + V k ( α n u n ( t n )) − ( Q k ( w n ) + V k ( w n )) | = | M k ( α n u n ( t n )) − M k ( w n ) | = | M k ( φ k ) − M k ( w n ) | . (2.18)From (2.14), it follows that ̺ →
0, as n → ∞ . Since0 ≤ | α n Q k ( w n ) + α n V k ( w n ) − ( a + b ) |≤ | α n Q k ( w n ) + α n V k ( w n ) − ( Q k ( w n ) + V k ( w n )) | + | ( Q k ( w n ) + V k ( w n )) − ( a + b ) |≤ ̺ + | Q k ( w n ) − a | + | V k ( w n ) − b | , (2.19)by using (2.14), we see that z n := α n Q k ( w n ) + α n V k ( w n ) −→ a + b, as n → ∞ . (2.20) Next we claim that ( α n ) is a bounded sequence. On the contrary, suppose ( α n ) isunbounded. Because Q k ( w n ) > Q k ( w n ) and V k ( w n ) are bounded we obtain, up toa subsequence, z n = α n ( α n Q k ( w n ) + V k ( w n )) −→ + ∞ , as n → ∞ , which contradicts (2.20).It is clear that by defining ˜ z n := ˜ α n Q k ( w n ) + ˜ α n V k ( w n ), the same analysis can beperformed to conclude that the sequence ( ˜ α n ) is also bounded.Therefore, there are subsequences of ( α n ) and ( ˜ α n ), which we still denote by ( α n ) and( ˜ α n ) such that α n −→ α , and ˜ α n −→ ˜ α , as n → ∞ . Taking the limit in z n and ˜ z n , it follows from (2.14) that α a + α b = a + b and ˜ α a + ˜ α b = a + b . This implies that α = 1 or α = − b + aa and ˜ α = 1 or ˜ α = − b + aa . The inequalities (2.17) imply that both sequences cannot converge to the same number.Thus we have either α = 1 or ˜ α = 1.(ii) In this case, we have a = 0 and, in view of assumption (H4) , b = 0. Thus f n and f are linear functions passing through the origin. It is then clear that there is asequence ( α n ) such that M k ( α n u n ( t n )) = f n ( α n ) = b = M k ( φ k ). As before, we concludethat z n := α n V k ( w n ) → b , as n → ∞ . Thus, | α n − || V k ( w n ) | ≤ | z n − b | + | V k ( w n ) − b | . (2.21)Since the right-hand side of (2.21) goes to zero and V k ( w n ) → b = 0, we obtain that α n →
1. The proof of the lemma is thus completed. (cid:3)
Next, we turn to the proof of Theorem 1.3 and assume, without loss of generality, thatthe sequence ( α n ) converges to 1. First we prove the following two claims. Claim 1: ρ ( u n ( t n ) , α n u n ( t n )) −→
0, as n → ∞ . In fact, by definition, ρ ( u n ( t n ) , α n u n ( t n )) = inf r ∈ R k u n ( · , t n ) − α n u n ( · + r, t n ) k H s / per ≤ k u n ( t n ) − α n u n ( t n ) k H s / per = | (1 − α n ) |k u n ( t n ) k H s / per . (2.22)On the other hand, since ρ ( u n ( t n ) , φ k ) = ε r ∈ R such that k u n ( t n ) k H s / per ≤ k u n ( t n ) − φ k ( · + r ) k H s / per + k φ k ( · + r ) k H s / per < ε + k φ k ( · + r ) k H s / per , which is to say that the sequence ( k u n ( t n ) k H s / per ) is uniformly bounded. Taking the limitin (2.22), as n → ∞ , and taking into account that α n −→
1, we obtain the claim.
Claim 2: ρ ( α n u n ( t n ) , φ k ) −→
0, as n → ∞ .In fact, from Claim 1 and (2.12), we see that ρ ( α n u n ( t n ) , φ k ) ≤ ρ ( α n u n ( t n ) , u n ( t n )) + ρ ( u n ( t n ) , φ k ) < ε ε ε < ε, TABILITY OF PERIODIC WAVES FOR DISPERSIVE EQUATIONS 11 for all n large enough. This means that for n large enough, α n u n ( t n ) ∈ U ε ( φ k ). Moreover,Lemma 2.4 implies that α n u n ( t n ) ∈ Σ k . By putting all this together, we obtain that α n u n ( t n ) ∈ Σ k ∩ U ε ( φ k ). Consequently, Lemma 2.3 implies ρ ( α n u n ( t n ) , φ k ) ≤ C | F k ( α n u n ( t n )) − F k ( φ k ) |≤ C | F k ( α n u n ( t n )) − F k ( u n ( t n )) | + C | F k ( u n ( t n )) − F k ( φ k ) | = C | F k ( α n u n ( t n )) − F k ( u n ( t n )) | + C | F k ( w n ) − F k ( φ k ) | . Taking the limit, as n → ∞ , in the last inequality, the continuity of F k and the boundednessof ( α n u n ( t n )) n ∈ N in H s / per ([0 , L ]) yield Claim 2.Finally, Claims 1 and 2 combine to give ε ρ ( u n ( t n ) , φ k ) ≤ ρ ( u n ( t n ) , α n u n ( t n )) + ρ ( α n u n ( t n ) , φ k ) −→ , as n → ∞ , which is a contradiction. The proof of Theorem 1.3 is thus established. (cid:3) Applications
In this section we apply Theorem 1.3 to prove the orbital stability, in the energy space,for some well known dispersive models. As is well known, one of the major difficulties inthe theory is to check that the spectral properties assumed in (H1)-(H2) hold. In manyof the interesting applications the function f in (1.1) is a power or a polynomial. Hereand in what follows we shall assume that it has this form.Let us brief recall the main spectral properties of L k . The spectrum of L k is formed bya sequence of eigenvalues, say, { λ m } ∞ m =0 satisfying λ ≤ λ ≤ λ ≤ . . . , and λ m → ∞ , as m → ∞ , where equality means multiplicity of an eigenvalue (see e.g., [5] and [23]). From(1.6) it easily follows that zero is an eigenvalue with associated eigenfunction φ ′ k . Thus,the task in general is to show that λ < λ = 0 < λ . Below we recall three different waysof determining this relation.(i) Lam´e’s type potential . In many situations when M is a second order differentialoperator and the periodic traveling wave under study depends on the Jacobian ellipticfunctions, L k turns out to be a Hill’s operator with a Lam´e type potential. In particular,studying the spectrum of L k is equivalent to studying the eigenvalue problem (cid:26) Λ ′′ ( x ) + (cid:2) h − n ( n + 1) · k SN ( x, k ) (cid:3) Λ( x ) = 0 , Λ(0) = Λ(2 K ( k )) , Λ ′ (0) = Λ ′ (2 K ( k )) , (3.1)where h is a real parameter and n is a non-negative integer. Depending on n , the firsteigenvalues of (3.1) are well known (see e.g., [26]). Many applications using this approachhave appeared in the literature (see e.g., [7], [8], [11], [17], [21], [25], [29], [37], [38] to citebut a few).(ii) Neves’ approach.
Assume that M is a second order differential operator. Let us firstrecall from Floquet’s theorem (see e.g., [35] page 4) that if y is any solution of L k y = 0,linearly independent of φ ′ k , then there exists a constant θ satisfying y ( x + L ) = y ( x ) + θφ ′ k ( x ) . (3.2)In particular, if y satisfies the initial condition y ′ (0) = 0 then by taking the derivativewith respect to x in both sides of (3.2) and evaluating the result at x = 0, we see that θ = y ′ ( L ) φ ′′ k (0) . (3.3) Under these conditions Theorem 3.1 in [39] (see also [36]) states that λ is simple if andonly if θ = 0. In addition, λ = 0 if and only if θ < Angulo and Natali’s approach . When f ( x ) = x p , for some integer p ≥
1, a differentapproach to check (H1)-(H2) was established in [5]. Such an approach is based on thetotal positivity theory (see e.g., [30]) and can be viewed as an extension to the periodiccase of the results in [1] and [2]. To give the precise statement, we recall that a sequence { α n } n ∈ Z of real numbers is said to be in the class P F (2) discrete if(i) α n >
0, for all n ∈ Z ;(ii) α n − m α n − m − α n − m α n − m >
0, for n < n and m < m .Assume that our assumptions on the operator M hold. Suppose in addition that φ k ispositive, even and such that b φ k > c φ pk belongs to the class P F (2) discrete, then L k satisfies (H1)-(H2) (see [5, Theorem 4.1]).Next we will give some applications of Theorem (1.3).3.1. The KdV equation.
This subsection is devoted to the study of the KdV equation u t + u xxx + ∂ x (cid:18) u (cid:19) = 0 , (3.4)which appears as an approximated equation for the propagation of unidirectional, one-dimensional, small-amplitude long waves in a nonlinear dispersive media and it was firstlyderived by Korteweg and de Vries in [33].Since M = − ∂ x . It is clear that our assumption on M are fulfilled with s = s = 2and γ = 0. Hence, our energy space is the Sobolev space of order 1. As an application ofthe quadrature method, it is well known that (3.4) has a periodic traveling-wave solutions u ( x, t ) = φ ( x − ct ) with φ ( y ) = φ k ( y ) = 12 k b CN ( by, k ) , k ∈ (0 , . (3.5)This means that φ k is a solution of − φ k + cφ k − φ k + A = 0 , (3.6)where c = 4 b (2 k −
1) and A = 24 b k (1 − k ) . Here b ∈ R is an arbitrary parameter. In order to obtain periodic solutions with a fixedperiod L >
0, for any k ∈ (0 , b := 2 K ( k ) L .
Since CN has fundamental period 2 K ( k ), with this choice of b , the functions in (3.5) turnout to be L -periodic. Also, to have our assumption c > − γ = 0 in (H0) , we need to restrictthe elliptic modulus to the interval J := ( k ∗ , k ∗ = √ /
2. This constructionsthen give the family of L -periodic solutions k ∈ J = ( k ∗ , φ k ∈ H per ([0 , L ]) . The assumption (H0) is thus fulfilled.Let us check (H1) and (H2) . Here we have L k = − ∂ x + c − φ k . It is not difficult tosee that studying the spectral problem (cid:26) L k f = λf,f (0) = f ( L ) , f ′ (0) = f ′ ( L ) , (3.7) TABILITY OF PERIODIC WAVES FOR DISPERSIVE EQUATIONS 13 is equivalent to study the problem (cid:26) Λ ′′ ( x ) + (cid:2) h − · k SN ( x, k ) (cid:3) Λ( x ) = 0 , Λ(0) = Λ(2 K ( k )) , Λ ′ (0) = Λ ′ (2 K ( k )) , (3.8)where h = (12 k b + λ − c ) /b . It is well known that the first three eigenvalues of (3.8)are simple and given by (see [11] and [26]) h = 2 + 5 k − p − k + 4 k , h = 4 + 4 k , h = 2 + 5 k + 2 p − k + 4 k . Note that h is an eigenvalue of (3.8) if and only if λ = 0 is an eigenvalue of (3.7). Since h < h < h , the relation between h and λ then implies that λ = 0 is a simple eigenvalueof L k , which in turn also implies that L k has a unique negative eigenvalue. Assumptions (H1)-(H2) are thus checked.To check (H3) we differentiate (3.6) to see thatΦ := (cid:28) L k (cid:18) ∂φ k ∂k (cid:19) , ∂φ k ∂k (cid:29) = − ∂c∂k ddk (cid:18)Z L φ k dx (cid:19) − ∂A∂k ddk (cid:18)Z L φ k dx (cid:19) . Using formulae 312.02 and 312.04 in [20], we obtain Z L φ k ( x ) dx = 48 K ( k ) L (cid:16) E ( k ) − (1 − k ) K ( k ) (cid:17) and Z L φ k ( x ) dx = 48 K ( k )3 L (cid:16) (2 − k + 3 k ) K ( k ) + (4 k − E ( k ) (cid:17) . Thus, using the expressions for c and A we see that Φ < · L ddk (cid:16) (2 k − K ( k ) (cid:17) ddk (cid:16) (2 − k + 3 k ) K ( k ) + (4 k − E ( k ) K ( k ) (cid:17) + 18432 L ddk (cid:16) k (1 − k ) K ( k ) (cid:17) ddk (cid:16) E ( k ) K ( k ) − (1 − k ) K ( k ) (cid:17) > . The positivity of this quantity can be checked numerically (easy) or analytically (hard)using Taylor expansions of the elliptic functions (see [21] for similar calculations). Thisshows (H3) .Finally, one can easily verify (H4) by noting that c and A have positive derivatives and φ k is non-negative.Thus an application of Theorem 1.3 gives the following result. Theorem 3.1.
For each k ∈ J = ( k ∗ , , the periodic traveling wave φ k given in (3.5) isorbitally stable in H per ([0 , L ]) . The result in Theorem 3.1 is not new; the orbital stability of the cnoidal waves hasalready appeared in [11] and [22] (see also [17], [27], [39]). In [11] the authors first showthat it is possible to choose the constant A in (3.6) such that the corresponding family ofcnoidal waves (which were written in a different way from that in (3.5)) has mean zeroover its fundamental period. Then, by a translation they show that it is possible to obtaina family with a fixed mean. The orbital stability with respect to small perturbations in H per ([0 , L ]) was then obtained as an adaptation of the ideas in [24]. In [22], the authorstake the advantage of the integrability of the KdV equation (and the results in [17]) toshow the orbital stability of the cnoidal waves with respect to subharmonic perturbationswhich respect the mean value of the solution. Here, for a fixed n ∈ N , subharmonicperturbations means perturbations in the space H per ([0 , nL ]). The modified KdV equation.
This subsection is devoted to study the modifiedKorteweg-de Vries (mKdV) equation u t + u xxx + ∂ x (2 u ) = 0 . (3.9)The constant 2 in f ( u ) = 2 u appears only for convenience. The periodic traveling wavesare also those solutions of the form u ( x, t ) = φ ( x − ct ), where φ must be a solution of theODE φ ′′ − cφ + 2 φ − A = 0 . (3.10)As in (1.17), multiplying (3.10) by φ ′ and integrating once we obtain( φ ′ ) = − φ + cφ + 2 Aφ + B =: P ( φ ) , (3.11)where B another integration constant. It is then seen that the solutions of (3.10) dependon the roots of the polynomial P . We will study two particular family of solutions of(3.10).3.2.1. Dnoidal type solutions.
By assuming that A = 0 and P has four real roots (notethat P is an even polynomial in this case), (3.10) admits a family of solutions given by φ ( y ) = φ k ( y ) = a DN( ax, k ) , k ∈ (0 , , (3.12)where a > c = a (2 − k ) . (3.13)In order to obtain periodic solutions with a fixed period L >
0, for any k ∈ (0 , a := 2 K ( k ) L .
By recalling that DN has fundamental period 2 K ( k ) we then see that the functions in(3.12) have fundamental period L . Since c > k ∈ (0 , (H0) holds with J = (0 , (H1) and (H2) . The linearized operator reads as L k = − ∂ x + c − φ k andthe spectral problem (cid:26) L k f = λf,f (0) = f ( L ) , f ′ (0) = f ′ ( L ) , (3.14)is equivalent to (cid:26) Λ ′′ ( x ) + (cid:2) h − · k SN ( x, k ) (cid:3) Λ( x ) = 0 , Λ(0) = Λ(2 K ( k )) , Λ ′ (0) = Λ ′ (2 K ( k )) , (3.15)where h = (6 a + λ − c ) /a . It is well known that the first three eigenvalues of (3.15) aresimple and given by (see [8] and [26]) h = 2(1 + k − p − k + k ) , h = 4 + k , h = 2(1 + k + p − k + k ) . It is easy to see that h correspond to the eigenvalue λ = 0 of (3.14). Since h < h < h ,the relation between h and λ then implies that λ = 0 is a simple eigenvalue and L k hasa unique negative eigenvalue. Assumptions (H1)-(H2) are hence checked.Next, since A = 0 we see from (3.11) thatΦ := (cid:28) L k (cid:18) ∂φ k ∂k (cid:19) , ∂φ k ∂k (cid:29) = − ∂c∂k ddk (cid:18)Z L φ k dx (cid:19) TABILITY OF PERIODIC WAVES FOR DISPERSIVE EQUATIONS 15
Using formulas 314.02 and 312.04 in [20], we obtain Z L φ k ( x ) dx = 8 K ( k ) E ( k ) L .
Since k K ( k ) E ( k ) is a strictly increasing function and ∂c∂k > < (H3) is fulfilled. It is clear that (H4) holds. As a consequence of Theorem1.3, we obtain the following theorem. Theorem 3.2.
For each k ∈ J = (0 , , the periodic traveling wave φ k given in (3.12) isorbitally stable in H per ([0 , L ]) . To the best of our knowledge, Theorem 3.2 has first appeared in [8] where the authorexploited his results established for the cubic Schr¨odinger equation. The approach toobtain the results was based on the classical ideas contained [12], [15], and [45]. From thepoint of view of Hamiltonian systems, the same result was established in [29].3.2.2.
Dnoidal-Snoidal type solutions.
We now assume that zero is a root of the polynomial P in (3.11). This immediately implies that the integration constant B must be zero. Byassuming that α < < α < α are the roots of P , applying the quadrature method andusing formula 257.00 in [20], we obtain the following L -periodic smooth curve of solutionsfor (3.10), φ k ( ξ ) = α ( k ) ( α ( k ) − α ( k )) DN (cid:16) K ( k ) L ξ, k (cid:17) ( α ( k ) − α ( k )) + ( α ( k ) − α ( k )) SN (cid:16) K ( k ) L ξ, k (cid:17) . where the constants α i ( k ) are given by (after some algebra) α ( k ) = − K ( k ) L (cid:18)q p k − k + 1 + 1 − k + 1 √ q p k − k + 1 − k (cid:19) ,α ( k ) = 2 K ( k ) L (cid:18)q p k − k + 1 + 1 − k − √ q p k − k + 1 − k (cid:19) ,α ( k ) = 4 K ( k ) √ L q p k − k + 1 − k . In addition c and A can also be expressed in terms of k as c ( k ) = 16 K ( k ) L p k − k + 1and A ( k ) = − K ( k )3 √ L (cid:16)p k − k + 1 − k + 1 (cid:17) q p k − k + 1 + 2 k − . After a few algebraic manipulations involving elliptic functions φ k can be written as φ k ( ξ ) = 4 K ( k ) √ g ( k ) L DN (cid:16) K ( k ) L ξ, k (cid:17) β SN (cid:16) K ( k ) L ξ, k (cid:17) , (3.16)where β = √ k − k + 1 + k − g ( k ) = q √ k − k + 1 − k + . Consequently, weobtain (H0) with J = (0 , Remark 3.3.
Formally, by making k → in (3.16) , the periodic function φ k looses itsperiodicity and degenerates to φ ( ξ ) = − α α − α + ( α − α )2 + ( α − α )2 cosh (2 ξ ) , which is a solitary wave for the mKdV equation. Orbital stability in the energy space ofthis solution was studied in [3] . In order to check (H1)-(H2) we will take the advantage of the results in [39], [40], and[36] as briefly described at the beginning of this section. Let ( n, z ) be the inertial indexof the linearized operator L k = − ∂ x + c ( k ) − φ k , that is, n denotes the dimension of thenegative subspace of L k and z denotes the dimension of ker( L k ). Under our constructions, L k is isonertial , that is, ( n, z ) does not depend on k and on L (see Theorem 3.1 in [40] orTheorem 3.1 [36]). Thus, it suffices to fix k ∈ (0 ,
1) and
L >
0. For the sake of simplicity,we fix k := 0 . L = 30. Since φ ′ k has two zeros in the interval [0 , L ) and L k ( φ ′ k ) = − φ ′′′ k + cφ ′ k − φ k φ ′ k , = (cid:0) − φ ′′ k + c ( k ) φ k − φ k (cid:1) ′ = 0 . we obtain that zero is the second or the third eigenvalue of L k . Theorem 3.2 in [36]establishes that zero is the second eigenvalue (which in turn is simple) provided that θ := y ′ ( L ) φ ′′ k (0) < , (3.17)where y is the unique solution of the IVP − y ′′ + (cid:2) c ( k ) − φ k (cid:3) y = 0 ,y (0) = − φ ′′ k (0) ,y ′ (0) = 0 . (3.18)The sign of θ can be obtained once we have the value y ′ ( L ). We can solve (numerically)(3.18) and, in particular, we deduce that θ ∼ = − . × . Next table illustrates some values of θ if we fix k and choose different values of L .Values of θ with k = 0 . L = 20 L = 50 L = 200 L = 1000 L = 1000000 θ ∼ = − θ ∼ = − . × θ ∼ = − . × θ ∼ = − . × θ ∼ = − . × This checks (H1)-(H2) .We now check (H3) . By deriving equation (3.10), with respect to k , we get L k (cid:18) ∂φ k ∂k (cid:19) = − ∂c∂k φ k − ∂A∂k . TABILITY OF PERIODIC WAVES FOR DISPERSIVE EQUATIONS 17
Thus, we can write Φ asΦ = − (cid:28) ∂c∂k φ k + ∂A∂k , ∂φ k ∂k (cid:29) = − (cid:28) M ′ k ( φ k ) , ∂φ k ∂k (cid:29) = − Z L ∂c∂k φ k ∂φ k ∂k + ∂A∂k ∂φ k ∂k dx = − Z L ∂c∂k ∂∂k φ k + ∂A∂k ∂φ k ∂k dx. In other words, Φ = − ∂c∂k ∂∂k (cid:18) Z L φ k dx (cid:19) − ∂A∂k ∂∂k Z L φ k dx = − ∂c∂k ∂∂k Q ( φ k ) − ∂A∂k ∂∂k V ( φ k ) . Since φ k is given in (3.16), we can compute Q ( φ k ) and V ( φ k ) to obtain Φ. Indeed, V ( φ k ) = Z L φ k ( x ) dx = Z L K ( k ) √ g ( k ) L DN (cid:16) K ( k ) L x, k (cid:17) β SN (cid:16) K ( k ) L x, k (cid:17) dx = 4 K ( k ) √ g ( k ) L Z L DN (cid:16) K ( k ) L x, k (cid:17) β SN (cid:16) K ( k ) L x, k (cid:17) dx. By making the change of variable y = 2 K ( k ) L x , we obtain V ( φ k ) = 4 K ( k ) √ g ( k ) L L K ( k ) Z K ( k )0 DN ( y, k )1 + β SN ( y, k ) dy, that is, V ( φ k ) = 4 √ g ( k ) Z K ( k )0 DN ( y, k )1 + β SN ( y, k ) dy =: 4 √ g ( k ) I . With similar arguments, we see that Q ( φ k ) = 4 K ( k ) g ( k ) L Z K ( k )0 DN ( y, k )(1 + β SN ( y, k )) dy =: 4 K ( k ) g ( k ) L I . Now, let α be such that − α = β it follows that 0 < − α < k and formula 410.04 in[20] implies that I = ( k − α ) G ( w, k ) p α (1 − α )( α − k ) , where, for k ′ = √ − k , G ( w, k ) = K ( k ) E ( w, k ′ ) − K ( k ) F ( w, k ′ ) + E ( k ) F ( w, k ′ ) = π ( w, k ) and w = sin − r α α − k = sin − β p k + β . The functions F and E are the incomplete elliptic integral of the first and second kind andthe function Λ ( w, k ) is known as Heuman’s Lambda function (see e.g., [20] for additionaldetails). By using the expression β = √ k − k + 1 + k −
1, we can rewrite I = p √ k − k + 1 + 2 k − G ( w, k ) q k − k + 1 + (2 k − √ k − k + 1 , and w = sin − s √ k − k + 1 + k − √ k − k + 1 + 2 k − . On the other hand, the relation DN = 1 − k SN yields I = Z K ( k )0 (1 − k SN ( y )) (1 + β SN ( y )) dy. Thus, by formula 410.8 in [20], we have I = 1 α (cid:16) k K ( k ) + 2 k ( α − k )Π( α , k ) + ( α − k ) V (cid:17) , where Π( α , k ) = k K ( k ) k − α − α G ( w, k ) p α (1 − α )( α − k )and V = η α E ( k ) + 2 k α − k + α ( k ′ ) k − α K ( k ) − α (2 α k + 2 α − α − k ) G ( w, k ) p α (1 − α )( α − k ) ! , with η = 1 / (2( α − k − α )) and w as before.In view of the above relations,Φ = − ∂c∂k ∂∂k (cid:18) K ( k ) g ( k ) L I (cid:19) − ∂A∂k ∂∂k (cid:18) √ g ( k ) I (cid:19) . Besides, ∂c∂k = 1 L ∂∂k (cid:16) K ( k ) p k − k + 1 (cid:17) =: 1 L m ( k )and ∂A∂k = 1 L ∂∂k − K ( k ) (cid:16) √ k − k + 1 − k + 1 (cid:17) √ q p k − k + 1 + 2 k − =: 1 L m ( k ) . Thus, Φ = − L m ( k ) ∂∂k (cid:18) K ( k ) g ( k ) L I (cid:19) − L m ( k ) ∂∂k (cid:18) √ g ( k ) I (cid:19) and finally, Φ = − L m ( k ) , TABILITY OF PERIODIC WAVES FOR DISPERSIVE EQUATIONS 19 where m ( k ) = m ( k ) ∂∂k (cid:18) K ( k ) g ( k ) I (cid:19) + m ( k ) ∂∂k (cid:18) √ g ( k ) I (cid:19) . Since m ( k ) depends only on k , we can check, at least numerically, that m ( k ) >
0, for all k ∈ (0 ,
1) (see Figure 1 below). Therefore, we conclude that Φ < Figure 1.
Graph of Φ as function of k , with k ∈ (0 , . k ∈ (0 , .
5) and k ∈ (0 , M k ( φ k ) > k ∈ (0 , k M k ( φ k ) (see Figure 2). In additionsince k c ( k ) is an increasing function, we see that (H4) holds. Figure 2.
Graph of M k ( φ k ) as function of k , with k ∈ (0 , . k ∈ (0 , . k ∈ (0 , Combining these informations with Theorem 1.3, we obtain
Theorem 3.4.
For each k ∈ J = (0 , , the periodic traveling wave φ k given in (3.16) isorbitally stable in H per ([0 , L ]) . The Gardner equation.
Our purpose in this section is to prove orbital stability ofperiodic waves for the Gardner equation v t + v xxx + avv x + bv v x = 0 , (3.19)where a and b are real parameters and b = 0. From the mathematical point of view,Gardner equations is seem as a mixed equation because it contains both KdV and mKdVnonlinearities. On the other hand, Gardner and mKdV equations appear as models for theflow of waves in plasma and solid environments. It also appears as a model in quantumfields (see e.g., [32], [43] and [44]).Our strategy consists in transferring the problem of orbital stability of periodic wavesfor (3.19) to that for the mKdV equation. Indeed, Gardner equation is close related withthe Focusing or Defocusing mKdV equation (F-mKdV or D-mKdV for simplicity) u t + u xxx + γ u u x = 0 , (3.20)where γ = sgn( b ). More precisely, there is a dipheomorfism T (between suitable spaces)that relates solutions of (3.19) to solutions of (3.20) given by( T v )( x, t ) := s b γ (cid:20) v (cid:18) x − a b t, t (cid:19) + a b (cid:21) . (3.21)Using these transformations, Alejo [3] studied the orbital stability of soliton-like solu-tions for (3.19). The author dealt with the case a = 6 σ and b = 6 and proved the orbitalstability of the solitary traveling waves φ ( σ,c ) ( x − c σ t ) = c σ + √ σ + c cosh( √ c ( x − c σ t )) , (3.22)where c σ = 6 σ + c . Here, the constant c satisfies c ∈ (0 , ∞ ) if b >
0, and c ∈ (0 , σ )if b <
0. For the stability of N -solitons we refer the reader to [4].Before proceeding, let us highlight a crucial difference between periodic and solitarywave solutions of (3.19). Assume that v is a solution of (3.19) with a = 0. Then, T v is asolution of F-mKdV or D-mKdV having the form α + βv , where α and β are real constantswith α = 0. In particular, if v ∈ C ( R , H ( R )) then T v does not belong to C ( R , H ( R )).On the other hand, if v ∈ C ( R , H per ([0 , L ])) then T v also belongs to C ( R , H per ([0 , L ])).Therefore, periodic traveling waves solutions of (3.19) relate in a better way with theperiodic traveling waves solution of (3.20) in the sense that once obtained spatially periodicsolutions of (3.19) we also obtain spatially periodic solutions of (3.20) (and vice-versa).This is the content of the next lemma. Lemma 3.5.
Let T be defined as in (3.21) . Then, T : C ( R , H per ([0 , L ])) → C ( R , H per ([0 , L ])) is a diffeomorphism, whose inverse is given by ( T − u )( x, t ) := r γb u (cid:18) x + a b t, t (cid:19) − a b . (3.23) In addition, there is a constant
C > such that, for any u, v ∈ C ( R , H per ([0 , L ])) , ρ ( u, v ) = Cρ ( T u, T v ) , where ρ is defined as in (2.1) with s = 2 . TABILITY OF PERIODIC WAVES FOR DISPERSIVE EQUATIONS 21
Proof.
The first part is immediate (see also [34]). Also, if u, v ∈ C ( R , H per ([0 , L ])) then ρ ( T u, T v ) = inf r ∈ R (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)s b γ (cid:20) u (cid:18) x − a b t, t (cid:19) + a b − v (cid:18) x − a b t + r, t (cid:19) − a b (cid:21)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) H per = s b γ inf r ∈ R (cid:13)(cid:13)(cid:13)(cid:13) u (cid:18) x − a b t, t (cid:19) − v (cid:18) x − a b t + r, t (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) H per = Cρ ( u, v ) , where C = q b γ . This completes the proof. (cid:3) Note that from Lemma 3.5, up to a constant, the diffeomorphism T is distance-preservingwhen measured in the pseudo-metric ρ .Although we are able to obtain the orbital stability of periodic traveling waves in cases b > b <
0, in what follows, we restrict ourselves to the case b > v ( x, t ) = ψ (˜ c, ˜ A ) ( x − ˜ ct ) are obtained assolutions of ψ ′′ − ˜ cψ + a ψ + b ψ − ˜ A = 0 (3.24)and periodic traveling waves of (3.20), say, u ( x, t ) = φ ( c,A ) ( x − ct ) are obtained as solutionsof φ ′′ − cφ + 2 φ − A = 0 . (3.25)Since T takes solutions of (3.19) to solutions of (3.20), T also takes solutions of (3.24)to solutions of (3.25) in the following way: if ψ (˜ c, ˜ A ) is a solution of (3.24), then φ ( c,A ) ( x ) := T ψ (˜ c, ˜ A ) ( x ) = s b γ h ψ (˜ c, ˜ A ) ( x ) + a b i is a solution of (3.25) with c = ˜ c + a b ,A = r b (cid:18) − ˜ c a b − a b + ˜ A (cid:19) . (3.26)Conversely, if φ ( c,A ) is a solution of (3.25), then ψ (˜ c, ˜ A ) ( x ) := T − φ ( c,A ) ( x ) = r b φ ( c,A ) ( x ) − a b is a solution of (3.24) with ˜ c = c − a b , ˜ A = r b A + c a b − a b . (3.27)Note that (3.26) and (3.27) bring and explicit relation among the constants c , ˜ c , A and˜ A . This is useful because the wave speed is also known explicitly. Having in mind the orbital stability results presented for the mKdV equation in Sub-section 3.2, we present two new results of orbital stability for the Gardner equation.
Theorem 3.6.
Let ψ k be defined by ψ k ( ξ ) = 2 √ K ( k ) √ bL DN (cid:18) K ( k ) L ξ, k (cid:19) − a b , (3.28) Then, ψ k ( ξ ) is a solution of (3.24) with ˜ c = c ( k ) = 4 K ( k ) L (2 − k ) − a b , ˜ A = A ( k ) = 2 aK ( k ) bL (2 − k ) − a b . In addition, v ( x, t ) = ψ k ( x − c ( k ) t ) is a periodic traveling solution of (3.19) , which isorbitally stable in H per ([0 , L ]) .Proof. The first statement follows from the fact that φ k ( ξ ) = T ( ψ ( c ( k ) ,A ( k )) ( ξ )) = 2 K ( k ) L DN (cid:18) K ( k ) L ξ, k (cid:19) (3.29)is a solution of (3.25) with (see (3.12) and (3.13)) c ( k ) = 4 K ( k ) L (2 − k ) ,A ( k ) = 0 . Now, let v ( t ) be a solution of (3.19) with initial data v . From Lemma 3.5, it followsthat ρ ( v ( t ) , ψ k ) = Cρ ( T v ( t ) , T ψ k ) , for all t ∈ R , (3.30)and T v ( t ) is a solution of the mKdV equation with initial data T v . Theorem 3.2 impliesthat the dnoidal solution (3.29) is orbitally stable by the mKdV flow in H per ([0 , L ]). Thus,given ε > δ > ρ ( T u , φ k ) < δ thensup t ∈ R ρ ( T v ( t ) , φ k ) < ε . (3.31)Next, let ε > ε > ε < ε/C . In addition, by choosing δ > δ < Cδ we see that if ρ ( v , ψ k ) < δ then ρ ( T v , φ k ) = 1 C ρ ( v , ψ k ) < C δ < δ . Hence, from (3.30) and (3.31), we obtainsup t ∈ R ρ ( v ( t ) , ψ k ) = C sup t ∈ R ρ ( T v ( t ) , φ k ) < ε, which proves the orbital stability of ψ k . The proof of the theorem is thus completed. (cid:3) Theorem 3.7.
Let ϕ k be defined as ϕ k ( ξ ) = 4 √ K ( k ) g ( k ) √ bL DN (cid:16) K ( k ) L ξ, k (cid:17) β SN (cid:16) K ( k ) L ξ, k (cid:17) − a b . (3.32) TABILITY OF PERIODIC WAVES FOR DISPERSIVE EQUATIONS 23
Then, ϕ k ( ξ ) is a solution of (3.24) with ˜ c = c ( k ) = 16 K ( k ) L p k − k + 1 − a b , ˜ A = A ( k ) = 8 aK ( k ) bL p k − k + 1 − √ K ( k ) p √ k − k + 1 + 2 k − √ bL (cid:16) √ k − k + 1 − k + 1 (cid:17) − − a b . In addition, v ( x, t ) = ϕ k ( x − c ( k ) t ) is a solution of (3.19) orbitally stable in H per ([0 , L ]) .Proof. The first statement now follows from the fact that φ k ( ξ ) = T ( ϕ k ( ξ )) = 4 K ( k ) √ g ( k ) L DN (cid:16) K ( k ) L ξ, k (cid:17) β SN (cid:16) K ( k ) L ξ, k (cid:17) (3.33)is a solution of (3.25) with c ( k ) = 16 K ( k ) L p k − k + 1 ,A ( k ) = − K ( k )3 √ L (cid:16)p k − k + 1 − k + 1 (cid:17) q p k − k + 1 + 2 k − . In fact, the periodic travelling wave φ k in (3.33) is exactly the solution of the mKdV whichwe proved to be orbitally stable in H per ([0 , L ]) (see Theorem 3.4). Therefore, the rest ofthe proof follows exactly the same arguments as in the previous theorem. (cid:3) The ILW equation.
Next we consider the Intermediate Long Wave (ILW) equation u t − M δ u + ∂ x ( u ) = 0 , (3.34)where u = u ( x, t ) is L -periodic in the spatial variable and the linear operator M δ is definedvia Fourier transform by [ M δ u ( m ) = (cid:18) πmL coth (cid:18) πmδL (cid:19) − δ (cid:19) b u ( m ) , δ > . Equation (3.34) is derived as a model equation for long, weakly nonlinear internal gravitywaves in a stratified fluid of finite depth. The parameter δ > M = M δ satisfies our assumptions with γ = 0 and s = s = 1. Theauthors in [9] shown that (3.34) has traveling wave solution of the form u ( x, t ) = φ k ( x − c ( k ) t ), with k ∈ J := (0 , k ) ⊂ (0 , c ( k ) := 1 δ − πK ( k ) L K ( k ′ ) − K ( k ) L (cid:20) Z ( α, k ′ ) + CN( α, k ′ )DN( α, k ′ )SN( α, k ′ ) (cid:21) > , α = 4 δK ( k ) L ,A ( k ) = 1 L Z L ϕ k ( x ) dx, and φ k ( x ) = − K ( k ) L Z ( δy, k ) − δπL K ( k ) K ( k ′ ) + 4 K ( k ) L DN ( y, k )CN( δy, k ′ )SN( δy, k ′ )DN( δy, k ′ )1 − DN ( y, k )SN ( δy, k ) . Here, y = K ( k ) xL , k ′ = √ − k and Z ( x, k ) stands for the Jacobian Zeta function definedby Z ( x, k ) = Z x (cid:18) DN( s, k ) − E ( k ) K ( k ) (cid:19) ds. This shows that (H0) holds. Also, by using the total positivity theorem as described inthe beginning of this section, it was shown that the linearized operator L k = M δ + c − φ k satisfies (H1)-(H2) . Assumptions (H3) and (H4) were also checked in [9] taking theadvantage of the explicit form of the solutions φ k . Consequently, one deduces the orbitalstability of the periodic wave φ k . For details we refer the reader to [9, Section 6].3.5. The Schamel equation.
This subsection is devoted to the so called Schamel equa-tion u t + ∂ x ( u xx + | u | / ) = 0 , (3.35)which governs the behaviour of weakly nonlinear ion-acoustic solitons that are modifiedby the presence of trapped electrons. Such equations was first derived by Schamel in [41],[42]. The existence of periodic solutions can be obtained by using the quadrature method.In particular, in [19] the authors established that φ k ( ξ ) = 6400 K ( k ) L (cid:20) − k + p − k + k + 3 k CN (cid:18) K ( k ) ξL ; k (cid:19)(cid:21) (3.36)= 6400 K ( k ) L (cid:20) k − p − k + k + 3DN (cid:18) K ( k ) ξL ; k (cid:19)(cid:21) , is an L -periodic traveling wave solution of (3.35), where for each k ∈ (0 ,
1) =: J , c = 64 K ( k ) L p − k + k , (3.37)and A = 204800 K ( k ) L h − k + 3 k + 3 k − − − k + k ) i . (3.38)This shows that (H0) holds with J = (0 , (H1)-(H2) can be establishedby studying the periodic eigenvalue problem − y ′′ + (cid:18) c ( k ) − φ k (cid:19) y = λy,y (0) = y ( L ) , y ′ (0) = y ′ ( L ) , (3.39)or using the same technique as in Subsection 3.2.2 (see [19] and [21] for details). Onthe other hand, (H3)-(H4) can be checked by using the explicit form of the quantitiesinvolved. Consequently, Theorem 1.3 may also be applied to obtain the orbital stabilityof the periodic traveling waves (3.36). We refer the reader to [19] for the details.4. Extension to regularized equations
In this section, we extend the theory developed in Section 2 to regularized equations ofthe form u t + M u t + ∂ x ( u + f ( u )) = 0 , (4.1)where M and f satisfy the assumptions posed in the introduction. Equations of this formarise as models of wave propagation in a variety of physical contexts. TABILITY OF PERIODIC WAVES FOR DISPERSIVE EQUATIONS 25
Traveling waves solutions for (4.1) are also special solutions having the form u ( x, t ) = φ ( x − ct ). By replacing this form of wave in (4.1), φ must solve c M φ + ( c − φ − f ( φ ) + A = 0 , (4.2)where again A is an integration constant.It is well-known that (4.1) has three conserved quantities, namely, E ( u ) = 12 Z L ( u M u − F ( u )) dx, with F ( u ) = Z u f ( s ) ds, (4.3) Q ( u ) = 12 Z L ( u + u M u ) dx, (4.4)and V ( u ) = Z L u dx. (4.5)Solutions of (4.2) are now critical points of the functional E + ( c − Q + AV.
With this in hand, our assumptions read as follows. (P0)
There are an interval J ⊂ R , C -functions k ∈ J c = c ( k ) and k ∈ J A = A ( k ), and a nontrivial smooth curve of L -periodic solutions for (4.2), k ∈ J φ k := φ ( c ( k ) ,A ( k )) ∈ H s per ([0 , L ]) with c = c ( k ) > (P1) The linearized operator L k := c M + ( c − − f ′ ( φ k ), defined on a dense subspaceof L per ([0 , L ]), has a unique negative eigenvalue, which is simple. (P2) Zero is a simple eigenvalue of L k with associated eigenfunction φ ′ k . (P3) The quantity Φ defined by Φ := D L k (cid:16) ∂φ k ∂k (cid:17) , ∂φ k ∂k E is negative. (P4) It holds M k ( φ k ) = − ∂c∂k Q ( φ k ), where M k ( u ) := ∂c∂k Q ( u ) + ∂A∂k V ( u ).The theory developed in Section 2 extends mutatis mutandis to the present situationand we can prove the following. Theorem 4.1 (Orbital stability) . Under assumptions (P0)-(P4) , for each k ∈ J , theperiodic traveling wave φ k is orbitally stable by the flow of (4.1) in H s / per ([0 , L ]) , that is,for any ε > , there exists δ > such that if u ∈ H s / per ([0 , L ]) satisfies k u − φ k H s / per < δ, then the solution u ( t ) of (4.1) , with initial data u , satisfies sup t ∈ R inf r ∈ R k u ( t ) − φ k ( · + r ) k H s / per < ε. Remark 4.2.
Note that in the proof of Lemma 2.4 the sign of the coefficients in f ( α ) = aα + bα is not relevant. So, the sign of the quantity Q k ( φ k ) does not change the arguments in theproof of Lemma 2.4. As a result, the proof of Theorem 4.1 is similar to that of Theorem1.3. The modified BBM equation.
This section is devoted to study the orbital stabilityof periodic waves for the modified BBM equation u t − u xxt + ( u + u ) x = 0 . (4.6)At least from the mathematical point of view, (4.6) can be viewed as a regularized versionof the mKdV equation (see also [14]). In such a case, the traveling wave φ must be asolution of the equation − cφ ′′ + ( c − φ − φ + A = 0 , (4.7)If it is assumed that A = 0 then (4.7) has a dnoidal-type solution of the form φ c ( ξ ) = η DN (cid:18) ηx √ c (cid:19) , (4.8)where η is a real constant. The stability of the solution (4.8) in the energy space H per ([0 , L ])was studied in [10]. In particular, the authors have shown the orbital stability of φ c . Themethod used to obtain the spectral properties was based on the total positivity theory;whereas the techniques to prove the stability itself was based on the classical method.Our goal here is to study a solution of (4.7) when A = 0. Indeed, by applying thequadrature method and using formula 257.00 in [20] we obtain a solution in terms of theelliptic functions given by φ k ( x ) = 4 √ cK ( k )DN (cid:16) K ( k ) L x, k (cid:17) g ( k ) L (cid:16) β SN (cid:16) K ( k ) L x, k (cid:17)(cid:17) , (4.9)where β = √ k − k + 1 + k − g ( k ) = q √ k − k + 1 − k + . Here, both c and A are also considered as functions of k . More precisely, c ( k ) = L L − K ( k ) √ k − k + 1and A ( k ) = 16 c √ cK ( k )( r ( k ) − √ L r ( k ) , where r ( k ) = p √ k − k + 1 + 2 k −
1. Note that since the function k ∈ (0 , K ( k ) √ k − k + 1 is strictly increasing, c ( k ) has a unique singular point, generallyclose to 1, which we shall call k L . Figure 3 illustrates the behaviour of the functions c ( k )and A ( k ) with k ∈ (0 , k L ).Notice the function c ( k ) is increasing on (0 , k L ) and c ( k ) >
1. Thus, we must have L L − π > . This inequality implies the period L of the profile φ k in (4.9) must satisfy L > π . As aconsequence, the condition ( P0 ) is fulfilled, with J := (0 , k L ).In what follows we check conditions (P1)-(P2) . We will use a similar analysis as inSubsection 3.2. Since, c ( k ) > L k = − ∂ x + ( c ( k ) − c ( k ) − c ( k ) φ k . Thus, because c and φ k depends smoothly on k , ˜ L k is isonertial, that is, the inertial index( n, z ) does not depend on k . Therefore, as before it suffices to fix k ∈ (0 , k L ) and L > TABILITY OF PERIODIC WAVES FOR DISPERSIVE EQUATIONS 27
Figure 3.
Left: Graph of k ∈ (0 , k L ) c ( k ). Right: Graph of k ∈ (0 , k L ) A ( k ). In both cases, L = 50 .Let us fix k := 0 . L = 50. Since φ ′ k has two zeros in the interval [0 , L ) and˜ L k ( φ ′ k ) = 0 , we obtain that zero is the second or the third eigenvalue of L k . In order toshow that zero is indeed the second one, it suffices to prove that θ := y ′ ( L ) φ ′′ k (0) < , (4.10)where y is the unique solution of − y ′′ + 1 c ( k ) (cid:2) ( c ( k ) − − φ k (cid:3) y = 0 ,y (0) = − φ ′′ k (0) ,y ′ (0) = 0 . (4.11)The constant θ can be determined by solving numerically (4.11). In particular, we deducethat θ ∼ = − . × . Thus, (P1)-(P2) are checked. Table 1 illustrates some values of θ if we fix k = 0 . L . Note that θ is always negative and increases, in absolutevalue, with L . Table 1.
Values of θ with k = 0 . L = 10 L = 20 L = 200 L = 1000 L = 100000 θ ∼ = − . θ ∼ = − . θ ∼ = − . × θ ∼ = − . × θ ∼ = − . × We now proceed to check (P3) . By deriving equation (4.7), with respect to k , we obtain L k (cid:18) ∂φ k ∂k (cid:19) = − ∂c∂k (cid:0) − φ ′′ k + φ k (cid:1) − ∂A∂k . Therefore, with a similar analysis as in Subsection 3.2, we can write Φ asΦ = − ∂c∂k ∂∂k (cid:18) Z L (cid:16)(cid:0) φ ′ k (cid:1) + φ k (cid:17) dx (cid:19) − ∂A∂k ∂∂k Z L φ k dx = − ∂c∂k ∂∂k Q ( φ k ) − ∂A∂k ∂∂k V ( φ k ) . We now can compute numerically the values of Q ( φ k ) and V ( φ k ) and obtain Φ <
0. Anillustration of the behaviour of Φ as function of k is given in Figure 4. Figure 4.
Left: Graph of Φ as function of k , with k ∈ (0 , k L ). Right:Graph of Φ as function of k , with k ∈ (0 , . L = 30.The expressions of φ k , Q and V also allow us to check, numerically, that Ψ := M k ( φ k ) + ∂c∂k Q ( φ k ) = 0 and condition (P4) hods. See Figure 5. Figure 5.
Left: Graph of Ψ for k ∈ (0 , k L ). Right: Graph of Ψ for k ∈ (0 , . L = 30.Table 2 below also gives the values of Φ and Ψ for different values of k . As an applicationof Theorem 4.1 we then have proved the following. TABILITY OF PERIODIC WAVES FOR DISPERSIVE EQUATIONS 29
Theorem 4.3.
For each k ∈ J = (0 , k L ) , the periodic traveling wave φ k given in (4.9) isorbitally stable in H per ([0 , L ]) . Table 2.
Φ and Ψ for some values of k . k Φ Ψ0.1 3 . × − The regularized Schamel equation.
Here we consider the so called regularizedSchamel equation u t − u xxt + ∂ x ( u + | u | / ) = 0 , (4.12)which can be viewed as a regularized version of (3.35) in much the same way that theBBM equation can be viewed as a regularized version of the KdV equation.Periodic traveling waves solutions for (4.12) may be obtained in view of the quadraturemethod. Indeed, in [21] it was shown that for L > π there exists k L ∈ (0 ,
1) such thatfor k ∈ (0 , k L ) the function φ k ( x ) = (cid:20) (cid:18) L − k − K ( k )˜ m ( k ) − (cid:19) + 80 k K ( k )˜ m ( k ) CN (cid:18) K ( k ) L x, k (cid:19)(cid:21) , (4.13)where ˜ m ( k ) = L − K ( k ) √ k − k + 1, is a traveling wave for (4.12) with c = L L − K ( k ) √ k − k + 1 (4.14)and A = − K ( k )27 ˜ m ( k ) (cid:20)(cid:16)p k − k + 1 − (2 k − (cid:17) (cid:16) p k − k + 1 + (2 k − (cid:17)(cid:21) . (4.15)The hypothesis (P0) is then fulfilled with J := (0 , k L ). Properties (P1)-(P2) can beobtained by studying the periodic eigenvalue problem associated with the operator L k = − c∂ x + ( c − − φ / k , which in turn is equivalent to study the eigenvalue problemassociated with a Lam´e type equation, namely, (cid:26) Λ ′′ ( x ) + (cid:2) h − · · k SN ( x, k ) (cid:3) Λ( x ) = 0 , Λ(0) = Λ(2 K ( k )) , Λ ′ (0) = Λ ′ (2 K ( k )) , (4.16)Once we known the eigenvalues of (4.16) explicitly, (P1)-(P2) are promptly obtained (see[21] and [26]).In view of the expression of the solution φ k , properties (P3)-(P4) are a little bit hardto be obtained. However, after some algebraic computations one can check that they stillhold here. In conclusion, Theorem 4.1 can be applied to obtain the orbital stability of φ k by the flow of (4.12). We refer the reader to [21] for the details. Acknowledgements
Part of this work was developed during the Ph.D Thesis of the first author, concludedat IMECC-UNICAMP under the guidance of the second author. The first author ac-knowledges the financial support from Capes and CNPq. The second author is partiallysupported by CNPq.
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