This is a preprint of: “Persistence of periodic traveling waves and Abelian integrals”, Armengol Gasull, Anna Geyer, Vı́ctor Mañosa, J. Differential Equations, vol. 293, 48–69, 2021. DOI: [10.1016/j.jde.2021.05.033] PERSISTENCE OF PERIODIC TRAVELING WAVES AND ABELIAN INTEGRALS ARMENGOL GASULL, ANNA GEYER AND VÍCTOR MAÑOSA Abstract. It is well known that the existence of traveling wave solutions (TWS) for many partial differential equations (PDE) is a consequence of the fact that an associated planar ordinary differential equation (ODE) has certain types of solutions defined for all time. In this paper we address the problem of persistence of TWS of a given PDE under small perturbations. Our main results deal with the situation where the associated ODE has a center and, as a consequence, the original PDE has a continuum of periodic traveling wave solutions. We prove that the TWS that persist are controlled by the zeroes of some Abelian integrals. We apply our results to several famous PDE, like the Ostrovsky, Klein-Gordon, sine-Gordon, Korteweg-de Vries, Rosenau-Hyman, Camassa-Holm, and Boussinesq equations. 1. Introduction Traveling wave solutions (TWS) are an important class of particular solutions of partial differential equations (PDE). These waves are special solutions which do not change their shape and which propagate at constant speed. They appear in fluid dynamics, chemical kinetics involving reactions, mathematical biology, lattice vibrations in solid state physics, plasma physics and laser theory, optical fibers, etc. In these systems the phenomena of dispersion, dissipation, diffusion, reaction and convection are the fundamental physical common facts. We re- fer the reader to some interesting sources to know more details about the first appearance of this kind of solutions in the works of Russell (1834), Boussinesq (1877), Korteweg and de Vries (1895), Luther (1906), Fisher (1937), Kolmogorov, Petrovskii and Piskunov (1937), and to find several examples of applications and further motivation to study them: see [12, 14, 15, 16, 18, 20, 23, 26, 28] and the references therein. When studying ordinary differential equations (ODE), especially when they are modeling real world phenomena, it is very important to take into account whether the ODE are structurally stable. In a few words this means that if we fix a compact set K in the phase space it is said that an ODE is structurally stable on K when any other close enough (in the C 1 -topology) differential equation has a conjugated phase portrait. This concept is relevant for applications because it implies that the observed behaviours are qualitatively robust with respect to small 2010 Mathematics Subject Classification. Primary: 35C07; 34CO8. Secondary: 34C23, 34C25, 37C27. Key words and phrases. Traveling wave, Abelian integral, Melnikov-Poincaré-Pontryagin function, periodic orbit, limit cycle, bifurcation. 1 2 ARMENGOL GASULL, ANNA GEYER AND VÍCTOR MAÑOSA changes of the model, see for instance [1, 24, 27] for more details, in particular concerning the planar case. Recall that the boundary of the sets of structurally stable differential equations is precisely where bifurcations (that is, qualitative changes of the phase portraits) may occur. It is well-known that for many PDE the existence of TWS is established by proving the existence of a particular solution of a planar ordinary differential equation. These particular solutions must be defined for all time and, in the light of the previous definition, can roughly be classified into two categories: • TWS created by a dynamical behaviour that is structurally stable. Exam- ples of this situation are hyperbolic limit cycles or heteroclinic connections where both critical points are hyperbolic and one of them is a node. • TWS created by a dynamical behaviour that is not structurally stable, as for instance continua of periodic orbits, or homoclinic or heteroclinic solutions connecting hyperbolic saddles. In the first situation, simply take as the set K a compact neighbourhood of the orbit that gives rise to the TWS for a given PDE. Then it can be easily seen that a small enough C 1 perturbation of the original PDE with the same order will still have a TWS. This is so because all the structurally stable phenomena in ODE are robust under C 1 -perturbations. The only condition that must be checked is that the C 1 -closeness between the two PDE’s is translated into a C 1 -closeness in K of the corresponding ODE. An example corresponding to the first situation is the Fisher-Kolmogorov PDE, ut = uxx +u(1−u), where the existence of several TWS of front type with different speeds is associated to the existence of a homoclinic connection between a hyper- bolic saddle and a node, see [2, 13] and references therein. Therefore, all PDE of the form ut = uxx + u(1 − u) + εg(u, ux , ut , ε) for ε small enough have such type of TWS. In fact, the same result holds for many perturbed Fisher-Kolmogorov PDE with a perturbation term of the form εg(u, ux , ut , uxx , uxt , utt , ε). As a sec- ond example of the first situation mentioned above, for some PDE of the form ut = uxx + h(u)ux + g(u) there are periodic TWS which are associated to the existence of hyperbolic limit cycles, see for instance [8, 22] and the references therein. In this paper we address the second, more delicate, situation. More specifically, we consider several PDE having a continuum of periodic TWS associated to a center of a second order ODE associated to the PDE, and we study which conditions have to be imposed on the perturbation of the PDE to be able to ensure that TWS persist and to quantify them. We split our main results into two theorems, which we state in Section 2 after giving some preliminary definitions and notations. Our first result deals with second order PDE, see Theorem A, and applies to a wide range of equations. Our second result, Theorem B, is more restrictive on the one hand because it only considers some special perturbations, but on the other hand it applies to higher order PDE. In Section 3 we study a particular class of Abelian integrals that will often appear in the analysis of the perturbations in Section 4. For these 3 Abelian integrals our main result is given in Theorem C. Finally, in Section 4 we detail some applications of our results. First, in Section 4.1, we apply Theorem A to perturbations of TWS of second order equations such as the Ostrovsky, Klein- Gordon and sine-Gordon equations. Afterwards, in Section 4.2 we use Theorem B to study perturbations of higher order PDE given by the Korteweg-de Vries, Rosenau-Hyman, Camassa-Holm, and Boussinesq equations. 2. Definitions and main results Consider m-th order partial differential equations of the form  ∂u ∂u ∂ 2 u ∂ 2 u ∂ 2 u ∂ mu ∂ mu ∂ mu  (1) P u, , , 2 , , , . . . , m , m−1 , . . . , m , ε = 0, ∂x ∂t ∂x ∂x∂t ∂t2 ∂x ∂x ∂t ∂t where W ⊂ R(m+1)(m+2)/2 is an open set, I is an open interval containing 0, P : W × I → R is a sufficiently smooth function and ε is a small parameter. Recall that the traveling wave solutions of (1) are particular solutions of the form u = U (x − ct) where U (s) is defined for all s ∈ R and satisfies certain bound- ary conditions at infinity. It is well-known that the existence of such solutions is equivalent to finding solutions defined for all s of the m-th order ordinary differential equation (2) Pc (U, U 0 , U 00 , . . . , U (m) , ε) :=  P U, U 0 , −cU 0 , U 00 , −cU 00 , c2 U 00 , . . . , U (m) , −cU (m) , . . . , (−c)m U (m) , ε = 0, satisfying these conditions. Here the prime denotes derivative with respect to s and Pc : Wc (ε) × I → R, where Wc (ε) is an open subset of Rm+1 . We will distinguish two cases according to whether (1) is a second order equa- tion (m = 2) or a higher order equation (m > 2). Second order equations. Our main result applies to a certain class of per- turbed PDE that satisfy three conditions (i)–(iii) that we detail below. Suc- cinctly, it requires the existence of a certain wave speed c ∈ R such that: (i) the associated ODE has the form U 00 = fc (U, U 0 ) + εgc (U, U 0 , ε); (ii) after a time reparameterization if necessary the planar system associated with this ODE can be written as a perturbation of a Hamiltonian system; and (iii) this Hamiltonian system has a center, and the Melnikov-Poincaré-Pontryagin function associated with the perturbation has ` simple zeroes, see [4, Part II] for further details. More precisely, we will say that the PDE (1) with m = 2 satisfies Property A if there exists c ∈ R such that the following three conditions hold: (i) There exist C 1 functions fc : Vc (ε) → R and gc : Vc (ε) × I ⊂ R3 → R, with Vc (ε) ⊂ R2 and Vc (ε) × I ⊂ R3 open sets, such that, for ε small enough, {(x, y) ∈ Vc (ε) : z = fc (x, y) + εgc (x, y, ε)} ⊂ {(x, y, z) ∈ Wc (ε) : Pc (x, y, z, ε) = 0}. Moreover, if Uc is the limit of the sets Vc (ε) when ε → 0, the only solution of fc (x, 0) = 0 in Uc is x = xc . 4 ARMENGOL GASULL, ANNA GEYER AND VÍCTOR MAÑOSA (ii) There exists a C 2 function Hc : Vc ⊂ R2 → R+ ∪{0} such that Hc (xc , 0) = 0, ∂Hc (x, y) y ∂Hc (x, y) fc (x, y) = , =− , ∂y sc (x, y) ∂x sc (x, y) for some C 1 function sc : Vc ⊂ R2 → R+ . Notice that sc is such that     ∂ y ∂ fc (x, y) + ≡ 0. ∂x sc (x, y) ∂y sc (x, y) (iii) For each h ∈ (0, hc ), where hc ∈ R+ ∪ {∞}, the set γc (h) := {(x, y) ∈ Vc : Hc (x, y) = h} is a closed oval surrounding (xc , 0) and the function Mc : (0, hc ) → R, defined as the line integral Z gc (x, y, 0) Mc (h) = dx, γc (h) sc (x, y) has ` ≥ 1 simple zeroes in (0, hc ). Theorem A. Assume that the second order PDE (3) P (u, ut , ux , utt , utx , uxx , ε) = 0, satisfies Property A for some c ∈ R. Then: (a) For ε = 0 the PDE (3) has a continuum of periodic TWS, u = Uh (x − ct), for h in an open real interval. (b) For ε small enough it has at least ` periodic TWS, u = Uhj (x − ct, ε), j = 1, 2, . . . , `. Proof of Theorem A. From the discussion at the very beginning of this section, a function U (s) is a TWS for the PDE (1) if it is defined for all time and (4) Pc (U (s), U 0 (s), U 00 (s), ε) = 0, where Pc is defined in (2). By using (i) of Property A we can write the above expression as U 00 (s) = fc (U (s), U 0 (s)) + εgc (U (s), U 0 (s), ε), for some suitable fc and gc . In other words, (x, y) = (U (s), U 0 (s)) is a solution of the planar ODE  dx   x0 = = y, ds   y 0 = dx = f (x, y) + εg (x, y, ε). c c ds By item (ii) of Property A we can parameterize U by a new time, say τ, with dτ /ds = sc (x, y), and then x = U (τ ) satisfies the equivalent planar ODE   dx dx dτ y ∂Hc (x, y)  ẋ = dτ = dτ ds = s (x, y) = , c ∂y (5)   dy dy dτ fc (x, y) gc (x, y, ε) ∂Hc (x, y) gc (x, y, ε) ẏ = = = +ε =− +ε . dτ dτ ds sc (x, y) sc (x, y) ∂x sc (x, y) 5 When ε = 0 the above system is Hamiltonian, and by (i) and (iii) of Property A the continuum of curves γc (h) for 0 < h < h are periodic orbits of system (5) with ε = 0 that surround the center (xc , 0). The functions Uh (s, c) = xh (τ (s), c), where (xh (τ, c), yh (τ, c)) is the parameterization of γc (h), give rise to the continuum of periodic traveling wave solutions of (3). When ε 6= 0 is small enough we are in the setting of the perturbations of Hamiltonian systems, [4, 9]. Recall that for general perturbed C 1 Hamiltonian systems,    ∂H(x, y) ẋ = + εR(x, y, ε), ∂y (6)  ẏ = − ∂H(x, y) + εS(x, y, ε),  ∂x its associated Melnikov-Poincaré-Pontryagin function is Z M (h) = S(x, y, 0) dx − R(x, y, 0) dy, γ(h) where the curves γ(h) form a continuum of ovals contained in {H(x, y) = h, for h ∈ (h0 , h1 )}. Then, it is known that each simple zero h∗ ∈ (h0 , h1 ) of M gives rise to a limit cycle of (6) that tends to γ(h∗ ) when ε → 0. For system (5), M (h) = Mc (h) and so, each simple zero hj ∈ (0, hc ), j = 1, 2, . . . , ` of Mc (h) gives rise to a limit cycle of system (5). Each of these limit cycles correspond to a periodic TWS of (3).  Higher order equations (m > 2). In this situation our approach only works for a particular class of differential equations. Again, fixed (c, k) ∈ R2 , we will define a property similar to Property A which will consist of four conditions. The first one, that we will call condition (o), is the most restrictive one and it is totally different to the ones imposed when m = 2. It states that the associated ODE can somehow be reduced to a second order equation, or that some of the solutions of the associated ODE are also solutions of a related second order ODE. The rest of the conditions are quite similar to the ones of the planar case. More precisely, we say that a PDE satisfies Property B if there exist c, k ∈ R such that: (o) There exists a function Qc : Wc × I → R, where Wc ⊂ R3 is open and Qc is sufficiently smooth, such that dm−2  m−2 Qc (U, U 0 , U 00 , ε) = Pc (U, U 0 , U 00 , . . . , U (m) , ε), ds where Pc is defined in (2) and U = U (s). (i) There exist two C 1 functions fc,k : Vc,k (ε) → R and gc,k : Vc,k (ε) × I → R with Vc,k (ε) ⊂ R2 and Vc,k (ε) × I ⊂ R3 open sets, such that, for ε small enough, {(x, y) ∈ Vc,k (ε) : z = k + fc,k (x, y) + εgc,k (x, y, ε)} ⊂ {(x, y, z) ∈ Wc,k (ε) : Qc (x, y, z, ε) = k}. 6 ARMENGOL GASULL, ANNA GEYER AND VÍCTOR MAÑOSA Moreover, if Uc,k is the limit of the sets Vc,k (ε) when ε → 0, the only solution of fc (x, 0) = k in Vc,k is x = xc,k . (ii) There exists a C 2 function Hc,k : Vc,k ⊂ R2 → R+ ∪ {0} such that Hc,k (xc,k , 0) = 0, ∂Hc,k (x, y) y ∂Hc,k (x, y) fc (x, y) + k = , =− , ∂y sc,k (x, y) ∂x sc,k (x, y) for some C 1 function sc,k : Vc,k ⊂ R2 → R+ . Notice that the function sc,k is such that     ∂ y ∂ fc (x, y) + k + ≡ 0. ∂x sc,k (x, y) ∂y sc,k (x, y) (iii) For each h ∈ (0, hc,k ), where hc,k ∈ R+ ∪ {∞}, the set γc,k (h) := {(x, y) ∈ Vc,k : Hc,k (x, y) = h} is a closed oval surrounding (xc,k , 0) and the function Mc,k : (0, hc,k ) → R, defined as the line integral Z gc,k (x, y, 0) Mc,k (h) = dx, γc,k (h) sc,k (x, y) has ` ≥ 1 simple zeroes in (0, hc,k ). Theorem B. Assume that the m-th order PDE (1), with m > 2, satisfies Prop- erty B, for some c ∈ R and k ∈ R. Then: (a) For ε = 0 the PDE (3) has a continuum of periodic TWS, u = Uh,k (x − ct) for h in an open real interval. (b) For ε small enough it has at least ` periodic TWS, u = Uhj ,k (x − ct, ε) for j = 1, 2, . . . , `. Proof of Theorem B. From condition (o) of Property B, if we restrict our atten- tion to the solutions of (2) contained in (7) Qc (U, U 0 , U 00 , ε) = k, for the given value of k ∈ R, we can find some TWS with speed c and associated to this particular value of k. Other values of k give different TWS with the same speed. Starting from equation (7), instead of equation (4), we can repeat all the steps of the proof of Theorem A, point by point, to get the desired conclusion.  3. Some particular Abelian integrals This section is devoted to studying a particular class of Abelian integrals for which we prove a result quantifying their zeros, see Theorem C. We will use this result in the next section when we study the persistence of TWS for several perturbed PDE, which is governed by the number of zeros of integrals of this type. 7 Proposition 3.1. Let A, B and D be analytic functions, defined in an open interval I ⊂ R and such that (x − x∗ )2  A(x) =a2 + O(x − x∗ ), B(x) = + O (x − x ∗ 3 ) , b2 D(x) =(x − x∗ )2n D0 (x) with D0 (x) = d + O(x − x∗ ), for certain x∗ ∈ I, where a, b, c are real constants with abd 6= 0 and n ∈ N ∪ {0}. Consider the Hamiltonian function H(x, y) = A(x)y 2 + B(x). Then, the following holds: (a) The Hamiltonian system ẋ = Hy (x, y), ẏ = −Hx (x, y), has a center at (x∗ , 0). We will denote by γ(h) the periodic orbits contained in {H(x, y) = h}, which exist when h ∈ (0, e h) for some eh ∈ R. e (b) For h ∈ (0, h) and p, n ∈ N, define the Abelian integral Z (8) Jp (h) = D(x)y p dx. γ(h) Then J2p (h) ≡ 0 and 2db2n+1 (2p − 1)!! (2n − 1)!! π p+n J2p−1 (h) ∼ 2p−1 p+n h at h = 0+ , a 2 (p + n)! where (2k − 1)!! = (2k − 1)(2k − 3) · · · 3 · 1 and (−1)!! = 1!! = 1. Proof. Without loss of generality we will assume that x∗ = 0. To prove (a), notice that the origin is a non-degenerate singular point of the vector field X = (Hy , −Hx ) because det(DX(0, 0)) = 2A(0)B 00 (0) = 4a2 /b2 > 0. Moreover, since a singular point of a Hamiltonian system can neither be a focus nor a node, it is a center. To study the Abelian integral Jp it is convenient to introduce the new variable w as h = w2 . Then, by the Weierstrass preparation theorem, see for instance [1, 3], in a neighbourhood of (0, 0) the only solutions of equation B(x) − w2 = x2 /b2 − w2 + O(x3 ) = 0 are x = x± (w) = ±bw + O(w2 ), where x± (w) are analytic functions at zero. Moreover, in this neighbourhood, (9) w2 − B(x) = (x − x− (w))(x+ (w) − x)U (x, w), 2 where U (0, 0) = 1/b p is also analytic at (0, 0). Notice that the points of the oval γ(h) satisfy y = ± (w2 − B(x))/A(x). When p is even the integral (8) vanishes because of symmetry with respect to y = 0. Hence  0,  when p is even, 2 Z +   p Jp (w ) = x (w) w2 − B(x) 2  2 D(x) dx, when p is odd. x− (w) A(x) 8 ARMENGOL GASULL, ANNA GEYER AND VÍCTOR MAÑOSA By using (9) we get that Z x+ (w)  2  2p−1 2 w − B(x) 2 J2p−1 (w ) =2 D(x) dx x− (w) A(x) Z x+ (w)   2p−1  2p−1 U (x, w) 2 =2 (x − x− (w))(x+ (w) − x) 2 D(x) dx x− (w) A(x) Z 1   2p−1 2p  2p−1 U (z, w) 2 =2(∆(w)) z(1 − z) 2 D(z, w) dz, 0 A(z, w) where in the integral we have introduced the change of variables z = (x − x− (w))/∆(w), being ∆(w) = x+ (w) − x− (w), and for any function E(x, w) or E(x), we denote E(z, w) = E ∆(w)z + x− (w), w or E(z, w) = E(∆(w)z + x− (w)). In particular,  2n − 2n 2n x− (w) D(z, w) = ∆(w)z + x (w) D0 (z, w) = (∆(w)) z+ D0 (z, w), ∆(w) with D0 (0, 0) = d. Hence, Z 1  2p−1 J2p−1 (w2 ) = (∆(w))2p+2n z(1 − z) 2 F (z, w) dz, 0 where  2n   2p−1 x− (w) U (z, w) 2 F (z, w) = 2D0 (z, w) z + . ∆(w) A(z, w) Since x± (w) = ±bw + O(w2 ), it holds that ∆(w) = 2bw + O(w2 ) and hence − (w) limw→0 x∆(w) = − 12 . Therefore for all z ∈ [0, 1] and w small enough the function F (z, w) is continuous, and as a consequence  2p+2n Z 1 J2p−1 (w2 ) ∆(w)  2p−1 lim = lim z(1 − z) 2 lim F (z, w) dz w→0 w 2p+2n w→0 w 0 w→0 Z 1  2p+2n 2d  2p−1 1 2n = (2b) z(1 − z) 2 z − dz. a2p−1 b2p−1 0 2 Now we claim that Z 1  2p−1  1 2n (2p − 1)!! (2n − 1)!! K(p, n) := z(1 − z) 2 z − dz = π, 0 2 8p+n (p + n)! and we observe that, from this claim, the result follows. To prove the claim we observe that by using integration by parts, one easily gets that 2p − 1 (10) K(p, n) = K(p − 1, n + 1). 2n + 1 Now the claim follows by using induction. First we prove that for any p ∈ N0 , K(p, 0) satisfies the claim. Indeed, if B is the Euler’s Beta function, and since 9 B(x, y) = Γ(x)Γ(y)/Γ(x + y), we have Z 1   p− 21 p− 12 1 1 (Γ(p + 12 ))2 K(p, 0) = z (1 − z) dz = B p + , p + = . 0 2 2 Γ(2p + 1) Since p is an integer number Γ(2p + 1) = (2p)!. On the other hand, it is well known that   1 (2p)! √ (2p − 1)!! √ Γ p+ = p π= π. 2 4 p! 2p Hence, (2p − 1)!! K(p, 0) = π 8p p! as we wanted to prove. Now we assume that for n > 0 and for all p ∈ N0 , K(p, n) satisfies the claim. By using the relation (10), we get that, 2n + 1 (2n + 1) (2p + 1)!! (2n − 1)!! K(p, n + 1) = K(p + 1, n) = π= 2p + 1 (2p + 1) 8p+n+1 (p + n + 1)! (2p − 1)!! (2n + 1)!! = π, 8p+n+1 (p + n + 1)! so the claim follows.  Before proving the main result of this section, Theorem C, and to motivate one of its hypotheses, we collect some simple observations in the following lemma. Lemma 3.2. Let γ(h) ⊂ {H(x, y) = h}, h ∈ (0, h) = L, be a continuum of periodic orbits surrounding a center, corresponding to h = 0, of the Hamiltonian system associated to a C 1 Hamiltonian function H(x, y) and assume that they have a clockwise time parameterization. For each p, q ∈ N ∪ {0}, consider the Abelian integral Z Jq,p (h) = xq y p dx. γ(h) The following holds. (a) When q is even and p is odd then Jq,p (h) > 0 for all h ∈ L. (b) When q is even and p is even and H(x, y) = H(x, −y) then Jq,p (h) ≡ 0 on the whole interval L. (c) When q is odd and p is odd and H(−x, y) = H(x, y) then Jq,p (h) ≡ 0 on L. Proof. Notice that by Green’s theorem Z ZZ q p Jq,p (h) = x y dx = pxq y p−1 dx dy, γ(h) Int(γ(h)) where Int(γ(h)) denotes the interior of the oval. Then, trivially (a) follows. The other two statements are consequence of the symmetries of H and the function xq y p−1 .  The next proposition will be one of the key results to prove Theorem C, which is stated below. 10 ARMENGOL GASULL, ANNA GEYER AND VÍCTOR MAÑOSA Proposition 3.3. ([6]) Set L ⊂ R an open real interval and let Fj : L → R, j = 0, 1, . . . , `, be ` + 1 linearly independent analytic functions. Assume also that one of them, say Fk , 0 ≤ k ≤ `, has constant sign on L. Then, P`there exist real constants dj , j = 0, 1, . . . , `, such that the linear combination j=0 dj Fj has at least ` simple zeroes in L. Notice that in the next theorem, and due to Lemma 3.2, the monomials of the Abelian integral that we consider are of the form x2q y 2p−1 . Theorem C. Let H(x, y) = A(x)y 2 + B(x), with A and B functions satisfy- ing the hypotheses of Proposition 3.1, and denote by γ(h), h ∈ (0, h), the peri- odic orbits surrounding the origin of the corresponding Hamiltonian system. For d0 , d1 , . . . , dn ∈ R and qj , pj ∈ N, j = 0, 1, . . . , ` consider the family of Abelian integrals Z X̀ J(h) = dj x2qj y 2pj −1 dx. γ(h) j=0 If all values mj = qj + pj , j = 0, 1, . . . , ` are different, there exit values of dj , j = 0, 1, . . . , `, such that the corresponding function J(h) has at least ` simple zeroes in (0, h). Proof. Notice that X̀ Z J(h) = dj Jj (h), where Jj (h) = x2qj y 2pj −1 dx. j=0 γ(h)  By Proposition 3.1, for each j = 0, 1, . . . , `, Jj (h) = kj hmj + o hmj and, by hypothesis, all these mj are different. This clearly implies that all these ` + 1 functions are linearly independent. Moreover, by item (a) of Lemma 3.2 we know that none of them vanish in (0, h). Hence we can apply Proposition 3.3 to this set of functions and L = (0, h) and the result follows.  4. Applications In this section we consider perturbations of several relevant PDE with continua of periodic TWS and prove that the perturbations can be tailored such that a prescribed number of TWS persist in these perturbed PDE. In many examples, for simplicity, we perturb the PDE with an additive term that only contains partial derivatives up to m − 1. For more general perturbations, even including terms of order m, most of the results can be adapted. 4.1. Second order PDE. We start with an illustrative toy example for which we give all the details on how a prescribed number of periodic TWS can be obtained. 11 4.1.1. A toy example. Consider the PDE (11) u + auxx + buxt + dutt + εg(ux , ut , ε) = 0, with g a C 1 function and take c such that a − bc + dc2 > 0. Then equation (2) can be written as U + (a − bc + dc2 )U 00 + εg(U 0 , −cU 0 , ε) = 0. We define C 2 = a − bc + dc2 and gc (U 0 , ε) = −g(U 0 , −cU 0 , ε)/C 2 . Then it is easy to see that this PDE satisfies Property A with Hc (x, y) = x2 /(2C 2 ) + y 2 /2, sc (x, y) ≡ 1 and (0, h) = (0, ∞). That is,  ẋ = ∂Hc (x, y) = y,   ∂y   ∂Hc (x, y) x  ẏ = − + εgc (x, y, ε) = − 2 + εgc (y, ε). ∂x C Moreover Z Mc (h) = gc (y, 0) dx, γ(h) where γ(h) is the ellipse {x2 /(2C√2 ) + y 2 /2 √ = h}. We parameterize the closed curves Hc (x, y) = h as (x, y) = (C 2h cos θ, 2h sin θ) for 0 ≤ θ ≤ 2π. Then √ Z 2π √ Mc (h) = −C 2h gc ( 2h sin θ, 0) sin θ dθ. 0 PN Assume for instance that gc (y, 0) = j=0 gj y j is a polynomial of degree N , and gj ∈ R. Then, √ N X √ j Z 2π  Mc (h) = −C 2h gj ( 2h) sinj+1 θ dθ . j=0 0 When j is even, by symmetry, the above integrals vanish. Hence  [(NX −1)/2]  Mc (h) = −2Ch g2i+1 2i I2i+2 hi , i=0 R 2π where [ · ] denotes the integer part and I2n = 0 sin2n θ dθ > 0. Removing the factor h, and taking suitable g2i+1 , the polynomial Mc (h)/h can be any arbitrary polynomial of degree [(N − 1)/2] in h. Hence, by applying Theorem C, for any ` ≤ [(N − 1)/2], there exist coefficients gj such that the function Mc (h) has ` simple zeros and, therefore, by applying Theorem A, the PDE (11) has at least ` periodic TWS. We remark that the above computations are essentially the same as the ones of the celebrated paper [21] where the authors present the first example of classical polynomial Liénard differential system of degree N with [(N − 1)/2] limit cycles. By doing similar computations we can consider more general perturbations in PDE (11), like for instance u + auxx + buxt + dutt + ε(uuxx + g(u, ux , ut , ε)) = 0, 12 ARMENGOL GASULL, ANNA GEYER AND VÍCTOR MAÑOSA and similar results hold. 4.1.2. Reduced Ostrovsky equation. We consider perturbations of the reduced Os- trovsky equation, introduced by L. Ostrovsky in 1978, which is a modification of the Korteweg-de Vries equation that models gravity waves propagating in a rotat- ing background under the influence of the Coriolis force when the high-frequency dispersion is neglected. More concretely, we take (12) (ut + uux )x − u + εg(u, ux , ut , ε) = 0, which satisfies Property A with c > 0, because its associated ODE is (U − c)U 00 + (U 0 )2 − U + εg(U, U 0 , −cU 0 , ε) = 0. Then, taking gc (U, U 0 , ε) = −g(U, U 0 , −cU 0 , ε)/(U − c); Vc = {x < c}; xc = 0; and sc (x, y) = (x − c)−2 , the system that has to be studied to find TWS is    ∂Hc (x, y) ẋ = , ∂y  ẏ = − ∂Hc (x, y) + (x − c)2 εgc (x, y, ε).  ∂x with (x − c)2 y 2 cx2 x3 Hc (x, y) = + − . 2 2 3 Consider also the Melnikov-Poincaré-Pontryagin function Z Mc (h) = (x − c)2 gc (x, y, 0) dx, h ∈ (0, c3 /3). γc (h) As in the toy example, it is not difficult to find a perturbation term g such that the the function Mc (h) has several simple zeroes in (0, c3 /3) which, by Theorem C, give rise to periodic TWS of the PDE (12). 4.1.3. Perturbed non-linear Klein-Gordon equation. The Klein-Gordon equation is a wave equation related to the Schrödinger equation, which is used to model spinless relativistic particles. It was introduced in 1926 in parallel by O. Klein, W. Gordon and V. Fock as a tentative to describe the relativistic electron dynamics. In the one-dimensional setting we look at a perturbation of this equation of the form utt − uxx + λup + εg(u, ux , ut , ε) = 0, with λ ∈ R+ and p an odd integer. It can readily be seen that it satisfies Property A and that the system that has to be studied to find TWS is    ∂Hc (x, y) ẋ = , ∂y  ẏ = − ∂Hc (x, y) + εgc (x, y, ε),  ∂x 13 where Cxp+1 y 2 Hc (x, y) = + , p+1 2 with C = λ/(c − 1), gc (x, y, ε) = −g(x, y, −cy, ε)/(c2 − 1). The associated 2 Melnikov-Poincaré-Pontryagin function is Z Mc (h) = gc (x, y, 0) dx, h ∈ (0, ∞). γc (h) The interested reader can take a look to the papers [5, 17] where perturbations of this Hamiltonian system and the zeros of its associated Melnikov-Poincaré- Pontryagin function are studied with two different approaches. In particular, the zeroes of the above first integral can be studied in a similar way to the toy example considered at the beginning of this section. Notice, how- ever, that when p ≥ 3, instead of using trigonometric functions to parametrize the invariant closed curves, one can use the generalized polar coordinates intro- duced by Lyapunov in 1893 in his study of the stability of degenerate critical points, [19]. All the details can be found in [5]. Again, Theorem A guaran- tees that the zeros of the function Mc (h) correspond with periodic TWS of the Klein-Gordon equation. 4.1.4. Perturbed sine-Gordon equation. The sine-Gordon equation first appeared in 1862 in the context of differential geometry. Specifically in a study by E. Bour on surfaces of constant negative curvature. The equation was rediscovered later by J. Frenkel and T. Kontorova in 1939, in their study of crystal dislocations. The equation is relevant to the community investigating integrable systems because it has soliton solutions. Its perturbation writes as utt − uxx + sin u + εg(u, ux , ut , ε) = 0. Again, it satisfies Property A for c > 1, and its associated planar system is    ∂Hc (x, y) ẋ = , ∂y  ẏ = − ∂Hc (x, y) + εgc (x, y, ε),  ∂x where y2 Hc (x, y) = C(1 − cos x) + , 2 with C = 1/(c2 − 1) > 0 and gc (x, y, ε) = g(x, y, −cy, ε)/(1 − c2 ). The Melnikov- Poincaré-Pontryagin function is Z Mc (h) = gc (x, y, 0) dx, h ∈ (0, 2C). γc (h) The above type integrals are studied for instance in [11]. There, several condition on g for obtaining many simple zeroes of Mc , and therefore periodic TWS of the considered PDE, are obtained. 14 ARMENGOL GASULL, ANNA GEYER AND VÍCTOR MAÑOSA 4.2. PDE with order greater than 2. In this section we study perturbations of several PDE with order m > 2. We start with the following result that helps us to characterize the existence of centers for the unperturbed Hamiltonian systems that will appear. Lemma 4.1. Consider a Hamiltonian system of the form    ∂H(x, y) ẋ = = y m(x, y), ∂y  ẏ = − ∂H(x, y) = f (x, y) m(x, y),  ∂x 2 ∂ ∂ where H ∈ C , m(x, y) > 0 and such that ∂x (ym(x, y)) + ∂y (f (x, y)m(x, y)) ≡ 0. Then, a singular point of the form (x∗ , 0) is a center if ∂ (13) f (x, y) < 0. ∂x (x∗ ,0) Furthermore, if m(x, y) depends only on x, condition (13) holds, and H is analytic, then the Hamiltonian H satisfies the hypotheses of Proposition 3.1. Proof. Consider the vector field X = (Hy , −Hx ). Since ∂ det(DX(x∗ , 0)) = −m2 (x∗ , 0) (f (x, y)) , ∂x (x∗ ,0) then equation (13) implies that det(DX(x∗ , 0)) > 0 and therefore (x∗ , 0) is a center (once more, remember that a singular point of a Hamiltonian system cannot be neither a focus nor a node). If m(x, y) = m(x), then H(x, y) = y 2 m(x)/2 + B(x) for some analytic func- tion B. Since m(x) > 0 we can write m(x) = 2a2 +O(x−x∗ ) near x = x∗ . Suppose that condition (13) holds, then Hxx (x∗ , 0) = −fx (x∗ , 0)m(x∗ ) > 0, and we can  write 1/b2 = B 00 (x∗ ) = Hxx (x∗ , 0), obtaining B(x) = (x − x∗ )2 /b2 + O (x − x∗ )3 . So H fulfills the hypotheses of Proposition 3.1.  Observe that condition (13) is equivalent to the fact that Hxx (x∗ , 0) > 0 and det(HH (x∗ , 0)) > 0 (where H is the hessian matrix), which implies that H has a non-degenerate local minimum at (x∗ , 0). 4.2.1. Perturbed generalized Korteweg-de Vries equation. We consider a pertur- bation of a family of PDE which for certain values of the parameters contains the celebrated Korteweg-de Vries and Benjamin-Bona-Mahony equations appearing in several domains of physics (non-linear mechanics, water waves, etc.). More concretely, we consider the family of PDE (14) ut + aux + buux + duut + puxxx + quxxt + ruxtt + suttt + ε∇g(u, ux , ut , ε) · (ux , uxx , uxt , 0)t = 0. Notice that the KdV equation corresponds to ε = 0 and a = d = q = r = s = 0, b = −6 and p = 1. The ODE associated to (14) is  b − dc 2 0 (a − c)U + U + CU 00 + εg(U, U 0 , −cU 0 , ε) = 0, 2 15 where C = p − qc + rc2 − sc3 . Notice that then, for any function U satisfying previous equation, it holds that there exists k ∈ R, such that b − dc 2 (15) (a − c)U + U + CU 00 + εg(U, U 0 , −cU 0 , ε) = k 2 Thus we have to study the equivalent planar system    ∂Hc,k (x, y) ẋ = y = , ∂y  ẏ = αc,k + βc x + γc x2 + εgc (x, y, ε) = − ∂Hc,k (x, y) + εgc (x, y, ε),  ∂x where βc γc 1 Hc,k (x, y) = −αc,k x − x2 − x3 + y 2 , 2 3 2 with k c−a dc − b αc,k = , βc = , γc = , C C C and gc (x, y, ε) = −g(x, y, −cy, ε)/C. Hence, using Lemma 4.1, it is not difficult to see that the PDE (14) satisfies Property B when equation αc,k + βc x + γc x2 = 0 has two different real solutions (that correspond to a center and a saddle of the planar system). Then, by Theorem B, the periodic TWS that persist for ε small enough correspond to the simple zeroes of the elliptic integral Z Mc,k (h) = gc (x, y, 0) dx γc,k (h) in a suitable open interval of energies. This kind of Abelian integrals are studied in detail in the classical paper of Petrov ([25]) and more recently in the Chapter 3 of Part II of the book [4]. Again, it is not difficult to impose conditions on g to get a prescribed number of TWS for (14) for ε small enough and different values of c and k. 4.2.2. Perturbed Rosenau-Hyman equation. The Rosenau-Hyman equation is a generalization of the KdV equation. It was introduced in 1993 by P. Rosenau and J.M. Hyman to show the existence of solitary waves with compact support (compactons) in the context of non-linear dispersive equations. We consider the perturbed equation ut + a(un )x + (un )xxx + ε∇g(u, ux , ut , ε) · (ux , uxx , uxt , 0)t = 0, where a ∈ R and n ∈ N. To find TWS for it we have to study the third order ODE −cU 0 + a(U n )0 + (U n )000 + ε∇g(U, U 0 , −cU 0 , ε) · (U 0 , U 00 , −cU 00 , 0)t 0 = − cU + aU n + (U n )00 + εg(U, U 0 , −cU 0 , ε) 0 = (−cU + aU n + n(n − 1)U n−2 U 0 + nU n−1 U 00 + εg(U, U 0 , −cU 0 , ε) = 0. Thus, we need to find solutions of the second order ODE −cU + aU n + n(n − 1)U n−2 U 0 + nU n−1 U 00 + εg(U, U 0 , −cU 0 , ε) = k 16 ARMENGOL GASULL, ANNA GEYER AND VÍCTOR MAÑOSA with k ∈ R. It writes as the planar system  0  x = y,  n n−2 2  y 0 = k + cx − ax − n(n − 1)x y + ε gc (x, y, ε) ,  nxn−1 nxn−1 where gc (x, y, ε) = −g(x, y, −cy, ε). With the new time τ, where dτ /ds = sc,k (x, y) and sc,k (x, y) = x2(1−n) /n, we get x = U (τ ) satisfies the equivalent planar ODE    ∂Hc,k (x, y) ẋ = , ∂y  ẏ = − ∂Hc,k (x, y) + εxn−1 gc (x, y, ε),  ∂x where n k c a Hc,k (x, y) = x2(n−1) y 2 − xn − xn+1 + x2n . 2 n n+1 2n By Lemma 4.1 (see the comment below its statement), if there exists a singular point (x∗ , 0) such that ∂ 2 Hc,k (x∗ , 0) = xn−2 ∗ (a (2n − 1) xn∗ − c n x∗ − k (n − 1)) > 0, ∂x2 then it is a center. Furthermore, since the Hypothesis of Proposition 3.1 are satisfied, we can apply Theorem B and the periodic TWS for the perturbed PDE correspond to simple zeroes of Z Mc,k (h) = xn−1 gc (x, y, 0) dx γc,k (h) in a suitable interval of the energy. To get examples of perturbations with several simple zeroes we can apply Theorem C. 4.2.3. Camassa-Holm equation and related PDE. The Camassa-Holm equation is a model for the propagation of shallow water waves of moderate amplitude. The horizontal component of the fluid velocity field at a certain depth within the fluid is described by the PDE ut + (2κ + 3u)ux − 2ux uxx + uuxxx − uxxt = 0, and the parameter κ is positive. Constantin and Lannes derived in [7] a similar PDE for surface waves also with moderate amplitude in the shallow water regime, ut + 1 + 6u − 6u2 + 12u3 )ux + 28ux uxx + 14uuxxx + uxxx − uxxt = 0, see also [10]. Similarly, the Degasperis-Procesi equation ut + 4uux − 3ux uxx − uuxxx − uxxt = 0, which was derived initially only for its integrability properties, has a similar role in hydrodynamics. 17 In fact, perturbations of the above equations can be written under the common expression ut + A0 (u)ux + bux uxx + duuxxx + puxxx + quxxt + ruxtt + suttt + ε∇g(u, ux , ut , ε) · (ux , uxx , uxt , 0)t = 0, where A is sufficiently smooth and b, d, p, q, r and s are real parameters. Its associated third order ODE is − cU 0 + A0 (U )U 0 + bU 0 U 00 + dU U 000 + CU 000 + ε∇g(U, U 0 , −cU 0 , ε) · (U 0 , U 00 , −cU 00 , 0)t   0 = Ac (U ) + b(U 0 )2 /2 + d U U 00 − (U 0 )2 /2 + CU 00 + εg(U, U 0 , −cU 0 , ε) = 0 where Ac (U ) = A(U ) − cU, with Ac (0) = 0, and C = p − qc + rc2 − sc3 . Hence, for any function U satisfying the previous equation, there exists k ∈ R, such that Ac (U ) + β(U 0 )2 + (C + dU )U 00 + εg(U, U 0 , −cU 0 , ε) = k, where β = (b − d)/2. The above equation can be written as the planar system  0 x = y,  2 y 0 = k − Ac (x) − βy + εgc (x, y, ε) ,  C + dx 0 0 where gc (x, y, ε) = −g(U, U , −cU , ε). Then, taking dτ /ds = sc (x), ( (C + dx)−2β/d when d 6= 0, sc (x) = −2βx/C e when d = 0, we get   ∂Hc,k (x, y)  ẋ = , ∂y   ∂H (x, y) gc (x, y, ε) ẏ = − c,k +ε , ∂x (C + dx)sc (x) with Z x y2 Ac (w) − k Hc,k (x, y) = + dw. 2sc (x) 0 (C + dw)sc (w) By Lemma 4.1, any singular point (x∗ , 0) such that Hxx (x∗ , 0) > 0, is a center. So by Theorem B, the periodic TWS of the perturbed equations correspond with the simple zeros of Z gc (x, y, 0) Mc.k (h) = dx. γc,k (h) (C + dx)sc (x) Again, for some particular examples, the zeroes of the above type of Abelian integrals can be obtained by using Theorem C. For instance, we observe P  that this ` 2i+1 is trivially the case if gc (x, y, 0) = (C + dx)sc (x) i=0 d2i+1 y . 18 ARMENGOL GASULL, ANNA GEYER AND VÍCTOR MAÑOSA 4.2.4. Boussinesq-type equations. The Boussinesq equation describes bi-directional surface water waves and reads utt + uuxx − uxx + (ux )2 − uxxxx = 0. Similarly, the modified Boussinesq equation is utt + uuxx − uxx + (ux )2 − uxxtt = 0, and appears in the modelling of non-linear waves in a weakly dispersive medium. We consider the following perturbation of the family of PDE (16) auxx + buxt + dutt + 2e(uuxx + (ux )2 ) + puxxxx + quxxxt + ruxxtt + suxttt + f utttt + εG = 0, , where a, b, d, e, p, q, r, s, f are suitable real parameters. We do not detail here the perturbation G, but it is a function of all the partial derivatives of u up to order four, and such that after replacing u by U (x − ct) it holds that there exists 00 a function gc such that G = gc (U, U 0 , ε) . Hence the ODE associated to (16) is 00 00 CU 00 + e(U 2 )00 + DU 0000 + ε gc (U, U 0 , ε) = CU + eU 2 + DU 00 + εgc (U, U 0 , ε) = 0, where C = a − bc + d2 c, D = p − qc + rc2 − sc3 + f c4 , and we have used that (u2 )xx = 2uux + 2(ux )2 . We are interested in solutions of the above fourth order ODE (17) CU + eU 2 + DU 00 + εgc (U, U 0 , ε) = k, for some k ∈ R. When D 6= 0 we are again under the situation covered by Theo- rem B. Notice that other solutions would satisfy CU +eU 2 +DU 00 +εgc (U, U 0 , ε) = k1 s + k2 , for some k1 6= 0, k2 ∈ R, but we do not consider them. In fact, from (17) we arrive at the same ODE that appears in the study done in Section 4.2.1 about the perturbed generalized Korteweg-de Vries equation, but with a different notation. Indeed, the above ODE is the same as (15) and it can be studied to get TWS for (16) exactly like in that case. Acknowledgements. The first and third authors are supported by Ministry of Science and Innovation–Research Agency of the Spanish Government by grants PID2019-104658GB-I00 (first author) and DPI2016-77407-P (AEI/FEDER, UE, third author) and by the grants AGAUR, Generalitat de Catalunya (2017-SGR- 1617, first author) and (2017-SGR-388, third author). Part of this work was carried out at the Erwin Schrödinger International In- stitute for Mathematics and Physics where authors participated in a Research in Teams project in 2018. References [1] Andronov, A. A., Leontovich, E. A., Gordon, I. I., and Maı̆er, A. G. Theory of bifurcations of dynamic systems on a plane. 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Departament de Matemàtiques, Edifici Cc, Universitat Autònoma de Barcelona, 08193 Cerdanyola del Vallès, Barcelona, Spain. Centre de Recerca Matemàtica, Edifici Cc, Campus de Bellaterra, 08193 Cer- danyola del Vallès, Barcelona, Spain. E-mail address: gasull@mat.uab.cat Delft Institute of Applied Mathematics, Faculty Electrical Engineering, Mathematics and Computer Science Delft University of Technology, Van Mourik Broekmanweg 6, 2628 XE Delft, The Netherlands E-mail address: A.Geyer@tudelft.nl Departament de Matemàtiques and Institut de Matemàtiques de la UPC-Bar- celonaTech (IMTech), Universitat Politècnica de Catalunya, Colom 11, 08222 Terrassa, Spain E-mail address: victor.manosa@upc.edu