Persistence of periodic traveling waves and Abelian integrals

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.


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 refer 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 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. Examples 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, u t = u xx +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 hyperbolic saddle and a node, see [2,13] and references therein. Therefore, all PDE of the form u t = u xx + u(1 − u) + εg(u, u x , u t , 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, u x , u t , u xx , u xt , u tt , ε). As a second example of the first situation mentioned above, for some PDE of the form u t = u xx + h(u)u x + 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 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.

Definitions and main results
Consider m-th order partial differential equations of the form 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 boundary 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 satisfying these conditions. Here the prime denotes derivative with respect to s and P c : is an open subset of R m+1 . We will distinguish two cases according to whether (1) is a second order equation (m = 2) or a higher order equation (m > 2).
Second order equations. Our main result applies to a certain class of perturbed PDE that satisfy three conditions (i)-(iii) that we detail below. Succinctly, it requires the existence of a certain wave speed c ∈ R such that: (i) the associated ODE has the form U ′′ = f c (U, U ′ ) + εg c (U, U ′ , 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 ℓ 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: open sets, such that, for ε small enough, Moreover, if U c is the limit of the sets V c (ε) when ε → 0, the only solution is a closed oval surrounding (x c , 0) and the function M c : (0, h c ) → R, defined as the line integral has ℓ ≥ 1 simple zeroes in (0, h c ).
Theorem A. Assume that the second order PDE where P c is defined in (2). By using (i) of Property A we can write the above expression as U ′′ (s) = f c (U(s), U ′ (s)) + εg c (U(s), U ′ (s), ε), for some suitable f c and g c . In other words, (x, y) = (U(s), U ′ (s)) is a solution of the planar ODE By item (ii) of Property A we can parameterize U by a new time, say τ, with dτ /ds = s c (x, y), and then x = U(τ ) satisfies the equivalent planar ODE When ε = 0 the above system is Hamiltonian, and by (i) and (iii) of Property A the continuum of curves γ c (h) for 0 < h < h are periodic orbits of system (5) with is the parameterization of γ c (h), give rise to the continuum of periodic traveling wave solutions of (3). When ε = 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, its associated Melnikov-Poincaré-Pontryagin function is Then, it is known that each simple zero h * ∈ (h 0 , h 1 ) of M gives rise to a limit cycle of (6) that tends to γ(h * ) when ε → 0. For system (5), 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) ∈ R 2 , 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 Q c : open sets, such that, for ε small enough, Proof of Theorem B. From condition (o) of Property B, if we restrict our attention to the solutions of (2) contained in 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.

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.
for certain x * ∈ I, where a, b, c are real constants with abd = 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 systemẋ = H y (x, y),ẏ = −H x (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, h) for some h ∈ R. (b) For h ∈ (0, h) and p, n ∈ N, define the Abelian integral 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 = (H y , −H x ) because det(DX(0, 0)) = 2A(0)B ′′ (0) = 4a 2 /b 2 > 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 J p it is convenient to introduce the new variable w as h = w 2 . Then, by the Weierstrass preparation theorem, see for instance [1,3], in a neighbourhood of (0, 0) the only solutions of equation where x ± (w) are analytic functions at zero. Moreover, in this neighbourhood, where U(0, 0) = 1/b 2 is also analytic at (0, 0). Notice that the points of the oval γ(h) satisfy y = ± (w 2 − B(x))/A(x). When p is even the integral (8) vanishes because of symmetry with respect to y = 0. Hence dx, when p is odd.
By using (9) we get that 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 ). In particular, .

Now we claim that
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 (10) K(p, n) = 2p − 1 2n + 1 K(p − 1, n + 1). Now the claim follows by using induction. First we prove that for any p ∈ N 0 , K(p, 0) satisfies the claim. Indeed, if B is the Euler's Beta function, and since B(x, y) = Γ(x)Γ(y)/Γ(x + y), we have Since p is an integer number Γ(2p + 1) = (2p)!. On the other hand, it is well known that Hence, as we wanted to prove. Now we assume that for n > 0 and for all p ∈ N 0 , K(p, n) satisfies the claim. By using the relation (10), we get that, 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. x q y p dx.
The following holds. (a) When q is even and p is odd then J q,p (h) > 0 for all h ∈ L. (b) When q is even and p is even and H(x, y) = H(x, −y) then J q,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 J q,p (h) ≡ 0 on L.

Proof. Notice that by Green's theorem
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 x q y p−1 .
The next proposition will be one of the key results to prove Theorem C, which is stated below. . . , ℓ, be ℓ + 1 linearly independent analytic functions. Assume also that one of them, say F k , 0 ≤ k ≤ ℓ, has constant sign on L. Then, there exist real constants d j , j = 0, 1, . . . , ℓ, such that the linear combination ℓ j=0 d j F j has at least ℓ 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 x 2q y 2p−1 .
Theorem C. Let H(x, y) = A(x)y 2 + B(x), with A and B functions satisfying the hypotheses of Proposition 3.1, and denote by γ(h), h ∈ (0, h), the periodic orbits surrounding the origin of the corresponding Hamiltonian system. For d 0 , d 1 , . . . , d n ∈ R and q j , p j ∈ N, j = 0, 1, . . . , ℓ consider the family of Abelian integrals If all values m j = q j + p j , j = 0, 1, . . . , ℓ are different, there exit values of d j , j = 0, 1, . . . , ℓ, such that the corresponding function J(h) has at least ℓ simple zeroes in (0, h).

Proof. Notice that
x 2q j y 2p j −1 dx.
By Proposition 3.1, for each j = 0, 1, . . . , ℓ, J j (h) = k j h m j + o h m j and, by hypothesis, all these m j are different. This clearly implies that all these ℓ + 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.

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) u + au xx + bu xt + du tt + εg(u x , u t , ε) = 0, with g a C 1 function and take c such that a − bc + dc 2 > 0. Then equation (2) can be written as

A toy example. Consider the PDE
We define C 2 = a − bc + dc 2 and g c (U ′ , ε) = −g(U ′ , −cU ′ , ε)/C 2 . Then it is easy to see that this PDE satisfies Property A with H c (x, y) = x 2 /(2C 2 ) + y 2 /2, s c (x, y) ≡ 1 and (0, h) = (0, ∞). That is, Assume for instance that g c (y, 0) = N j=0 g j y j is a polynomial of degree N, and g j ∈ R. Then, When j is even, by symmetry, the above integrals vanish. Hence where [ · ] denotes the integer part and I 2n = 2π 0 sin 2n θ dθ > 0. Removing the factor h, and taking suitable g 2i+1 , the polynomial M c (h)/h can be any arbitrary polynomial of degree [(N − 1)/2] in h. Hence, by applying Theorem C, for any ℓ ≤ [(N − 1)/2], there exist coefficients g j such that the function M c (h) has ℓ simple zeros and, therefore, by applying Theorem A, the PDE (11) has at least ℓ 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 + au xx + bu xt + du tt + ε(uu xx + g(u, u x , u t , ε)) = 0, and similar results hold.

Reduced Ostrovsky equation.
We consider perturbations of the reduced Ostrovsky equation, introduced by L. Ostrovsky in 1978, which is a modification of the Korteweg-de Vries equation that models gravity waves propagating in a rotating background under the influence of the Coriolis force when the high-frequency dispersion is neglected. More concretely, we take (12) (u t + uu x ) x − u + εg(u, u x , u t , ε) = 0, which satisfies Property A with c > 0, because its associated ODE is Then, taking g c (U, U ′ , ε) = −g(U, U ′ , −cU ′ , ε)/(U − c); V c = {x < c}; x c = 0; and s c (x, y) = (x − c) −2 , the system that has to be studied to find TWS is

Consider also the Melnikov-Poincaré-Pontryagin function
As in the toy example, it is not difficult to find a perturbation term g such that the the function M c (h) has several simple zeroes in (0, c 3 /3) which, by Theorem C, give rise to periodic TWS of the PDE (12). 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 u tt − u xx + λu p + εg(u, u x , u t , 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 with C = λ/(c 2 − 1), g c (x, y, ε) = −g(x, y, −cy, ε)/(c 2 − 1). The associated Melnikov-Poincaré-Pontryagin function is 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, however, that when p ≥ 3, instead of using trigonometric functions to parametrize the invariant closed curves, one can use the generalized polar coordinates introduced 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 guarantees that the zeros of the function M c (h) correspond with periodic TWS of the Klein-Gordon equation. Again, it satisfies Property A for c > 1, and its associated planar system is with C = 1/(c 2 − 1) > 0 and g c (x, y, ε) = g(x, y, −cy, ε)/(1 − c 2 ). The Melnikov-Poincaré-Pontryagin function is The above type integrals are studied for instance in [11]. There, several condition on g for obtaining many simple zeroes of M c , and therefore periodic TWS of the considered PDE, are obtained.

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.
where H ∈ C 2 , 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 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
, 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).
So H fulfills the hypotheses of Proposition 3.1.
Observe that condition (13) is equivalent to the fact that H xx (x * , 0) > 0 and det(H H (x * , 0)) > 0 (where H is the hessian matrix), which implies that H has a non-degenerate local minimum at (x * , 0).

Perturbed generalized Korteweg-de Vries equation.
We consider a perturbation 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) u t + au x + buu x + duu t + pu xxx + qu xxt + ru xtt + su ttt + ε∇g(u, u x , u t , ε) · (u x , u xx , u xt , 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 where C = p − qc + rc 2 − sc 3 . Notice that then, for any function U satisfying previous equation, it holds that there exists k ∈ R, such that Thus we have to study the equivalent planar system and g c (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 α c,k + β c x + γ c x 2 = 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 where a ∈ R and n ∈ N. To find TWS for it we have to study the third order ODE Thus, we need to find solutions of the second order ODE −cU + aU n + n(n − 1)U n−2 U ′ + nU n−1 U ′′ + εg(U, U ′ , −cU ′ , ε) = k with k ∈ R. It writes as the planar system      x ′ = y, y ′ = k + cx − ax n − n(n − 1)x n−2 y 2 nx n−1 + ε g c (x, y, ε) nx n−1 , where g c (x, y, ε) = −g(x, y, −cy, ε). With the new time τ, where dτ /ds = s c,k (x, y) and s c,k (x, y) = x 2(1−n) /n, we get x = U(τ ) satisfies the equivalent planar ODE By Lemma 4.1 (see the comment below its statement), if there exists a singular point (x * , 0) such that 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 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, see also [10]. Similarly, the Degasperis-Procesi equation which was derived initially only for its integrability properties, has a similar role in hydrodynamics.
In fact, perturbations of the above equations can be written under the common expression u t + A ′ (u)u x + bu x u xx + duu xxx + pu xxx + qu xxt + ru xtt + su ttt + ε∇g(u, u x , u t , ε) · (u x , u xx , u xt , 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 where A c (U) = A(U) − cU, with A c (0) = 0, and C = p − qc + rc 2 − sc 3 . Hence, for any function U satisfying the previous equation, there exists k ∈ R, such that g c (x, y, 0) (C + dx)s c (x) dx.
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 that this is trivially the case if g c (x, y, 0) = (C + dx)s c (x) ℓ i=0 d 2i+1 y 2i+1 .
Similarly, the modified Boussinesq equation is u tt + uu xx − u xx + (u x ) 2 − u xxtt = 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) au xx + bu xt + du tt + 2e(uu xx + (u x ) 2 ) + pu xxxx + qu xxxt + ru xxtt + su xttt + f u tttt + 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 a function g c such that G = g c (U, U ′ , ε) ′′ . Hence the ODE associated to (16) is CU ′′ + e(U 2 ) ′′ + DU ′′′′ + ε g c (U, U ′ , ε) ′′ = CU + eU 2 + DU ′′ + εg c (U, U ′ , ε) ′′ = 0, where C = a − bc + d 2 c, D = p − qc + rc 2 − sc 3 + f c 4 , and we have used that (u 2 ) xx = 2uu x + 2(u x ) 2 . We are interested in solutions of the above fourth order ODE (17) CU + eU 2 + DU ′′ + εg c (U, U ′ , ε) = k, for some k ∈ R. When D = 0 we are again under the situation covered by Theorem B. Notice that other solutions would satisfy CU +eU 2 +DU ′′ +εg c (U, U ′ , ε) = k 1 s + k 2 , for some k 1 = 0, k 2 ∈ 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.