PERIODIC TRAVELING WAVES IN A GENERALIZED BBM EQUATION WITH WEAK BACKWARD DIFFUSION AND DISSIPATION TERMS

In this paper, we consider a generalized BBM equation with weak backward diffusion, dissipation and Marangoni effects, and study the existence of periodic and solitary waves. Main attention is focused on periodic and solitary waves on a manifold via studying the number of zeros of some linear combination of Abelian integrals. The uniqueness of the periodic waves is established when the equation contains one coefficient in backward diffusion and dissipation terms, by showing that the Abelian integrals form a Chebyshev set. The monotonicity of the wave speed is proved, and moreover the upper and lower bounds of the limiting wave speeds are obtained. Especially, when the equation involves Marangoni effect due to imposed weak thermal gradients, it is shown that at most two periodic waves can exist. The exact conditions are obtained for the existence of one and two periodic waves as well as for the co-existence of one solitary and one periodic waves. The analysis is mainly based on Chebyshev criteria and asymptotic expansions of Abelian integrals near the solitary and singularity.

1. Introduction.Traveling waves in nonlinear wave equations can model many nonlinear complex phenomena in physics, chemistry, biology, mechanics, optics, etc.The wave profiles of long waves in shallow water with different conditions can be modeled by the famous Korteweg-de Vries (KdV) [26], Benjamin-Bona-Mahony [4], the Green-Naghdi [15] and Camassa-Holm [6] equations.In solving real world problems, certain relatively weak influences due to the existence of uncertainty or perturbation are unavoidable, for example in describing the shallow water waves in nonlinear dissipative media [8] and dispersive media [23].In other words, one should add certain type of small terms in modelling the problems.Topper and Kawahara [41] studied the wave motions on a liquid layer over an inclined plane and established the following Partial Differential Equation (PDE), for which the wave motion is assumed depending only on the gradient direction.
When the inclined plane is relatively long and the surface tension is relatively weak, the u xx and u xxxx terms are relatively small, and the following equation is more appropriate for describing the real situation, where 0 < ε 1 represents small perturbations to the system.When ε = 0, the backward diffusion (u xx ) and dissipation (u xxxx ) vanish and (2) becomes the classical KdV equation [26], and so (2) is usually called a perturbed KdV equation.The KdV equation has played an important role in describing various physical problems, and many researchers have studied this equation and particularly paid attention to solitary and periodic waves.In 1993, Derks and Gils [9] discussed the uniqueness of traveling waves of equation (2).A year later, Ogawa [36] studied the existence of solitary and periodic waves of equation (2).
When the Marangoni effect is considered on the surface of a thin layer, additional nonlinearity in the form of (uu x ) x appears, see [12,20].For this model, Velarde et.al.[43] showed the consistent way of incorporating the Marangoni effect (heating the liquid layer from the air side) into the one-way long-wave assumption and derived the following equation: which contains the nonlinear term (uu x ) x due to the Marangoni effect, describing the opposite to the Bénard convection [35,44].For the sake of completeness here, we notice that different cases by setting some parameters α i = 0 in Eq. ( 3) have been considered in many other works, for example [10,24,27,28].In particular, Mansour [31] studied the existence of solitary in Eq. ( 3) with all small non-vanishing parameters α 2 , α 4 and α 5 , and in addition established the existence of solitary for the following equation [32].
It has been noted that solitary and traveling waves with periodic spatial profiles are very sensitive to weak external influence.For example, stationary periodic patterns in thermal convection may not be observed in a weakly windy circumstance [5].However, the weak Marangoni effect may destabilize the waves [21], and different perturbations may generate different dynamics of systems, leading to, for example, broking the periodic traveling waves, changing its stability and yielding quasi-periodic motions on invariant tori, etc.One efficient way to deal with such problems is to apply bifurcation techniques from the view point of dynamical systems by taking the weak external effects as perturbations, and many good results have been obtained for certain nonlinear wave problems, see [48,17,22,13].
In general, perturbations to a dynamical system may be classified into three types: periodic or quasi-periodic forcing, singular perturbation and regular perturbation.When a PDE is perturbed by quasi-periodic forcing terms, one method developed to investigate dynamics (quasi-periodic motion on some invariant tori) of the system is based on an infinite dimensional KAM theory.This theory is an extension of the well-known classical KAM theory, which was established by Kolmogorov [25], Anorld [2] and Moser [34].It asserts that the majority of tori is persistent under perturbations if the Kolmogorov non-degenerate condition is satisfied.When a perturbed system can be reduced to a singularly perturbed system, the first question is about the existence of traveling wave solutions of the system.There are lots of publications on this topic, such as singularly perturbed KdV equations [36,3,16,30,40], the perturbed dispersive-dissipative equations and reaction-diffusion systems [1,49,42].One classical method to deal with singular perturbations is to apply Fenichel's theory (e.g.see [11]), which assures the existence of an invariant manifold and then the problem is reduced to a regular perturbed system on this manifold, see [36,3,16,30,40].In these cases, the perturbation always has only one or two terms with lower degrees on the invariant manifold, see above mentioned references and also the works of Derks and Gils [9] and Ogawa [36,38,37].
However, very few problems can be directly reduced to regularly perturbed systems.Thus, perturbations are usually not restricted on manifolds.Moreover, there exist fewer mathematical tools which can be used to study the dynamics of perturbed systems, and yet, the analysis and computation based on these approaches are difficult to be used for proving the existence of periodic traveling waves.Thus, when Zhou et al studied the Burgers-Huxley equation [52] and Burgers-Fisher equation [53], they assumed that one coefficient in the equation and the wave speed are small so that these two small terms can be treated as two perturbations, which greatly simplifies the analysis and the proof on the existence of periodic waves [52,53].In general, if three or more perturbation terms are involved, the analysis becomes much more difficult.
After the works of Derks and Gils [9] and Ogawa [36], in 2014 Yan et al. [47] investigated the perturbed generalized KdV equation, When = 0 and n = 2, the above equation is reduced to the classical KdV equation, Yan et al. [47] proved that there exists one periodic wave by choosing some wave speed c for sufficiently small > 0. However, the uniqueness of the periodic wave is still open.
Another well-known model describing the propagation of surface water waves in a uniform channel is the Benjamin-Bona-Mahony (BBM) equation, This model describes surface waves of long wavelength in liquids, acoustic-gravity waves in compressible fluids, hydromagnetic waves in cold plasma, acoustic waves in harmonic crystals, see [33].Due to its wide applications and rich dynamics, researchers have developed many different forms of BBM equations which are usually called generalized BBM equations, see [46,39,51] and the reference therein.Wazwaz studied the following generalized BBM equation [46], and found its compaction of dispersive structures.More recently, Chen et al. [7] investigated a perturbed generalized BBM equation, (u and established the existence of solitary waves and uniqueness of periodic waves.Both of the works [47] and [7] studied the perturbation problems restricted on manifolds, by using geometric singular perturbation theory.In [7], the authors applied Picard-Focus equations to determine the existence of periodic waves, and developed a good approach to prove that the dominating factor of the Melnikov function is monotonic, see Lemma 4.10 in [7].Using the same approach, they also proved that the perturbed generalized defocusing mKdV equation, has a unique periodic wave.However, this approach failed to deal with the unperturbed equation having a nilpotent saddle or more degenerate cusp, corresponding to m > 2 in (4).This is because, taking m = 3 for example, one needs to consider more terms in Lemma 4.10 in [7] in order to find some combination of the terms in order to prove the monotonicity of the dominating part of the Melnikov function.However, in general this is very difficult in higher degenerate cases, which is similar to dealing with the cases when more than two perturbation terms are involved in the equations.
In this paper, we study the BBM equation ( 4) for m = 3, n = 4 and l = 1 with two different kinds of weak dissipative effects P 1 and P 2 , described by where in which α 0 , α 1 and α 2 are bounded parameters.P 1 describes the weak second and fourth derivative diffusions without Marangoni effect.P 2 describes a generalized Marangoni effect.The unperturbed problem has a more degenerate singularity which is a nilpotent saddle.Especially, for the case with weak Marangoni effect P 2 , the problem is not restricted on a manifold but is reduced to a regular problem with more parameters.It can be seen from our reduction of the problem with P 1 that the weak Marangoni effect P 2 is equivalent to the compound dissipative-Marangoni effect γ 0 u xx + γ 1 u xxxx + γ 2 (uu x ) x .The main mathematical tools we will use in this paper are based on the relatively new theory of weak Hilbert's 16th problem and bifurcation theory.The rest of this paper is organized as follows.In section 2, we give a reduction analysis and state our main results.In section 3, we present some perturbation theories and derive a special form of Abelian integral, also called Melnikov function, for periodic and solitary waves.It will be shown that our method without using Picard-Focus Equation is more effective compared to that used in the existing works.In section 4, we study the problem with perturbation P 1 by applying the Chebyshev criteria [14].In section 5, we investigate the problem with perturbation P 2 and obtain the conditions on the existence of periodic waves.In particular, we derive the exact conditions on the existences of one and two periodic waves.Further, we establish a criterion on the co-existence of one solitary wave and one unique periodic wave.Conclusion is drawn in section 6.
2. Main results.In this section, we present our main results for system (6).First, we consider system (6) with perturbation P 1 , that is, Taking ξ = x − ct into (8) yields Then, integrating this equation and omitting the integral constant, we obtain Further introducing the transformations ξ = τ c 3 2 and u = cµ into (9) yields Similarly, system (6) with perturbation P 2 can be transformed to where Because α i s are independent, we will use a i s in our analysis for convenience.
Correspondingly, the unperturbed system of ( 6) (with = 0) is given by −µ 3 (τ )+ µ 4 (τ ) + µ (τ ) = 0, which is equivalent to the system, which has a hyper-elliptic Hamiltonian function, given by satisfying 4 ), depicted in Figure 1, shows a family of closed orbits surrounded by a homoclinic loop Γ 0 , with a nilpotent saddle of order 1 at the origin.In order to state our main results clearly and systematically, we use the following notations: Γ h denoting the closed curve defined by H(µ, ν) = h; µ(τ, h) representing the closed orbit of system (12) corresponding to Γ h ; µ(τ, , h, c( , h)) being the periodic wave of system (10) near Γ h under the condition c = c( , h); µ(τ, , h) denoting the traveling wave of system (11) near Γ h .Our main results are given in the following Theorem 2.1 and Theorem 2.2.
(i) For any sufficiently small > 0 and any h ∈ (− 1 20 , 0), there exists a smooth function c( , h) in and h such that system (6) has one unique isolated periodic wave in a sufficiently small neighborhood of Γ h , given by u = cµ(τ, , h, c( , h)), satisfying where c(h) is a monotonically increasing function in h satisfying 1 < c(h) < 39 25 3 .(ii) For any sufficiently small > 0, there exists a critical wave speed c = ( 3925 ) 1 3 + O( ) such that system (6) has one solitary wave in a sufficiently small neighborhood of Γ 0 .Theorem 2.2.For any sufficiently small > 0, the perturbed BBM equation ( 6) with perturbation P 2 has at most two isolated periodic waves.More precisely, we have the following results.
(i) The Abelian integral M(h) (given in (3.11)) has one unique simple zero for any , where κ(h, λ) is defined in (45).Therefore, for any sufficiently small > 0, system (6) has one unique isolated periodic wave u = cµ(τ, , h * ), in sufficiently small neighborhood of any closed curve Γ h * by taking The Abelian integral M(h) has exactly two simple zeros h 1 and h 2 if and only if under which, for any sufficiently small > 0, system (6) has exactly two isolated periodic waves u 1 = cµ(τ, , h 1 ) and u 2 = cµ(τ, , h 2 ), in sufficiently small neighborhoods of the closed curves Γ h1 and Γ h2 by choosing The Abelian integral M(h) has a unique zero at h = 0 if and only if and further, under (14), M(h) has a unique simple zero in (− 1 20 , 0) if and only if a 1 = λ * * ∈ (− 16  11 , − 10  11 ).Therefore, for any sufficiently small > 0, system (6) can have a solitary wave by taking a 0 = − Before further analysis, in the next section we will present some definitions and lemmas in perturbation theory, which are needed for proving Theorems 2.1 and 2.2.
3. Perturbation theory and analysis.Lemma 3.1 (Fenichel Criteria).Consider the system where is an open interval containing zero.Assume that for = 0, system (15) has a compact normally hyperbolic manifold M 0 which is contained in the set f 1 (x, y, 0) = 0.The manifold M 0 is said to be normally hyperbolic if the linearization of (15) at each point in M 0 has exactly dim(M 0 ) eigenvalues on the imaginary axis.Then, for any 0 < r < +∞, there exists a manifold M for sufficiently small such that the following conclusions hold.
(i) M is locally invariant under the flow of (15).
(ii) M is C r in x, y and .
(iii) M = {(x, y)|x = h (y)} for some C r function h , and y in some compact set K. (iv) There exist locally invariant stable and unstable manifolds W s (M ), W u (M ), that lie within O( ) of, and are diffeomorphic to W s (M 0 ) and W u (M 0 ).Definition 3.2.Suppose f 0 (x), f 1 (x), . . ., f n−1 (x) are analytic functions on an real open interval J.
(i) The family of sets {f 0 (x), f 1 (x), . . ., f n−1 (x)} is called a Chebyshev system (T-system for short) provided that any nontrivial linear combination, has at most n − 1 isolated zeros on J. (ii) An ordered set of n functions {f 0 (x), f 1 (x), . . ., f n−1 (x)} is called complete Chebyshev system (CT-system for short) provided any nontrivial linear combination, Moreover it is called extended complete Chebyshev system (ECT-system for short) if the multiplicities of zeros are taken into account.(iii) The continuous Wronskian of {f 0 (x), , where f (x) is the first order derivative of f (x) and f (i) (x) is the ith order derivative of f (x), i ≥ 2. The definitions imply that the function tuple } is an ECT-system on J, so it is a CT-system on J, and thus a T-system on J. However the inverse is not true.
Let res(f 1 , f 2 ) denote the resultant of f 1 (x) and f 2 (x), where f 1 (x) and f 2 (x) are two univariate polynomials of x on rational number field Q.As it is known, res(f 1 (x), f 2 (x)) = 0 if and only if f 1 (x) and f 2 (x) have at least one common root.
Let res(f, g, x) and res(f, g, z) denote respectively the resultants between f (x, z) and g(x, z) with respect to x and z, where f (x, z) and g(x, z) are two polynomials in {x, z} on rational number field.res(f, g, x) is a polynomial in z and res(f, g, z) is a polynomial in x.About the relation between the common roots of two polynomials and their resultants, the following result can be found in many works on polynomial algebra, such as [45].For completeness we give a short proof.
(ii) Let res(f, g, z) have a unique real root on some open interval (α, β), and res(f, g, x) have a unique real root on some open interval (γ, θ).Then, there exists at most one common real root of f (x, z) and g(x, z) on (α, β) × (γ, θ).
Proof.(ii) is obvious if (i) is true.So we only prove (i).A two-variable polynomial can be treated as one univariate polynomial of one variable with the other treated as a parameter.Taking f (x, z) = f z (x) and g(x, z) = g z (x) which are polynomials of x with parameter z.Let x 0 be the common root of f z0 (x) and g z0 (x).Then, res(f z0 (x), g z0 (x)) = res(f, g, z 0 ) = 0, where z 0 is the common root of f x0 (z) and g x0 (z), and therefore, res 2 be an analytic function.Assume there exists a punctured neighborhood P of the origin foliated by ovals Γ h ⊆ {(x, y)|H(x, y) = h, h ∈ (0, h 0 ), h 0 = H(∂P)}.The projection of P on the x-axis is an interval (x l , x r ) with x l < 0 < x r .Under these assumptions it is easy to verify that xA (x) > 0 for all x ∈ (x l , x r )\{0}, and A(x) has a zero of even multiplicity at x = 0 and there exists an analytic involution z(x), defined by A(x) = A(z(x)), for all x ∈ (x l , x r ).Let where f i (x), i = 0, 1, . . ., n−1, are analytic functions on (x l , x r ) and s ∈ N .Further, define .

Lemma 3.5 ([29]
).Under the above assumption, if the following conditions are satisfied: has k zeros on (0, x r ) with multiplicities counted, and (iii) s > n + k − 2, then, any nontrivial linear combination of {I 0 , I 1 , • • • , I n−1 } has at most n + k − 1 zeros on (0, h 0 ) with multiplicities counted.In this case, {I 0 , I 1 , • • • , I n−1 } is called a Chebyshev system with accuracy k on (0, h 0 ), where W [l 0 , l 1 , . . ., l i ] denotes the Wronskian of {l 0 , l 1 , . . ., l i }.Now we consider system (10), which can be rewritten in the form of Introducing the time scaling σ = τ into (17) yields When > 0, system ( 17) is equivalent to (18).System ( 17) is called the slow system, while system ( 18) is called the fast system.The slow system (17) determines its critical manifold, which is a two-dimensional submanifold in R 3 : The Jacobian matrix of the fast system (18) restricted on M 0 is given by   which has three eigenvalues λ 1 = λ 2 = 0, λ 3 = −c −3/2 , with λ 1 and λ 2 being on the imaginary axis.Therefore, M 0 is normally hyperbolic.Consequently, it follows from Lemma 3.1 that for > 0 sufficiently small, there exists a two-dimensional submanifold M in R 3 , which is invariant under the flow of system (17), within the Hausdorff distance of M 0 .Let where η(µ, ν, ) is smooth in µ, ν and , satisfying η(µ, ν, 0) = 0, and expanded as Substituting (19) into the last equation of ( 17) and comparing its coefficients yields The dynamics of ( 17) on M is determined by For any h ∈ (− 1 20 , 0), H(µ, ν) = h defines a periodic orbit Γ h of (12) (or the system (20) with = 0).Let (α(h), 0) denote the intersection point of Γ h and the positive µ-axis, T the period of Γ h .Further, let Γ h, be the positive orbit of (20) starting from the point (α(h), 0) at time τ = 0, and (β(h, ), 0) the first intersection point of Γ h, with the positive µ-axis at time τ = τ * ( ).Let H * (µ, ν) denote the small perturbation of H(µ, ν).Then, the difference between the two points is given by By continuousness theorem, we have lim and thus, where M (h) is called Abelian integral or Melnikov function, given by It has been noted that compared with the application of Picard-Focus equation method (eg.see [7]) which is often used to derive the Abelian integral, here our approach developed above is much simpler.
Similarly, for system (11), we take µ (τ ) = ν and follow the above procedure to obtain the following regular perturbation problem which is not restricted on a manifold, Let (α * (h), 0) be the intersection point of Γ h and the positive µ-axis, T the period of Γ h , Γ * h, the positive orbit of ( 23) starting from the point (α * (h), 0) at time τ = 0, and (β * (h, ), 0) the first intersection point of Γ h, with the positive µ-axis at time τ = τ * ( ).Then, the difference between the two points (α * (h), 0) and (β * (h, ), 0) can be expressed as where the Abelian integral M(h) is given by To investigate the existence of periodic and solitary waves for the two perturbation problems, we need study the zeros of the functions H * (β(h, ), 0) − H * (α(h), 0) and H * (β * (h, ), 0) − H * (α * (h), 0) and their distributions.It follows from ( 21) and ( 24) that it suffices to consider the Abelian integrals M (h) and M(h).
It follows from Lemma 4.1 that the following ratio is well defined, Then, In the remaining of this section we mainly prove the following proposition, which is needed for proving Theorem 2.1.
To prove Proposition 1, we need the following lemmas.
Computing the resultant of Q 1 and q with respect to z, and applying Sturm's Theorem, we can show that Q 1 and q have no common zeros for µ ∈ (0, 1).Therefore, W [L 0 ] does not vanish for µ ∈ (0, 1).
Similarly, computing the resultant of S 12 and q with respect to z, and applying Stuum's Theorem, we can prove that S 12 and q have no common zeros for µ ∈ (0, 1).Therefore, S 12 does not vanish for µ ∈ (0, 1).Solving S 12 ( µ, z)λ − S 11 ( µ, z) = 0 gives for which we have the following result.
Lemma 5.4 implies the following proposition.

Conclusion.
In this paper, we have used bifurcation theory to study the existence of periodic and solitary waves in a BBM equation under weak dissipative influences and Marangoni effect.A special transformation given in (32) is introduced so that the Chebyshev criteria can be applied to overcome the difficulty arising from higher-order degenerate singularities, and then the exact condition on the number of periodic waves is obtained for the case with regular multiple-parameter perturbations.Also, the condition on the co-existence of one solitary and one periodic waves is derived.The methodologies developed in this paper include the reduction of three generating elements to special two ones, asymptotic expansion of Abelian integrals, and asymptotic analysis on the dominating part of the Abelian integrals.
Combination of these methods is not only useful in the study of other types of wave equations, but also has potential to be generalized to consider perturbations on hyper-elliptic Hamiltonian systems.

4 .
With the expansion of J i (h) near h = − 1 20 , given in(31), a direct computation shows that