# Existence and Uniqueness of Periodic and Solitary Waves for a Perturbed Generalized KdV Equation

Authors: Zhengyong Ouyang; Weihua Huang; Minzhi Wei
Journal: Journal of Mathematics
 (2022)

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## Abstract

This paper establishes the existence and uniqueness of periodic and solitary waves for a perturbed generalized KdV equation. We prove that the periodic wave persists when the wave speed is in a certain interval, and the wavelength of the periodic wave is an increasing function. Furthermore, under the condition of the unique periodic wave exists, the interval of wave speedc becomes smaller as the power n increases.


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## Body

## 1. Introduction

The Korteweg-de Vries (KdV) equationut−6uux+uxxx=0 was initially arose from the experiment by Scott Russell in 1834. It is a model that governs the one-dimensional propagation of small amplitude and weakly dispersive waves. In the equation, the nonlinear term uux causes steepening of wave form, and the linear dispersion term uxxx gives rise to the spread of the wave. In 1871, Lord Rayleigh and Boussinesq theoretically investigated the KdV equation. In 1895, Korteweg and De Vries integrated the former equation, see [1]. The KdV equation plays a central role in modeling the shallow water, and many variations have been proposed to simulate the shallow water within a certain physical environment, such as the modified KdV equation, generalized KdV equation, KdV-Burger equation, and 5th-order KdV. On the other hand, some weak terms should be considered in the equation when solving real-world problems due to externally uncertain factors, such as weakly dissipative effects [2]. Hence, it is natural to take the perturbed equation into consideration.In this paper, we study the following perturbed generalized KdV equations:(1)ut+unux+uxxx+εuxx+uxxxx=0,which is generalized from the perturbed KdV equation(2)ut+uux+uxxx+εuxx+uxxxx=0,where ε is sufficiently small. Equation (2) was first proposed to model the shallow water on an inclined thinner and relatively long layer by Derks and van Gils [3] and Ogawa [4]. Equation (2) is also mentioned in the study of the solitary wave in the nonlinear dispersive-dissipative solids [5, 6]. For the perturbed generalized KdV equation (1), the existence of the solitary wave was established in [7, 8], and there are some more related researches on this topic, see [9–13]. However, the existence of the periodic wave was not investigated before. Therefore, the presented paper tries to establish the existence of the solitary wave and periodic wave and especially proves the uniqueness of the periodic wave by applying an identical argument.The rest of the paper is organized as follows. In Section2, we conduct the model reduction and present the criteria for the monotonicity of the ratio of two Abelian integrals, which is the main tool for tracking the solitary and periodic waves on a critical manifold. Then, we establish the existence and uniqueness of periodic wave for the power n=1,2,3, respectively, in Section 3. Next, we discuss the existence problem for an unfixed integer n in Section 4. Finally, a conclusion is drawn in Section 5.


## 2. System Reduction and Poincaré Bifurcation

In order to check whether the traveling waves of the perturbed system equation (1) still exhibit, we make the normal traveling wave ansatz(3)ux,t=φξ,ξ=x−ct,for ξ∈−∞,+∞, satisfying(4)limξ⟶∞φξ=m,limξ⟶−∞φξ=n,where c is the propagation speed of a wave. By substituting equation (3) into (1), we obtain(5)−cφ′ξ+φnξφ′ξ+φ‴ξ+εφ″ξ+φ″″ξ=0.Integrating the above equation with respect toξ and taking the integral constant to be zero, we have(6)−cφξ+1n+1φn+1ξ+φ″ξ+εφ′ξ+φ‴ξ=0.By introducing the scale transformationsφ=cϕn and ξ=τc into equation (6), we obtain the following third-order ordinary differential equation:(7)−ϕτ+1n+1ϕn+1τ+ϕ″τ+ε1cϕ′τ+cϕ‴τ=0.By bifurcation theory of planar dynamical systems, we know that, ifm=n in equation (4), φξ is a solitary wave, and it corresponds to a homoclinic orbit of equation (7). If m≠n, φξ is a kink or antikink solution corresponding to a heteroclinic orbit (or so-called connecting orbit) of equation (7). A periodic orbit of equation (7) corresponds to a periodic traveling wave solution of equation (1). Therefore, we only investigate the corresponding orbits in equation (7) to study the solitary and periodic waves in equation (1). However, it is difficult to investigate equation (7) directly, so we shall apply the geometric singular perturbation theory [14, 15] to reduce the system equation (7) on a slow manifold, on which it is a regular perturbed problem. For completeness, we introduce the main geometric singular perturbation theorem as the following Lemma 1.Lemma 1 (see [14]).
Consider the following fast-slow system:(8)x′=fx,y,ε,y′=εgx,y,ε,where x∈Rn, y∈Rl and ε is a real parameter, f,g are C∞ on the set V×I, where V⊂Rn+l and I is an open interval containing 0. Assume that for ε=0, the system has a compact normally hyperbolic manifold M0 which is contained in the set fx,y,0=0. The manifold M0 is said to be normally hyperbolic if the linearization of equation (8) at each point in M0 has exactly l eigenvalues on the imaginary axis Rλ=0. Then, for any 0<r<+∞, if ε>0 sufficiently small, there exists a manifold Mε such that the following conclusions hold.(1)
Mε is locally invariant under the flow of equation (8)(2)
Mε is Cr in x,y, and ε(3)
Mε=x,y|x=hεy for some Cr function hϵ and y in some compact set K(4)
There exist locally invariant stable and unstable manifoldsWsMε, WuMε, that lie within Oε of, and are diffeomorphic to WsM0 and WuM0We introducedu/dτ=v and dv/dτ=w, and then, the ODE equation (7) becomes(9)dϕdτ=v,dvdτ=w,εcdwdτ=ϕτ−1n+1ϕn+1τ−wτ−ε1cvτ.By the basic theory of geometric singular perturbation, there exists a slow manifold(10)Sε=ϕ,v,w|w=ϕτ−1n+1ϕn+1τ+εcϕnv−1+1cv+Oε2.By Lemma1, the main interesting portraits of system equation (9) projected on Sε have the following form:(11)dϕdτ=y,dydτ=ϕτ−1n+1ϕn+1τ+εcϕny−1+1cy+Oε2.System equation (11) is a regular perturbation problem. The unperturbed system equation (11)ε=0 is integrable and has the Hamiltonian function (first integral).(12)Hϕ,y=y22−12ϕ2+1n+1n+2ϕn+2.Ifn is even, it is easy to see that the unperturbed system equation (11)ε=0 has three singular points 0,0, n+1n,0, and −n+1n,0, where 0,0 is a saddle point, and n+1n,0 and −n+1n,0 are two centers. If n is odd, it is also easy to see that the unperturbed system equation (11)ε=0 has two singular points 0,0, n+1n,0, where 0,0 is a saddle point, and n+1n,0 is a center. By the theory of planar dynamical systems (e.g., see [16–18]), for both n is even and odd, the phase portraits of system equation (11)ε=0 can be classified into three cases with the orbits defined by the following functions:(13)nis even:Hϕ,y=hah∈−nn+12/n2n+2,0,two periodic orbits,bh=0,two homoclinic orbits,ch>0,one big periodic orbit,and(14)nis odd:Hϕ,y=hah∈−nn+12/n2n+2,0,one periodic orbit,bh=0,one homoclinic orbit,ch>0,open orbit.The phase portrait of equation (11)ε=0 is shown in Figure 1 for odd n and Figure 2 for even n.Figure 1
The phase portrait of equation (11)ε=0 for odd n.Figure 2
The phase portrait of equation (11)ε=0 for even n.Under perturbation, lots of periodic orbits of equation (11)ε=0 is broken, and only a few periodic orbits persist as limit cycles of equation (11). Now, we investigate whether the periodic orbit is persistent. Suppose one orbit of system equation (11) starts from a point Mh=ah,0 on the ϕ-axis, where the positive ϕ-axis can be parameterized by h and 0<ah<n+11/n. Let Nh=bh,0 be the first intersection point after 2π of the orbit starting from Mh with the ϕ-axis. Then, the displacement function of system equation (11) can be obtained as [18](15)dh,ε=Nh−Mh=∫MN^dH=εIh,c+Oε,and(16)Ih,c=∮Γhcϕn−1−1cydϕ,(17)=∮Γh−c1+1cydϕ+∮Γhcϕnydϕ,(18)=−c1+1cI1h+cI2h,(19)=−cI1h1+1c−I2hI1h,with I1h=∮ydϕ and I2=∮ϕnydϕ. By Poincaré bifurcation theory [18], the number of zeros of Ih,c corresponds to the number of limit cycles of system equation (11), which is the number of isolated periodic traveling waves. If the ratio I2h/I1h is monotonic, then Ih,c has at most one zero. In order to prove the monotonicity of I2h/I1h, we will apply a result obtained by Li and Zhang [19] in the study of the weak Hilbert’s 16th problem. In [19], a criterion is given for determining the monotonicity of the ratio(20)I2hI1h≔∮Γhf2ϕgydϕ∮Γhf1ϕgydϕ,where Γh is a family of ovals described by(21)Hϕ,y=Φϕ+Ψy,which surrounds a point a,b, and Φ and Ψ are polynomials. Then, Γh represents a closed orbit of the Hamiltonian system(22)ϕ˙=Ψ′y,y˙=−Φ′ϕ.Here, we describe the theorem in [19] as follows. Consider the equation (72), where Φ∈C2α,A, Ψ∈C2β,B. Let Γh be the compact component of the level curve ϕ,y|Hx,y=h,h1<h<h2. Suppose that there exist a∈α,A and b∈β,B, such that the following hypothesis is satisfied:(H1) (i)Φ′ϕϕ−a>0or<0 for ϕ∈α,A∖a, (ii) Ψ′yy−b>0or<0 for y∈β,B∖b(H2)f1ϕf1μ>0 for ϕ∈α,a(H3)g′yg′y˜>0 for y∈β,bIn the hypothesis (H2) and (H3), the symbolsμ and y˜ match with ϕ and y in one-to-one mappings ϕ⟶μ∈a,A and y⟶y˜∈b,B, and they satisfy Φϕ=Φμ and Ψy=Ψy˜, respectively. For each h∈h1,h2, two functions ζϕ and ηy displayed in the Lemma are defined as the following forms:(23)ζϕ=f2ϕΦ′μ−f2μΦ′ϕf1ϕΦ′μ−f1μΦ′ϕ,and(24)ηy=gy˜−gyΨ′y˜Ψ′yg′y˜Ψ′y−g′yΨ′y˜.Lemma 2 (see [19]).
Assume thatHϕ,y has the form equation (72), and the hypotheses (H1), (H2), and (H3) are satisfied. For ϕ∈α,a and y∈β,b, we have(i)
Ifζ′ϕη′y>0, then I2h/I1h is monotonically increasing on h1,h2(ii)
Ifζ′ϕη′y<0, then I2h/I1h is monotonically decreasing on h1,h2


## 3. Existence and Uniqueness of the Periodic Wave

### 3.1. The Casen=1 for Equation (1)

First of all, we consider the casen=1, and we have the following result.Lemma 3.
For system equation (11), if n=1, then the ratio I2h/I1h in equation (19) is strictly decreasing on h∈−2/3,0.Proof.
Whenn=1, the unperturbed system equation (11)ε=0 has two singular points 0,0 and 2,0, where 0,0 is a saddle point and 2,0 is a center. According to equation (12), the Hamiltonian h at these two points are h1=H2,0=−2/3 and h2=H0,0=0, respectively. For each h∈h1,h2, the expression equation (12) corresponds to a periodic orbit. The Abelian integral of equation (11)n=1 is given by equation (16)n=1. Now, we use Lemma 2 to prove that I2/I1 is monotonic on h∈−2/3,0. Since 2,0 is a center, we choose a,b=2,0. In this situation, it is easy to know α,A=0,3, β,B=−23/3,23/3, f1ϕ=1, f2ϕ=ϕ, and gy=y in Lemma 2. Moreover, notice that Φϕ=−1/2ϕ2+1/6ϕ3 and Ψy=1/2y2. If ϕ∈0,3 and ϕ≠2, we have(25)Φ′ϕϕ−2=12ϕϕ−22>0.
Ify∈−23/3,23/3 and y≠0, we have(26)Ψ′yy−0=y2>0.
Therefore, the condition (H1) in Lemma2 is verified.
Furthermore, forϕ∈0,2,(27)f1ϕf1μ=1>0,and for y∈−23/3,0,(28)g′yg′y˜=1>0.
So the conditions (H2) and (H3) are verified.
Now, we decide the sign ofζ′ϕη′y. According to equations (23) and (24), by direct calculation, we get(29)ζϕ=f2ϕΦ′μ−f2μΦ′ϕf1ϕΦ′μ−f1μΦ′ϕ=ϕ−μ+1/2μ2−μ−ϕ+1/2ϕ2−μ+1/2μ2−−ϕ+1/2ϕ2=ϕμμ+ϕ−2,and(30)ηy=gy˜−gyΨ′y˜Ψ′yg′y˜Ψ′y−g′yΨ′y˜=y˜−yy˜yy−y˜=−y˜y.
SinceΦϕ=Φμ and Ψy=Ψy˜, which implies that(31)3μ2−μ3=3ϕ2−ϕ3andy˜=−y.
So it is easy to get that(32)ηy=y2,and(33)η′y=2y<0,since y∈−23/3,0.
On the following, we considerζ′ϕ. From the first equation of equation (31), μ=μϕ can be regarded as a function of ϕ, and taking the derivative of this equation on both sides with respect to ϕ yields(34)μ′ϕ=2ϕ−ϕ22μ−μ2.
Again, by direct calculation, we obtain the derivative ofζϕ with respect to ϕ as(35)ζ′ϕ=μ2−2μ+ϕ2−2ϕμ′μ+ϕ−22.
By substituting equation (34) into (35), we get(36)ζ′ϕ=μ2−2μ2+ϕ2−2ϕ2μ+ϕ−22μ2−2μ>0,since ϕ∈0,2 and μ∈2,3. From equations (33) and (36), it is obvious that(37)ζ′ϕη′y<0.
Therefore,I2h/I1h is monotonically decreasing on −2/3,0, and this completes the proof of Lemma 3.Theorem 1.
Ifn=1 and ϵ is sufficiently small in equation (1), then, for each wave speed c∈1,7/5, there exists only a definite Hamiltonian hc∈−2/3,0, which is a strictly increasing function of c, such that equation (1) has a unique periodic wave. Moreover, there exists a unique solitary wave if and only if the wave speed c=7/5.Proof.
SinceI2h/I1h is monotonically decreasing on −2/3,0, we can calculate its range of values. When h=0, I2h/I1h takes its minimum(38)I2I1|min=∮Γ0ϕydϕ∮Γ0ydϕ=127.
Whenh=−2/3, I2h/I1h takes its maximum(39)I2I1|max=∮Γ−2/3ϕydϕ∮Γ−2/3ydϕ,where Γ−2/3 is the center, and the integrals make no sense. In this case, another way in [20] is used to get I2−2/3/I1−2/3. From equation (19), we know that(40)I−23,c=−c1+1cI1−23+cI2−23=∂∂y−c1+1cy+cϕy2,0=−c1+1c+2c.
Comparing the coefficients of the above equation, we haveI1−2/3=1 and I2−2/3=2, that is, I2/I1max=2. Consequently, I2h/I1h∈12/7,2, according to equation (19), only if 12/7<1+1/c<2 can be made Ih,c=0. Namely, for each c∈1,7/5, we can find the corresponding Hamiltonian h∈−2/3,0, such that there exists a unique periodic orbit for the perturbed system equation (1). From equation (19), we know that I0,c=0 if and only if c=7/5; hence, there exists a unique homoclinic orbit.
Ash∈−2/3,0 and c∈1,7/5, due to the monotonicity of I2h/I1h, we can also build a one by one and increasing mapping h↦c such that(41)c=1I2h/I1h−1=I1hI2h−I1h.
Conversely, it is obvious thathc is a monotonically increasing function of c.
Ifn=1 and ε is sufficiently small in equation (1), for wave speed c∈1,7/5 and c=7/5, we give the graphical solutions of periodic and solitary waves, respectively, for a meaningful illustration (see Figures 3 and 4). Similarly, we can draw the numerical graphical solutions when n takes other values.Figure 3
The periodic wave of equation (1) for 1<c<7/5.Figure 4
The solitary wave of equation (1) for c=7/5.


### 3.2. The Casen=2 for Equation (1)

In this subsection, we will discuss the casen=2 for equation (1). We have the following result of monotonicity.Lemma 4.
For system equation (11), if n=2, then the ratio I2h/I1h in equation (19) is strictly decreasing on h∈−3/4,0.Proof.
Whenn=2, the unperturbed system equation (11)ϵ=0 has three singular points 0,0, −3,0, and 3,0, where 0,0 is a saddle point, and both −3,0 and 3,0 are centers. According to equation (12), the Hamiltonian h at these three points are h1=H±3,0=−3/4 and h2=H0,0=0, respectively. For each h∈h1,h2, the expression equation (12) corresponds to two periodic orbits. As the phase portrait of equation (12)n=2 is symmetric, we only consider the case for ϕ>0, and the case for ϕ<0 is paralleled. The Abelian integral of equation (11)n=2 is given by equation (16)n=2. With the similar method, we use Lemma 2 to prove that I2/I1 is monotonic on h∈−3/4,0. Because of 3,0 is a center, we choose a,b=3,0. In this case, α,A=0,6, β,B=−6/2,6/2, f1ϕ=1, f2ϕ=ϕ2 and gy=y in Lemma 2. Moreover, notice that Φϕ=−1/2ϕ2+1/12ϕ4 and Ψy=1/2y2. If ϕ∈0,6 and ϕ≠3, we have(42)Φ′ϕϕ−3=13ϕϕ+3ϕ−32>0.
Ify∈−6/2,6/2 and y≠0, we have(43)Ψ′yy−0=y2>0.
Therefore, the condition (H1) in Lemma2 is verified.
Moreover, forϕ∈0,3,(44)f1ϕf1μ=1>0,and for y∈−6/2,0,(45)g′yg′y˜=1>0.
So the conditions (H2) and (H3) are verified.
Now, we aim to find the sign ofζ′ϕη′y. The computation of η′y=2y<0−6/2<y<0 is the same as in Section 3. Thus, according to equation (23), we mainly focus on the sign of ζ′ϕ by direct calculation, and we obtain(46)ζϕ=f2ϕΦ′μ−f2μΦ′ϕf1ϕΦ′μ−f1μΦ′ϕ=ϕ2−μ+1/3μ3−μ2−ϕ+1/3ϕ3−μ+1/3μ3−−ϕ+1/3ϕ3=3ϕμ+ϕ2μ2μ2+μϕ+ϕ2−3.
SinceΦϕ=Φμ, which implies that(47)μ2−ϕ26−μ2−ϕ2=0,that is.(48)μ2+ϕ2−6=0,as ϕ≠μ. From the above equation (48), μ=μϕ can be regarded as a function of ϕ, and taking the derivative of this equation on both sides with respect to ϕ yields(49)μ′ϕ=−ϕμ.
Then, according to equation (46), we obtain the derivative of ζϕ with respect to ϕ as(50)ζ′ϕ=2ϕμ4+ϕ2μ3−6ϕμ2+3μ3−3ϕ2μ−9μ+2ϕ4μ+ϕ3μ2−6ϕ2μ+3ϕ3−3ϕμ2−9ϕμ′μ2+μ+ϕ2−32.
By substituting equation (49) into (50), we obtain(51)ζ′ϕ=2ϕμ4+ϕ2μ3−6ϕμ2+3μ3−3ϕ2μ−9μμ−2ϕ4μ+ϕ3μ2−6ϕ2μ+3ϕ3−3ϕμ2−9ϕϕμ2+μ+ϕ2−32μ.
Letζ′ϕ=Fϕ,μ/Gϕ,μ, since ϕ∈0,3 and μ∈3,6, it is obvious that Gϕ,μ>0. Notice that ϕ and μ satisfy equation (48), letting Wϕ,μ=ϕ2+μ2−6, with the aid of Maple, we get the resultant F,W with respect to ϕ is(52)F,W=4μ2−36.
It is clear that4μ2−36≠0 as μ∈3,6, it means Fϕ,μ≡0; meanwhile, Fϕ,μ>0. Consequently, we have(53)ζ′ϕη′y<0.
Therefore,I2h/I1h is monotonically decreasing on −3/4,0. This completes the proof of Lemma 4.Theorem 2.
Ifn=2 and ϵ is sufficiently small in equation (1), then, for each wave speed c∈1/2,5/7, there exists only a definite Hamiltonian hc∈−3/4,0, which is a strictly increasing function of c, such that equation (1) has a unique periodic wave for ux,t>0 or ux,t<0. Moreover, there exists a unique solitary wave for ux,t>0 or ux,t<0 if and only if the wave speed c=5/7.Proof.
SinceI2h/I1h is monotonically decreasing on −3/4,0, we can calculate its range of values. When h=0, I2h/I1h, taking its minimum(54)I2I1|min=∮Γ0ϕ2ydϕ∮Γ0ydϕ=125.
Whenh=−3/4, I2h/I1h, taking its maximum(55)I2I1|max=∮Γ−3/4ϕ2ydϕ∮Γ−3/4ydϕ,where Γ−3/4 is the center, and the integrals make no sense. In this case, the same method is used to get I20/I10. From equation (19), we know that(56)I−34,c=−c1+1cI1−34+cI2−34=∂∂y−c1+1cy+cϕ2y3,0=−c1+1c+3c.
Comparing the coefficients of the above equation, we haveI1−3/4=1 and I2−3/4=3; that is, I2/I1max=3. Consequently, only if 12/5<1+1/c<3, that is, 1/2<c<5/7, there exists one unique periodic orbit on each side of the origin for the perturbed system. It is clear that there exists one unique homoclinic orbit on each side of the origin if and only if c=5/7.
Ash∈−3/4,0 and c∈1/2,5/7, due to the monotonicity of I2h/I1h, we can build the increasing and one-to-one mapping c↦h such that the Hamiltonian hc is a strictly increasing function of c.


### 3.3. The Casen=3 for Equation (1)

In this section, we discuss the casen=3 for equation (1). Similarly, we have the result of monotonicity as follows.Lemma 5.
For system equation (11), if n=3, then the ratio I2h/I1h in equation (19) is strictly decreasing on h∈−161/3/4,0.Proof.
Whenn=3, the unperturbed system equation (11)ε=0 has two singular points 0,0 and 41/3,0. 0,0 is a saddle point, and 41/3,0 is a center. According to equation (12), the Hamiltonian h at these two points are h1=H41/3,0=−161/3/4 and h2=H0,0=0, respectively. For any h∈h1,h2, the expression equation (12) corresponds to one periodic orbit. With similar steps, we use Lemma 2 to prove that, for n=3, I2/I1 is monotonic on h∈−161/3/4,0. Because 41/3,0 is a center, we choose a,b=41/3,0. In this situation, α,A=0,101/3, β,B=−151/241/3/5,151/241/3/5, f1ϕ=1, f2ϕ=ϕ3 and gy=y in Lemma 2. Moreover, notice that Φϕ=−1/2ϕ2+1/20ϕ5 and Ψy=1/2y2. It is easy to verify that the hypotheses (H1), (H2), and (H3) hold.
In the following, we will judge the sign ofζ′ϕη′y. The sign of η′y=2y<0 is the same as in Sections 2 and 3; thus, we focus on ζ′ϕ. According to equation (23), by direct calculation, we obtain(57)ζϕ=f2ϕΦ′μ−f2μΦ′ϕf1ϕΦ′μ−f1μΦ′ϕ=ϕ3−μ+1/4μ4−μ3−ϕ+1/4ϕ4−μ+1/4μ4−−ϕ+1/4ϕ4=ϕ3μ3+4ϕ2μ+4ϕμ2μ3+μϕ2+μ2ϕ+ϕ3−4.
SinceΦϕ=Φμ, which implies that(58)μ5−10μ2−ϕ5+10ϕ2=μ−ϕμ4+μ3ϕ+μ2ϕ2+μϕ3+ϕ4−10μ−10ϕ=0.
From equation (58), μ=μϕ can be expressed by a function of ϕ, and taking derivative of this equation both sides with respect to ϕ yields(59)μ′ϕ=ϕ4−4ϕμ4−4μ.
Then, according to equation (57), we obtain the derivative of ζϕ with respect to ϕ as(60)ζ′ϕ=Fϕ,μ+Fμ,ϕμ′Gϕ,μ,where(61)Fϕ,μ=μ3ϕ2μ2+2ϕ3μ+ϕ4+4μ+8ϕμ3−4,and(62)Gϕ,μ=μ2+μ+ϕ2−32.
By substituting equation (59) into (60), we get(63)ζ′ϕ=Fϕ,μμ4−4μ+Fμ,ϕϕ4−4ϕGϕ,μμ4−4μ.
Sinceϕ∈0,41/3 and μ∈41/3,101/3, it is obvious that the denominator Gϕ,μμ4−4μ>0 in equation (63). Now, we need to prove that the numerator Fϕ,μμ4−4μ+Fμ,ϕϕ4−4ϕ is positive in equation (63). Notice that ϕ and μ satisfy equation (58), letting(64)Wϕ,μ=μ4+μ3ϕ+μ2ϕ2+μϕ3+ϕ4−10μ−10ϕ,and denoting R the resultant with respect to ϕ,(65)R≔Fϕ,μμ4−4μ+Fμ,ϕϕ4−4ϕ,Wϕ,μ,and then, using Maple to compute, we know that the resultant R is(66)R=81μ6μ30−56μ27+1472μ24−21664μ21+174768μ18−650368μ15+230784μ12+2427392μ9+11173376μ6−6193152μ3+3096576μ3−102μ3−42.R is a polynomial on μ, it can be verified by Sturm theorem that R has no zero on the open interval 41/3,101/3. This means that, for μ∈41/3,101/3, we have(67)Fϕ,μμ4−4μ+Fμ,ϕϕ4−4ϕ≡0.
Meanwhile, it is easy to verify(68)Fϕ,μμ4−4μ+Fμ,ϕϕ4−4ϕ>0.
Further, we have(69)ζ′ϕη′y<0.
Therefore,I2h/I1h is monotonically decreasing on −161/3/4,0, and this completes the proof of Lemma 5.Theorem 3.
Ifn=3 and ϵ is sufficiently small in equation (1), then, for each wave speed c∈1/3,13/27, there exists only a definite Hamiltonian hc∈−161/3/4,0, which is a strictly increasing function of c, such that equation (1) has a unique periodic wave. Moreover, there exists a unique solitary wave if and only if the wave speed c=13/27.Proof.
SinceI2h/I1h is monotonically decreasing on −161/3/4,0, we can calculate its range of values. When h=0, I2h/I1h takes its minimum(70)I2I1|min=∮Γ0ϕ3ydϕ∮Γ0ydϕ=4013.
Whenh=−161/3/4, I2h/I1h takes its maximum, and the same way is used to get I2−161/3/4/I1−161/3/4. From equation (16), we know that(71)I−161/34,c=−c1+1cI1−161/34+cI2−161/34=∂∂y−c1+1cy+cϕ3y41/3,0=−c1+1c+4c.
Comparing the coefficients of the above equation, we haveI1−161/3/4=1 and I2−161/3/4=4, that is, I2/I1|max=4. Consequently, only if 40/13<1+1/c<4, that is, 1/3<c<13/27, there exists a unique periodic orbit for the perturbed system. There exists a unique homoclinic orbit if and only if c=13/27.
Ash∈−161/3/4,0 and c∈1/3,13/27, due to the monotonicity of I2h/I1h, we can build an increasing and one-to-one mapping c↦h such that the Hamiltonian hc is a strictly increasing function of c.


## 3.1. The Casen=1 for Equation (1)

First of all, we consider the casen=1, and we have the following result.Lemma 3.
For system equation (11), if n=1, then the ratio I2h/I1h in equation (19) is strictly decreasing on h∈−2/3,0.Proof.
Whenn=1, the unperturbed system equation (11)ε=0 has two singular points 0,0 and 2,0, where 0,0 is a saddle point and 2,0 is a center. According to equation (12), the Hamiltonian h at these two points are h1=H2,0=−2/3 and h2=H0,0=0, respectively. For each h∈h1,h2, the expression equation (12) corresponds to a periodic orbit. The Abelian integral of equation (11)n=1 is given by equation (16)n=1. Now, we use Lemma 2 to prove that I2/I1 is monotonic on h∈−2/3,0. Since 2,0 is a center, we choose a,b=2,0. In this situation, it is easy to know α,A=0,3, β,B=−23/3,23/3, f1ϕ=1, f2ϕ=ϕ, and gy=y in Lemma 2. Moreover, notice that Φϕ=−1/2ϕ2+1/6ϕ3 and Ψy=1/2y2. If ϕ∈0,3 and ϕ≠2, we have(25)Φ′ϕϕ−2=12ϕϕ−22>0.
Ify∈−23/3,23/3 and y≠0, we have(26)Ψ′yy−0=y2>0.
Therefore, the condition (H1) in Lemma2 is verified.
Furthermore, forϕ∈0,2,(27)f1ϕf1μ=1>0,and for y∈−23/3,0,(28)g′yg′y˜=1>0.
So the conditions (H2) and (H3) are verified.
Now, we decide the sign ofζ′ϕη′y. According to equations (23) and (24), by direct calculation, we get(29)ζϕ=f2ϕΦ′μ−f2μΦ′ϕf1ϕΦ′μ−f1μΦ′ϕ=ϕ−μ+1/2μ2−μ−ϕ+1/2ϕ2−μ+1/2μ2−−ϕ+1/2ϕ2=ϕμμ+ϕ−2,and(30)ηy=gy˜−gyΨ′y˜Ψ′yg′y˜Ψ′y−g′yΨ′y˜=y˜−yy˜yy−y˜=−y˜y.
SinceΦϕ=Φμ and Ψy=Ψy˜, which implies that(31)3μ2−μ3=3ϕ2−ϕ3andy˜=−y.
So it is easy to get that(32)ηy=y2,and(33)η′y=2y<0,since y∈−23/3,0.
On the following, we considerζ′ϕ. From the first equation of equation (31), μ=μϕ can be regarded as a function of ϕ, and taking the derivative of this equation on both sides with respect to ϕ yields(34)μ′ϕ=2ϕ−ϕ22μ−μ2.
Again, by direct calculation, we obtain the derivative ofζϕ with respect to ϕ as(35)ζ′ϕ=μ2−2μ+ϕ2−2ϕμ′μ+ϕ−22.
By substituting equation (34) into (35), we get(36)ζ′ϕ=μ2−2μ2+ϕ2−2ϕ2μ+ϕ−22μ2−2μ>0,since ϕ∈0,2 and μ∈2,3. From equations (33) and (36), it is obvious that(37)ζ′ϕη′y<0.
Therefore,I2h/I1h is monotonically decreasing on −2/3,0, and this completes the proof of Lemma 3.Theorem 1.
Ifn=1 and ϵ is sufficiently small in equation (1), then, for each wave speed c∈1,7/5, there exists only a definite Hamiltonian hc∈−2/3,0, which is a strictly increasing function of c, such that equation (1) has a unique periodic wave. Moreover, there exists a unique solitary wave if and only if the wave speed c=7/5.Proof.
SinceI2h/I1h is monotonically decreasing on −2/3,0, we can calculate its range of values. When h=0, I2h/I1h takes its minimum(38)I2I1|min=∮Γ0ϕydϕ∮Γ0ydϕ=127.
Whenh=−2/3, I2h/I1h takes its maximum(39)I2I1|max=∮Γ−2/3ϕydϕ∮Γ−2/3ydϕ,where Γ−2/3 is the center, and the integrals make no sense. In this case, another way in [20] is used to get I2−2/3/I1−2/3. From equation (19), we know that(40)I−23,c=−c1+1cI1−23+cI2−23=∂∂y−c1+1cy+cϕy2,0=−c1+1c+2c.
Comparing the coefficients of the above equation, we haveI1−2/3=1 and I2−2/3=2, that is, I2/I1max=2. Consequently, I2h/I1h∈12/7,2, according to equation (19), only if 12/7<1+1/c<2 can be made Ih,c=0. Namely, for each c∈1,7/5, we can find the corresponding Hamiltonian h∈−2/3,0, such that there exists a unique periodic orbit for the perturbed system equation (1). From equation (19), we know that I0,c=0 if and only if c=7/5; hence, there exists a unique homoclinic orbit.
Ash∈−2/3,0 and c∈1,7/5, due to the monotonicity of I2h/I1h, we can also build a one by one and increasing mapping h↦c such that(41)c=1I2h/I1h−1=I1hI2h−I1h.
Conversely, it is obvious thathc is a monotonically increasing function of c.
Ifn=1 and ε is sufficiently small in equation (1), for wave speed c∈1,7/5 and c=7/5, we give the graphical solutions of periodic and solitary waves, respectively, for a meaningful illustration (see Figures 3 and 4). Similarly, we can draw the numerical graphical solutions when n takes other values.Figure 3
The periodic wave of equation (1) for 1<c<7/5.Figure 4
The solitary wave of equation (1) for c=7/5.


## 3.2. The Casen=2 for Equation (1)

In this subsection, we will discuss the casen=2 for equation (1). We have the following result of monotonicity.Lemma 4.
For system equation (11), if n=2, then the ratio I2h/I1h in equation (19) is strictly decreasing on h∈−3/4,0.Proof.
Whenn=2, the unperturbed system equation (11)ϵ=0 has three singular points 0,0, −3,0, and 3,0, where 0,0 is a saddle point, and both −3,0 and 3,0 are centers. According to equation (12), the Hamiltonian h at these three points are h1=H±3,0=−3/4 and h2=H0,0=0, respectively. For each h∈h1,h2, the expression equation (12) corresponds to two periodic orbits. As the phase portrait of equation (12)n=2 is symmetric, we only consider the case for ϕ>0, and the case for ϕ<0 is paralleled. The Abelian integral of equation (11)n=2 is given by equation (16)n=2. With the similar method, we use Lemma 2 to prove that I2/I1 is monotonic on h∈−3/4,0. Because of 3,0 is a center, we choose a,b=3,0. In this case, α,A=0,6, β,B=−6/2,6/2, f1ϕ=1, f2ϕ=ϕ2 and gy=y in Lemma 2. Moreover, notice that Φϕ=−1/2ϕ2+1/12ϕ4 and Ψy=1/2y2. If ϕ∈0,6 and ϕ≠3, we have(42)Φ′ϕϕ−3=13ϕϕ+3ϕ−32>0.
Ify∈−6/2,6/2 and y≠0, we have(43)Ψ′yy−0=y2>0.
Therefore, the condition (H1) in Lemma2 is verified.
Moreover, forϕ∈0,3,(44)f1ϕf1μ=1>0,and for y∈−6/2,0,(45)g′yg′y˜=1>0.
So the conditions (H2) and (H3) are verified.
Now, we aim to find the sign ofζ′ϕη′y. The computation of η′y=2y<0−6/2<y<0 is the same as in Section 3. Thus, according to equation (23), we mainly focus on the sign of ζ′ϕ by direct calculation, and we obtain(46)ζϕ=f2ϕΦ′μ−f2μΦ′ϕf1ϕΦ′μ−f1μΦ′ϕ=ϕ2−μ+1/3μ3−μ2−ϕ+1/3ϕ3−μ+1/3μ3−−ϕ+1/3ϕ3=3ϕμ+ϕ2μ2μ2+μϕ+ϕ2−3.
SinceΦϕ=Φμ, which implies that(47)μ2−ϕ26−μ2−ϕ2=0,that is.(48)μ2+ϕ2−6=0,as ϕ≠μ. From the above equation (48), μ=μϕ can be regarded as a function of ϕ, and taking the derivative of this equation on both sides with respect to ϕ yields(49)μ′ϕ=−ϕμ.
Then, according to equation (46), we obtain the derivative of ζϕ with respect to ϕ as(50)ζ′ϕ=2ϕμ4+ϕ2μ3−6ϕμ2+3μ3−3ϕ2μ−9μ+2ϕ4μ+ϕ3μ2−6ϕ2μ+3ϕ3−3ϕμ2−9ϕμ′μ2+μ+ϕ2−32.
By substituting equation (49) into (50), we obtain(51)ζ′ϕ=2ϕμ4+ϕ2μ3−6ϕμ2+3μ3−3ϕ2μ−9μμ−2ϕ4μ+ϕ3μ2−6ϕ2μ+3ϕ3−3ϕμ2−9ϕϕμ2+μ+ϕ2−32μ.
Letζ′ϕ=Fϕ,μ/Gϕ,μ, since ϕ∈0,3 and μ∈3,6, it is obvious that Gϕ,μ>0. Notice that ϕ and μ satisfy equation (48), letting Wϕ,μ=ϕ2+μ2−6, with the aid of Maple, we get the resultant F,W with respect to ϕ is(52)F,W=4μ2−36.
It is clear that4μ2−36≠0 as μ∈3,6, it means Fϕ,μ≡0; meanwhile, Fϕ,μ>0. Consequently, we have(53)ζ′ϕη′y<0.
Therefore,I2h/I1h is monotonically decreasing on −3/4,0. This completes the proof of Lemma 4.Theorem 2.
Ifn=2 and ϵ is sufficiently small in equation (1), then, for each wave speed c∈1/2,5/7, there exists only a definite Hamiltonian hc∈−3/4,0, which is a strictly increasing function of c, such that equation (1) has a unique periodic wave for ux,t>0 or ux,t<0. Moreover, there exists a unique solitary wave for ux,t>0 or ux,t<0 if and only if the wave speed c=5/7.Proof.
SinceI2h/I1h is monotonically decreasing on −3/4,0, we can calculate its range of values. When h=0, I2h/I1h, taking its minimum(54)I2I1|min=∮Γ0ϕ2ydϕ∮Γ0ydϕ=125.
Whenh=−3/4, I2h/I1h, taking its maximum(55)I2I1|max=∮Γ−3/4ϕ2ydϕ∮Γ−3/4ydϕ,where Γ−3/4 is the center, and the integrals make no sense. In this case, the same method is used to get I20/I10. From equation (19), we know that(56)I−34,c=−c1+1cI1−34+cI2−34=∂∂y−c1+1cy+cϕ2y3,0=−c1+1c+3c.
Comparing the coefficients of the above equation, we haveI1−3/4=1 and I2−3/4=3; that is, I2/I1max=3. Consequently, only if 12/5<1+1/c<3, that is, 1/2<c<5/7, there exists one unique periodic orbit on each side of the origin for the perturbed system. It is clear that there exists one unique homoclinic orbit on each side of the origin if and only if c=5/7.
Ash∈−3/4,0 and c∈1/2,5/7, due to the monotonicity of I2h/I1h, we can build the increasing and one-to-one mapping c↦h such that the Hamiltonian hc is a strictly increasing function of c.


## 3.3. The Casen=3 for Equation (1)

In this section, we discuss the casen=3 for equation (1). Similarly, we have the result of monotonicity as follows.Lemma 5.
For system equation (11), if n=3, then the ratio I2h/I1h in equation (19) is strictly decreasing on h∈−161/3/4,0.Proof.
Whenn=3, the unperturbed system equation (11)ε=0 has two singular points 0,0 and 41/3,0. 0,0 is a saddle point, and 41/3,0 is a center. According to equation (12), the Hamiltonian h at these two points are h1=H41/3,0=−161/3/4 and h2=H0,0=0, respectively. For any h∈h1,h2, the expression equation (12) corresponds to one periodic orbit. With similar steps, we use Lemma 2 to prove that, for n=3, I2/I1 is monotonic on h∈−161/3/4,0. Because 41/3,0 is a center, we choose a,b=41/3,0. In this situation, α,A=0,101/3, β,B=−151/241/3/5,151/241/3/5, f1ϕ=1, f2ϕ=ϕ3 and gy=y in Lemma 2. Moreover, notice that Φϕ=−1/2ϕ2+1/20ϕ5 and Ψy=1/2y2. It is easy to verify that the hypotheses (H1), (H2), and (H3) hold.
In the following, we will judge the sign ofζ′ϕη′y. The sign of η′y=2y<0 is the same as in Sections 2 and 3; thus, we focus on ζ′ϕ. According to equation (23), by direct calculation, we obtain(57)ζϕ=f2ϕΦ′μ−f2μΦ′ϕf1ϕΦ′μ−f1μΦ′ϕ=ϕ3−μ+1/4μ4−μ3−ϕ+1/4ϕ4−μ+1/4μ4−−ϕ+1/4ϕ4=ϕ3μ3+4ϕ2μ+4ϕμ2μ3+μϕ2+μ2ϕ+ϕ3−4.
SinceΦϕ=Φμ, which implies that(58)μ5−10μ2−ϕ5+10ϕ2=μ−ϕμ4+μ3ϕ+μ2ϕ2+μϕ3+ϕ4−10μ−10ϕ=0.
From equation (58), μ=μϕ can be expressed by a function of ϕ, and taking derivative of this equation both sides with respect to ϕ yields(59)μ′ϕ=ϕ4−4ϕμ4−4μ.
Then, according to equation (57), we obtain the derivative of ζϕ with respect to ϕ as(60)ζ′ϕ=Fϕ,μ+Fμ,ϕμ′Gϕ,μ,where(61)Fϕ,μ=μ3ϕ2μ2+2ϕ3μ+ϕ4+4μ+8ϕμ3−4,and(62)Gϕ,μ=μ2+μ+ϕ2−32.
By substituting equation (59) into (60), we get(63)ζ′ϕ=Fϕ,μμ4−4μ+Fμ,ϕϕ4−4ϕGϕ,μμ4−4μ.
Sinceϕ∈0,41/3 and μ∈41/3,101/3, it is obvious that the denominator Gϕ,μμ4−4μ>0 in equation (63). Now, we need to prove that the numerator Fϕ,μμ4−4μ+Fμ,ϕϕ4−4ϕ is positive in equation (63). Notice that ϕ and μ satisfy equation (58), letting(64)Wϕ,μ=μ4+μ3ϕ+μ2ϕ2+μϕ3+ϕ4−10μ−10ϕ,and denoting R the resultant with respect to ϕ,(65)R≔Fϕ,μμ4−4μ+Fμ,ϕϕ4−4ϕ,Wϕ,μ,and then, using Maple to compute, we know that the resultant R is(66)R=81μ6μ30−56μ27+1472μ24−21664μ21+174768μ18−650368μ15+230784μ12+2427392μ9+11173376μ6−6193152μ3+3096576μ3−102μ3−42.R is a polynomial on μ, it can be verified by Sturm theorem that R has no zero on the open interval 41/3,101/3. This means that, for μ∈41/3,101/3, we have(67)Fϕ,μμ4−4μ+Fμ,ϕϕ4−4ϕ≡0.
Meanwhile, it is easy to verify(68)Fϕ,μμ4−4μ+Fμ,ϕϕ4−4ϕ>0.
Further, we have(69)ζ′ϕη′y<0.
Therefore,I2h/I1h is monotonically decreasing on −161/3/4,0, and this completes the proof of Lemma 5.Theorem 3.
Ifn=3 and ϵ is sufficiently small in equation (1), then, for each wave speed c∈1/3,13/27, there exists only a definite Hamiltonian hc∈−161/3/4,0, which is a strictly increasing function of c, such that equation (1) has a unique periodic wave. Moreover, there exists a unique solitary wave if and only if the wave speed c=13/27.Proof.
SinceI2h/I1h is monotonically decreasing on −161/3/4,0, we can calculate its range of values. When h=0, I2h/I1h takes its minimum(70)I2I1|min=∮Γ0ϕ3ydϕ∮Γ0ydϕ=4013.
Whenh=−161/3/4, I2h/I1h takes its maximum, and the same way is used to get I2−161/3/4/I1−161/3/4. From equation (16), we know that(71)I−161/34,c=−c1+1cI1−161/34+cI2−161/34=∂∂y−c1+1cy+cϕ3y41/3,0=−c1+1c+4c.
Comparing the coefficients of the above equation, we haveI1−161/3/4=1 and I2−161/3/4=4, that is, I2/I1|max=4. Consequently, only if 40/13<1+1/c<4, that is, 1/3<c<13/27, there exists a unique periodic orbit for the perturbed system. There exists a unique homoclinic orbit if and only if c=13/27.
Ash∈−161/3/4,0 and c∈1/3,13/27, due to the monotonicity of I2h/I1h, we can build an increasing and one-to-one mapping c↦h such that the Hamiltonian hc is a strictly increasing function of c.


## 4. Discussion

We have studied the equation (1) for the special cases n=1,2,3, these cases are more likely meaningful in certain realistic background. From the view of theoretical mathematics, it is more interesting to prove I2h/I1h is monotonic on the interval h∈−nn+12/n/2n+2,0 for all integers n>3. However, we can verify that I2h/I1h is decreasing on the interval h∈−nn+12/n/2n+2,0 for any fixed n. For example, we can verify that the conclusion holds for n=3,4,…,10 and so on, and here, we skip the details. We conjecture the ratio I2h/I1h is monotonic on the interval h∈−nn+12/n/2n+2,0 for all integers n>3.Conjecture 1.
For system equation (11) and any integer n>3, then the ratio I2h/I1h in equation (19) is strictly decreasing on h∈−nn+12/n/2n+2,0.If Conjecture1 holds, then we have the following result for any positive integer n.Proposition 1.
Letn be any positive integer and ϵ is sufficiently small in equation (1). Then, for each wave speed c∈1/n,3n+4/2n2+3n, there exists a only definite Hamiltonian hc∈−nn+12/n/2n+2,0, which is a strictly increasing function of c, such that(i)
Ifn is odd, equation (1) has a unique periodic wave(ii)
Ifn is even, equation (1) has a unique periodic wave for ux,t>0 or ux,t<0
Moreover, there exists a unique solitary wave if and only if the wave speedc=3n+4/2n2+3n for odd n and for even n with ux,t>0 or ux,t<0.Proof.
SinceI2h/I1h is monotonically decreasing on −nn+12/n/2n+2,0, we can calculate its range of values. When h=0, I2h/I1h takes its minimum, according to Lemma 2 in [21],(72)I2I1|min=∮Γ0ϕnydϕ∮Γ0ydϕ=2n+1n+23n+4.
Whenh=−nn+12/n/2n+2, I2h/I1h takes its maximum, and the same way is used to get I2/I1|max. From equation (18), we know that(73)I−nn+12/n2n+2,c=−c1+1cI1−nn+12/n2n+2+cI2−nn+12/n2n+2=∂∂y−c1+1cy+cϕnynn+1,0=−c1+1c+n+1c.
Comparing the coefficients of the above equation, we have(74)I1−nn+12/n2n+2=1,and(75)I2−nn+12/n2n+2=n+1,that is, I2/I1|max=n+1. Consequently, only if 2n+1n+2/3n+4<1+1/c<n+1, that is 1/n<c<3n+4/2n2+3n, there exists a unique periodic orbit for the perturbed system for odd n, and for even n, there exists only one unique periodic orbit on each side of saddle point 0,0. Only if c=3n+4/2n2+3n, there exists a unique homoclinic orbit for odd n and on each side of origin for even n.
Ash∈−nn+12/n/2n+2,0 and correspondingly c∈1/n,3n+4/2n2+3n, due to the monotonicity of I2h/I1h in Conjecture 1 we can build the increasing one to one mapping h⟶c such that(76)c=1I2h/I1h−1=I1hI2h−I1h.
This completes the proof of Proposition1.


## 5. Conclusion

In this paper, we study a perturbed generalized KdV equation (1). We prove that, for the special cases n=1,2,3 and ε small enough, the perturbed equation possesses a unique periodic wave if the wave speed lies on a certain small interval. We also show that the amplitude of a periodic wave is an increasing function of the wave speed. On the other hand, with the parameter n increasing from 1 to 3, the interval of the wave speed of the periodic wave becomes smaller. However, for all integers n>3, Conjecture 1 remains open for further study. The method used in this paper is simpler than that in relative references [4, 21], and some results therein are extended. Of course, the method shown in this paper can be used to deal with some other perturbed differential equations.



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