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 speed becomes smaller as the power increases.
1. Introduction
The Korteweg-de Vries (KdV) equation 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 causes steepening of wave form, and the linear dispersion term 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:which is generalized from the perturbed KdV equationwhere 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 Section 2, 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 , respectively, in Section 3. Next, we discuss the existence problem for an unfixed integer 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 ansatzfor , satisfyingwhere is the propagation speed of a wave. By substituting equation (3) into (1), we obtain
Integrating the above equation with respect to and taking the integral constant to be zero, we have
By introducing the scale transformations and into equation (6), we obtain the following third-order ordinary differential equation:
By bifurcation theory of planar dynamical systems, we know that, if in equation (4), is a solitary wave, and it corresponds to a homoclinic orbit of equation (7). If , 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:where , and is a real parameter, are on the set , where and is an open interval containing 0. Assume that for , the system has a compact normally hyperbolic manifold which is contained in the set . The manifold is said to be normally hyperbolic if the linearization of equation (8) at each point in has exactly eigenvalues on the imaginary axis . Then, for any , if sufficiently small, there exists a manifold such that the following conclusions hold.(1) is locally invariant under the flow of equation (8)(2) is in , and (3) for some function and in some compact set (4)There exist locally invariant stable and unstable manifolds , , that lie within of, and are diffeomorphic to and
We introduce and , and then, the ODE equation (7) becomes
By the basic theory of geometric singular perturbation, there exists a slow manifold
By Lemma 1, the main interesting portraits of system equation (9) projected on have the following form:
System equation (11) is a regular perturbation problem. The unperturbed system equation (11) is integrable and has the Hamiltonian function (first integral).
If is even, it is easy to see that the unperturbed system equation (11) has three singular points , , and , where is a saddle point, and and are two centers. If is odd, it is also easy to see that the unperturbed system equation (11) has two singular points , , where is a saddle point, and is a center. By the theory of planar dynamical systems (e.g., see [16–18]), for both is even and odd, the phase portraits of system equation (11) can be classified into three cases with the orbits defined by the following functions:and
The phase portrait of equation (11) is shown in Figure 1 for odd and Figure 2 for even .
Under perturbation, lots of periodic orbits of equation (11) 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 on the -axis, where the positive -axis can be parameterized by and . Let be the first intersection point after 2 of the orbit starting from with the -axis. Then, the displacement function of system equation (11) can be obtained as [18]andwith and . By Poincaré bifurcation theory [18], the number of zeros of corresponds to the number of limit cycles of system equation (11), which is the number of isolated periodic traveling waves. If the ratio is monotonic, then has at most one zero. In order to prove the monotonicity of , 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 ratiowhere is a family of ovals described bywhich surrounds a point , and and are polynomials. Then, represents a closed orbit of the Hamiltonian systemHere, we describe the theorem in [19] as follows. Consider the equation (72), where , . Let be the compact component of the level curve . Suppose that there exist and , such that the following hypothesis is satisfied: (H1) (i) for , (ii) for (H2) for (H3) for
In the hypothesis (H2) and (H3), the symbols and match with and in one-to-one mappings and , and they satisfy and , respectively. For each , two functions and displayed in the Lemma are defined as the following forms:and
Lemma 2 (see [19]). Assume that has the form equation (72), and the hypotheses (H1), (H2), and (H3) are satisfied. For and , we have(i)If , then is monotonically increasing on (ii)If , then is monotonically decreasing on
3. Existence and Uniqueness of the Periodic Wave
3.1. The Case for Equation (1)
First of all, we consider the case , and we have the following result.
Lemma 3. For system equation (11), if , then the ratio in equation (19) is strictly decreasing on .
Proof. When , the unperturbed system equation (11) has two singular points and , where is a saddle point and is a center. According to equation (12), the Hamiltonian at these two points are and , respectively. For each , the expression equation (12) corresponds to a periodic orbit. The Abelian integral of equation (11) is given by equation (16). Now, we use Lemma 2 to prove that is monotonic on . Since is a center, we choose . In this situation, it is easy to know , , , , and in Lemma 2. Moreover, notice that and . If and , we haveIf and , we haveTherefore, the condition (H1) in Lemma 2 is verified.
Furthermore, for ,and for ,So the conditions (H2) and (H3) are verified.
Now, we decide the sign of . According to equations (23) and (24), by direct calculation, we getandSince and , which implies thatSo it is easy to get thatandsince .
On the following, we consider . From the first equation of equation (31), can be regarded as a function of , and taking the derivative of this equation on both sides with respect to yieldsAgain, by direct calculation, we obtain the derivative of with respect to asBy substituting equation (34) into (35), we getsince and . From equations (33) and (36), it is obvious thatTherefore, is monotonically decreasing on , and this completes the proof of Lemma 3.
Theorem 1. If and is sufficiently small in equation (1), then, for each wave speed , there exists only a definite Hamiltonian , which is a strictly increasing function of , such that equation (1) has a unique periodic wave. Moreover, there exists a unique solitary wave if and only if the wave speed .
Proof. Since is monotonically decreasing on , we can calculate its range of values. When , takes its minimumWhen , takes its maximumwhere is the center, and the integrals make no sense. In this case, another way in [20] is used to get . From equation (19), we know thatComparing the coefficients of the above equation, we have and , that is, . Consequently, , according to equation (19), only if can be made . Namely, for each , we can find the corresponding Hamiltonian , such that there exists a unique periodic orbit for the perturbed system equation (1). From equation (19), we know that if and only if ; hence, there exists a unique homoclinic orbit.
As and , due to the monotonicity of , we can also build a one by one and increasing mapping such thatConversely, it is obvious that is a monotonically increasing function of .
If and is sufficiently small in equation (1), for wave speed and , 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 takes other values.
3.2. The Case for Equation (1)
In this subsection, we will discuss the case for equation (1). We have the following result of monotonicity.
Lemma 4. For system equation (11), if , then the ratio in equation (19) is strictly decreasing on .
Proof. When , the unperturbed system equation (11) has three singular points , , and , where is a saddle point, and both and are centers. According to equation (12), the Hamiltonian at these three points are and , respectively. For each , the expression equation (12) corresponds to two periodic orbits. As the phase portrait of equation (12) is symmetric, we only consider the case for , and the case for is paralleled. The Abelian integral of equation (11) is given by equation (16). With the similar method, we use Lemma 2 to prove that is monotonic on . Because of is a center, we choose . In this case, , , , and in Lemma 2. Moreover, notice that and . If and , we haveIf and , we haveTherefore, the condition (H1) in Lemma 2 is verified.
Moreover, for ,and for ,So the conditions (H2) and (H3) are verified.
Now, we aim to find the sign of . The computation of is the same as in Section 3. Thus, according to equation (23), we mainly focus on the sign of by direct calculation, and we obtainSince , which implies thatthat is.as . From the above equation (48), can be regarded as a function of , and taking the derivative of this equation on both sides with respect to yieldsThen, according to equation (46), we obtain the derivative of with respect to asBy substituting equation (49) into (50), we obtainLet , since and , it is obvious that . Notice that and satisfy equation (48), letting , with the aid of Maple, we get the resultant with respect to isIt is clear that as , it means ; meanwhile, . Consequently, we haveTherefore, is monotonically decreasing on . This completes the proof of Lemma 4.
Theorem 2. If and is sufficiently small in equation (1), then, for each wave speed , there exists only a definite Hamiltonian , which is a strictly increasing function of , such that equation (1) has a unique periodic wave for or . Moreover, there exists a unique solitary wave for or if and only if the wave speed .
Proof. Since is monotonically decreasing on , we can calculate its range of values. When , , taking its minimumWhen , , taking its maximumwhere is the center, and the integrals make no sense. In this case, the same method is used to get . From equation (19), we know thatComparing the coefficients of the above equation, we have and ; that is, . Consequently, only if , that is, , 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 .
As and , due to the monotonicity of , we can build the increasing and one-to-one mapping such that the Hamiltonian is a strictly increasing function of .
3.3. The Case for Equation (1)
In this section, we discuss the case for equation (1). Similarly, we have the result of monotonicity as follows.
Lemma 5. For system equation (11), if , then the ratio in equation (19) is strictly decreasing on .
Proof. When , the unperturbed system equation (11) has two singular points and . is a saddle point, and is a center. According to equation (12), the Hamiltonian at these two points are and , respectively. For any , the expression equation (12) corresponds to one periodic orbit. With similar steps, we use Lemma 2 to prove that, for , is monotonic on . Because is a center, we choose . In this situation, , , , and in Lemma 2. Moreover, notice that and . It is easy to verify that the hypotheses (H1), (H2), and (H3) hold.
In the following, we will judge the sign of . The sign of is the same as in Sections 2 and 3; thus, we focus on . According to equation (23), by direct calculation, we obtainSince , which implies thatFrom equation (58), can be expressed by a function of , and taking derivative of this equation both sides with respect to yieldsThen, according to equation (57), we obtain the derivative of with respect to aswhereandBy substituting equation (59) into (60), we getSince and , it is obvious that the denominator in equation (63). Now, we need to prove that the numerator is positive in equation (63). Notice that and satisfy equation (58), lettingand denoting the resultant with respect to ,and then, using Maple to compute, we know that the resultant is is a polynomial on , it can be verified by Sturm theorem that has no zero on the open interval . This means that, for , we haveMeanwhile, it is easy to verifyFurther, we haveTherefore, is monotonically decreasing on , and this completes the proof of Lemma 5.
Theorem 3. If and is sufficiently small in equation (1), then, for each wave speed , there exists only a definite Hamiltonian , which is a strictly increasing function of , such that equation (1) has a unique periodic wave. Moreover, there exists a unique solitary wave if and only if the wave speed .
Proof. Since is monotonically decreasing on , we can calculate its range of values. When , takes its minimumWhen , takes its maximum, and the same way is used to get . From equation (16), we know thatComparing the coefficients of the above equation, we have and , that is, . Consequently, only if , that is, , there exists a unique periodic orbit for the perturbed system. There exists a unique homoclinic orbit if and only if .
As and , due to the monotonicity of , we can build an increasing and one-to-one mapping such that the Hamiltonian is a strictly increasing function of .
4. Discussion
We have studied the equation (1) for the special cases , these cases are more likely meaningful in certain realistic background. From the view of theoretical mathematics, it is more interesting to prove is monotonic on the interval for all integers . However, we can verify that is decreasing on the interval for any fixed . For example, we can verify that the conclusion holds for and so on, and here, we skip the details. We conjecture the ratio is monotonic on the interval for all integers .
Conjecture 1. For system equation (11) and any integer , then the ratio in equation (19) is strictly decreasing on .
If Conjecture 1 holds, then we have the following result for any positive integer .
Proposition 1. Let be any positive integer and is sufficiently small in equation (1). Then, for each wave speed , there exists a only definite Hamiltonian , which is a strictly increasing function of , such that(i)If is odd, equation (1) has a unique periodic wave(ii)If is even, equation (1) has a unique periodic wave for or Moreover, there exists a unique solitary wave if and only if the wave speed for odd and for even with or .
Proof. Since is monotonically decreasing on , we can calculate its range of values. When , takes its minimum, according to Lemma 2 in [21],When , takes its maximum, and the same way is used to get . From equation (18), we know thatComparing the coefficients of the above equation, we haveandthat is, . Consequently, only if , that is , there exists a unique periodic orbit for the perturbed system for odd , and for even , there exists only one unique periodic orbit on each side of saddle point . Only if