Solitary Waves of the Perturbed KdV Equation with Nonlocal Effects

In this paper, the Korteweg–de Vries (KdV) equation is considered, which is a shallow water wave model in fluid mechanic fields. First the existence of solitary wave solutions for the original KdV equation and geometric singular perturbation theory are recalled. Then the existence of solitary wave solutions is established for the equation with two types of delay convolution kernels by using the method of dynamical system, especially the geometric singular perturbation theory, invariant manifold theory and Melnikov method. Finally, the asymptotic behaviors of solitary wave solution are discussed by applying the asymptotic theory. Moreover, an interesting result is found for the equation without backward diffusion effect, there is no solitary wave solution in the case of local delay, but there is a solitary wave solution in the case of nonlocal delay.


Introduction
Significant progress has been made in the study of nonlinear evolution equations, which are widely used to describe complex physical phenomena. The KdV equation, a classical nonlinear evolution equation, is an integrable shallow water wave model to describe the uni-directional propagation of waves at the free surface of shallow water [1]. which was actually derived by Boussinesq [2,3] in 1872 [1]. u(x, t) represents the fluid velocity and can be regarded as the wave height above a flat bottom.
Solitary wave, consisting of a single elevation, which propagates without change of form, was first observed scientifically by Russell in 1844 [4]. In 1947, Lavrentiev [5] presented a proof of the existence of permanent solitary waves, in which these waves are considered as limits of periodic waves with indefinitely increasing wave length. Meanwhile Friedrichs and Hyers [6] proved the existence of solitary waves in a direct manner in 1954.
The KdV equation is well known to play an important role in the development of the soliton theory. The nonlinear term uu x in Eq. (1) causes the steepening of wave form, whereas the dispersion effect term u xxx makes the wave form spread, solitons are produced from the interaction between them [7,8]. The orbital and asymptotic stability of solitons in the KdV equation on the energy space have also attracted some interests [9,10]. When backward diffusion and dissipation effects are taken into consideration, the model is called the following Kortewegde Vries-Kuramoto-Sivashinsky (KdV-KS) equation where R is the parameter corresponding to the Reynolds number, which was derived by the method of asymptotic expansion and truncation from the two dimensional Navier Stokes equation on an slightly inclined plane [11]. If the dispersion is very strong compared to the backward diffusion and dissipation, the equation could be scaled to the following form where 0 < ≪ 1 , which can be viewed as a perturbed KdV equation. The persistence of solitary and periodic waves of Eq. (2) was studied in [12]. The monotonicity and boundedness of wave velocity were obtained, and the relationship between the wavelength and the amplitude were derived. The perturbation terms, the backward diffusion u xx and dissipation u xxxx , are usually called the KS perturbation. In recent years the problem of persistence of solitary waves in shallow water wave equations with the KS perturbation has received much attention [13][14][15].
After rescaling and applying a Galilean transformation, the KdV equation can also be obtained from the family of third-order dispersive PDE conservation laws of the form in [16] which also is a shallow water wave model. By using geometric singular perturbation and invariant manifold theory, Ge and Du [14] studied the existence of solitary wave solutions for Eq. (3) with KS perturbation. Even more interesting is that Eq.
(3) contains the other two integral shallow water wave equations: Camassa-Holm (1) u t + 6uu x + u xxx = 0, u(x, 0) = u 0 (x), (CH) and Degasperis-Procesi (DP) equations. The CH equation was first derived by Fuchssteiner and Fokas [17] as an abstract bi-Hamiltonian equation with infinitely many conservation laws, and later re-derived by Camassa and Holm [18] from physical principles. The DP equation was derived by Degasperis and Procesi [16] using the method of asymptotic integrability. The peakon, which means the traveling wave with slope discontinuities, is used to distinguish them from general traveling wave solutions since they have a corner at the peak of height c. The CH equation has the peakon of the form u(x, t) = ce −|x−ct| [18], and the DP equation also has the peakon solution [19]. There are also other remarkable properties about the CH and DP equations: the isospectral problem [18,19], wave breaking [20,21], integrability [17,18], see also [22]. Moreover, the Fokas-Olver-Rosenau-Qiao (FORQ) equation is an integrable equation with cubic nonlinearity [23,24]. There have been various results about the FORQ equation in mathematical and physical fields [25][26][27]. Convolution in space and time was first introduced by Britton [28] into the diffusive model. He discussed the following model for a single biological population To account for the drift of individuals to their present position from all possible positions at previous times, the nonlocal delay and spatial effects were introduced, in the form of convolution in time and space. In addition to the KS perturbation, perturbation in such form of spatiotemporal convolution frequently appears in differential equations and receives more attention. Recently, solitary waves of water wave equations with the delay and diffusion perturbation were considered in [29,30].
Motivated by the aforementioned results, the present work is devoted to the study of existence of solitary wave solutions for perturbed Eq. (1) of the form where 0 < ≪ 1 , f * u is the spatial-temporal convolution representing distributed delay and nonlocal spatial effect, and f is a kernel function. u xx is regarded as viscosity term perturbation. Throughout the paper, we discuss the f * u in three cases: (i) without delay (ii) with the local distributed delay (iii) with the nonlocal distributed delay Although the addition of convolution has more practical significance for the discussion of these equations, it breaks the integrability, that will be solved by a double singular perturbation reduction according to the special structure of the KdV equation. The main tools to prove the existence of perturbed solitary waves for Eq. (4) are the geometric singular perturbation (GSP) theory [31,32] and Melnikov method [33].
The remaining part is organized as follows. In Sect. 2, we recall the general theory of GSP and the Melnikov integral, and give a short discussion on the existence of solitary waves for Eq. (4) without delay and perturbation. In Sect. 3, the existence of solitary wave solutions is proved for Eq. (4) with two types: local delay and nonlocal delay. In Sect. 4, the asymptotic behavior of the solitary waves for Eq. (4) is analyzed by applying the standard asymptotic theory. In Sect. 5, some conclusions on the existence of solitary wave solutions are drawn.

Preliminaries
The general theory of GSP and Melnikov method have been well developed. The singular perturbation theorem on invariant manifolds of GSP comes from Theorem 9.1 in [31]. For convenience, we use another more detailed version of this theorem from Theorem 1-2 in [32]. Robinson [33] presented the following Melnikov integral.
Assume equations above satisfy the saddle connection assumption, and that f 0 is divergence free as a function of X, i.e. trDf 0 (X, ) ≡ 0 , where D is the derivative with respect to X. Then where satisfies ( ) � = r(X 0 (t), 0 ), and = 0 , at t = 0. Now we will present a short discussion on Eq. (4) without delay and perturbation. Eq. (4) can be reduced to Eq. (1) as → 0 . In fact, let the kernel function f = 1 e − t , then we have and u xx → 0 . Similarly, this result also holds for the case of nonlocal delay. It is a Hamiltonian system with the Hamiltonian function from which and phase portrait analysis, we get the following Theorem 1 In the ( , ) phase plane, system (7) has a homoclinic orbit (see Fig. 1) to the equilibrium point (0, 0). This connection is confined in 0 ≤ ≤ c 2 .
Proof It is easy to find that system (7) has two equilibra E 1 (0, 0) and E 2 ( c 3 , 0) . Moreover, E 1 is a saddle and E 2 is a center. Notice that system (7) is a Hamiltonian system with the Hamiltonian function H. Consider the level curve of the form H = k . When k = 0 , it includes a homoclinic orbit to −c + 3 2 + �� = 0, So there is a homoclinic orbit to E 1 , then there exists a solitary wave solution for Eq. (4) without delay and perturbation. ◻

Solitary Wave Solution for Equation (4)
In this section, we will investigate the existence of solitary wave solutions for Eq. (4) in two cases: local distributed delay and nonlocal distributed delay.

The Equation with Local Delay
Consider solitary wave solutions for Eq. (4) in the case that the convolution f * u is defined by where the kernel f ∶ [0, ∞) → [0, ∞) satisfies the following normalization assumption The average time delay for the distributed delay kernel f(t) is defined as = ∫ ∞ 0 tf (t)dt . Usually we use the Gamma distributed delay kernel where > 0 is a constant, n is an integer, with average delay = n > 0 . Two special cases are frequently used in the delay differential equation. The first of two above kernels is sometimes called the weak generic kernel, and the other is the strong case. We discuss the strong generic delay kernel in this section (similarly for the weak case), i.e.
Consider the case of averaging delay 0 < ≪ 1 . Solitary wave solutions of Eq. (4) take the form u( Differentiating with respect to , we get Furthermore, we obtain Therefore, the solitary wave solution to Eq. (8) can be replaced by the following system It is worthwhile to note that when → 0 , we obtain → and arrive at the nondelay Eq. (5), for which the existence of the solitary wave solution was already discussed in Sect. 2. For > 0 , system (9) defines a system of ODEs whose solutions evolve in the five dimensional ( , , , , ) phase space. However, when = 0 , system (9) does not define a dynamical system in ℜ 5 . This problem may be overcome by the transformation = z , under which the system becomes where ⋅ denotes the derivative with respect to z. Generally speaking, system (9) is referred to as the slow system since the time scale is slow, while system (10) is referred to as the fast system since the time scale z is fast. The two systems are equivalent when > 0 . To study the persistence of the homoclinic orbit for sufficiently small > 0 , it suffices to restrict to a neighborhood of the unperturbed homoclinic orbit which satisfies 0 ≤ ≤ * . If is set to zero in system (9), then the flow of that system is restricted to the set which is a three dimensional manifold of equilibrium for system (10) with = 0 . In order to obtain an invariant manifold for sufficiently small > 0 by using GSP theory, it suffices to verify the normal hyperbolicity of M 0 . The linearization matrix of system (10) with = 0 is From Definition 1 in [32], we know that the manifold M 0 is normally hyperbolic with two unstable normal directions. According to Theorem 1-2 in [32], M 0 persists for 0 < ≪ 1 , i.e. there exists a locally invariant slow manifold M , which can be written in the form where the functions g and h are smooth functions defined on a compact domain, and satisfy When → 0 , system (9) reduces to the non-delay Eq. (5). According to Definition 2 in [32], we know that M is locally invariant under the flow of system (10), so and (10) Substituting system (10) into system (11), we get To compute series expansions for M , let us Taylor expand the righthand side of system (12) with respect to Comparing the coefficient of for systems (13) and (14), we have Therefore restricted to M , system (9) becomes the following system of regular perturbation Because both normal directions are unstable, the persistent homoclinic orbit (if ever exists) must be completely contained in M . So system (15) is exactly the system to study for the existence of a homoclinic orbit.
Note that, for the reduced system (15), which has a line of equilibrium given by

Remark 1
This key information can not be drawn from solely investigation of system (15) because of the o( ) term, but from the special structure of the KdV equation. It is this property that makes possible another application of the method of singular perturbation.
Integrate once for non-delayed Eq. where C is a constant, that is to say, With the above analysis, next we change system (15) into the standard form of singular perturbation. Now consider the following variable transformations Then ̃,̃,̃ satisfy the following system We suppress the superscript title for the sake of readability Since system (15)  , = 0, | |≤ } for some small constant > 0 independent of , which is easily checked to be normally hyperbolic with one stable and one unstable normal directions. Applying Theorem 1-2 in [32] again, M 0 persists for 0 < ≪ 1 , denoted by M , which is C 1 O( ) close to M 0 . Because of the existence of the curve of equilibrium, M is nothing else but the same as M 0 . Fenichel's theory indicates the existence of two dimensional stable and unstable manifolds W s and W u of M , being C 1 O( ) close to the corresponding stable and unstable manifolds, W s 0 and W u 0 , of M 0 , respectively. Viewing as parameter, when = 0 , system (16) is a Hamiltonian system, with non-transversal intersection of W s 0 and W u 0 and a two dimensional homoclinic manifold. We need to study the separation of W s and W u along the unperturbed homoclinic orbit Γ , which was already founded in Theorem 1. Restricted on the plane { = 0} , the O( ) signed distance (Dist) between W s and W u on a transverse cross section at ( , ) = ( c 2 , 0) is given by Because the ( , ) sub-vector field is divergence free, then from Lemma 1, we can know that −c + 3 2 + = C.
where , are evaluated along Γ , and satisfies Δ is calculated by integration by parts, Since d d ( ) = 6c 2 + c − 3 2 − is bounded and exponentially small at ±∞ , increase at most sub-exponentially at ±∞ . On the other hand, is exponentially small at ±∞ , the first term in Δ vanishes. So Based on the analysis above, it is clear that Δ is continuous about c for c > 0 , whose figure can be well depicted by numerical simulation, see Fig. 2.
Owing to this, we can obviously find the fact that when c > 0 , Δ has a tendency to increase first and then decrease, which finally tends to negative infinity. Let Δ = 0 in (18), we can obtain that there is only a c * =  After entering the neighborhood V, Γ intersects some submanifold W s ( ) and follows the evolution of the submanifold, whose existence and properties are explained by Theorem 9.1 in [31]. In addition, W s ( ) is C 1 O close to Γ , which belongs to the stable manifold of 0 = . Notice that the slow manifold M consists of true equilibrium. Therefore base points on M is not moving at all, and the 'moving invariant' submanifold W s ( ) is actually invariant.
As a result, for all forward time, Γ is C 1 O close to the unperturbed homoclinic Γ . The backward time is the same. which is clearly negative for c > 0 , there is no solitary wave solution. Easy calculation shows that if the original KdV equation (1) is only perturbed by u xx without local delay, the corresponding Melnikov integral is which is clearly positive for c > 0 , then there is no solitary wave solution, either. So we can understand that the delay effect drives Δ negative while the perturbation drives Δ positive. The joint effect generates the unique solitary wave.

The Equation with Nonlocal Delay
In this section, we consider solitary wave solutions for Eq.
where � = d d . In terms of the system of first order ODEs, we get Let = √ , =̃ , then the above system is transformed to which is equivalent to the following system where ⋅ = d dz and = z . The time scale given by is said to be slow whereas that for z is fast, as long as ≠ 0 the two systems are equivalent. Thus we call system (21) the fast system and system (20) the slow system. If is set to zero in system (20), then the flow of that system is restricted to the set which is a three dimensional manifold of equilibrium for system (21) with = 0 . Note that when → 0 , we get → 0 , → and arrive at the non-delay Eq. (5). We want to investigate the persistence of the homoclinic orbit for small > 0 . In order to obtain an invariant manifold for sufficiently small > 0 by using GSP theory, it suffices to verify the normal hyperbolicity of M 0 . The linearization of system (21) Following the line of last subsection, we calculate the corresponding Melnikov integral We can know from the analysis that Δ is monotonic about c for c > 0 . It is clear that Δ is continuous about c for c > 0 , whose figure can be well depicted by numerical simulation, see Fig. 3. Owing to this, we can obviously find the fact that when c > 0 , Δ has a tendency to increase first and then decrease, which finally tends to negative infinity. Let Δ = 0 in (23), we can obtain that there exists a unique c * > 0 such that Then from the implicit function theorem, for each small nonzero value of , there is a unique value c = c( ) such that Dist = 0 . So we have the following theorem.

Theorem 3
For sufficiently small > 0 , there exists a unique speed c such that Eq. (4) with the weak generic nonlocal delay kernel has a solitary wave solution in the sense that the corresponding system (19) has a solution Γ which is heteroclinic to two equilibrium O 1 and O 2 lying O close to the origin, and that Γ lies uniformly C 1 O close to the unperturbed homoclinic orbit Γ and approaches Γ in C 1 norm as → 0.
The proof of Theorem 3 is exactly the same as that of Theorem 2.

Remark 3
Without the perturbation term u xx for Eq. (4), corresponding Melnikov integral is calculated to be let the above Δ = 0 , we can obtain that there is only c * = ( 35 128 ) 2 > 0 such that Then from the implicit function theorem, for each small nonzero value of , at present there is a unique value c = c( ) such that Dist = 0 . Thus, we find an interesting phenomenon that there is still a solitary wave solution for Eq. (4) with only the weak generic nonlocal delay kernel, which is different from the case of Eq. (4) with only the strong generic local delay kernel in Remark 2. Easy calculation shows that if the original KdV equation (1) is the perturbation only by u xx without nonlocal delay, corresponding Melnikov integral is which is clearly positive for c > 0 , then there is no solitary wave solution, which is the same as the case of the perturbed equation (4) without local delay in Remark 2.

Asymptotic Behavior
In this section, we shall deal with the asymptotic behavior of solitary waves obtained in the above sections by using the standard asymptotic theory.

Analysis for Equation (4) Without Delay and Perturbation
The Thus the following asymptotic behavior as → −∞ is derived where 1 is a constant and 1 can not be zero simultaneously [34]. If the first component of eigenvector g 1 is zero, the matrix A implies that the other components are zero, which implies the 1 ≠ 0.
Again since and from (27), we get 1 = 0 , and Thus we deduce the following asymptotic behavior as → +∞ where 2 is a constant and 2 can not be zero simultaneously. Then we claim that 2 ≠ 0 similarly.

Theorem 4
For any sufficiently small > 0 , there exist positive constants K and L such that Eq. (4) without delay and perturbation has a solitary wave Φ( ) with the following asymptotic properties and

Analysis for Equation (4) with Local Delay and Perturbation
In  We differentiate system (28) with respect to and denote Φ � ( ) = (̃( ),̃( ),̃( )) T , then we have The limiting system for system (29) as → −∞ is which is equivalent to which is a first-order system of ordinary differential equation, denote where m i (i = 1, 2, 3, 4) are arbitrary constants and f i (i = 1, 2, 3, 4) are eigenvectors of matrix B corresponding to eigenvalues i . Since So we can obtain from the form (32) that m 2 = 0, and As a result, the following asymptotic behavior as → −∞ is derived where i ,̃i,̂i (i = 1, 3, 4) and ii ,̃i i ,̂i i (i = 3, 4) are constants, and i (i = 1, 3, 4) can not be zero simultaneously. Similar to the analysis of Sect. 4.1, we obtain Similar to the discussion above, when → +∞ , the limiting system for system (29) is which is equivalent to which is a first-order system of ordinary differential equation, denote Z + = (̃+,̃1 + ,̃+,̃+) T , then system (34) can be rewritten as So we can obtain from the form (35) that n 1 = n 3 = n 4 = 0, and As a result, the following asymptotic behavior as → +∞ is derived where q 2 ,q 2 and q 2 ≠ 0 are constants, and p 2 can not be zero simultaneously.
In summary, we obtain the following theorem.

Remark 4
The asymptotic properties of solitary wave solutions for Eq. (4) are described in two cases: without delay and perturbation, with local delay and perturbation. The asymptotic properties of solitary wave solutions for Eq. (4) with nonlocal delay and perturbation could also be obtained by similar ways.

Conclusion
The existence of solitary wave solutions for the perturbed KdV equation with distributed delay is investigated in the work. Three cases are focused on: without delay and perturbation, with local delay and perturbation, with nonlocal delay and perturbation, the existence of solitary wave solutions is established in these cases. Finally, the asymptotic behaviors of solitary wave solution for Eq. (4) with the third case are also obtained. Specifically, constructing a Hamiltonian system with Hamiltonian function, the existence of solitary wave solution for Eq. (4) with the first case could be established. By using geometric singular perturbation theory and Melnikov method, the existence of solitary wave solutions is showed for Eq. (4) with the second and third cases. In addition, the asymptotic behaviors of solitary wave solution for Eq. (4) are analyzed by applying the standard asymptotic theory.
From Remark 2, if Eq. (4) is perturbed only by u xx without local delay, then there is no solitary wave solution. Moreover, if it is only local delayed without perturbation, then there is no solitary wave solution, either. In the case of local delay this velocity with which the original solitary wave to the KdV persists is determined by the balance between the two perturbations. From Remark 3, if Eq. (4) is only nonlocal delayed without perturbation, then the solitary wave solution can still persist. if Eq. (4) is perturbed only by u xx without nonlocal delay, which is the same as the case of perturbation and without local delay in Remark 2, and there is still no solitary wave solution.
On the basis of the above analysis, we find a new interesting result for Eq. (4) with only delay and without perturbation. Because the local delay is not integrable about the spatial variable x, it is just the integration over time t, then there is no solitary wave solution. While nonlocal delay is not only the integral over time t, but also the integral over space x, then there is a solitary wave solution. In short, the existence of solitary wave solution without perturbation depends on whether convolution is an integral about the space x. This new finding is different from that obtained in [29,30]. These results are helpful for understanding the effects of the addition of delay and perturbation on the existence of solitary wave solutions in the KdV equation.