Solitary wave solutions for a generalized KdV – mKdV equation with distributed delays ∗

In the past three decades, traveling wave solutions to the Korteweg–de Vries equation have been studied extensively and a large number of theoretical issues concerning the KdV equation have received considerable attention. These wave solutions when they exist can enable us to well understand the mechanism of the complicated physical phenomena and dynamical processes modeled by these nonlinear evolution equations. One can easily find abundant reports about it, such as [1–7]. And many powerful methods to construct exact solutions of KdV have been established and developed. Among these methods we mainly cite, for example, the bifurcation method of dynamical systems [8], (G′/G)-expansion method [9–11] and the sub-ODE method [12], Lie group theoretical methods [13] and so on. In this paper, we will use the geometrical singular perturbation theory and the linear chain trick to investigate solitary wave solutions of the generalized KdV–mKdV equation. Let us consider a generalized Korteweg–de Vries–modified Korteweg–de Vries (KdV– mKdV) equation Ut + ( α+ βU + γU ) Ux + Uxxx = 0, (1)


Introduction
In the past three decades, traveling wave solutions to the Korteweg-de Vries equation have been studied extensively and a large number of theoretical issues concerning the KdV equation have received considerable attention.These wave solutions when they exist can enable us to well understand the mechanism of the complicated physical phenomena and dynamical processes modeled by these nonlinear evolution equations.One can easily find abundant reports about it, such as [1][2][3][4][5][6][7].And many powerful methods to construct exact solutions of KdV have been established and developed.Among these methods we mainly cite, for example, the bifurcation method of dynamical systems [8], (G /G)-expansion method [9][10][11] and the sub-ODE method [12], Lie group theoretical methods [13] and so on.In this paper, we will use the geometrical singular perturbation theory and the linear chain trick to investigate solitary wave solutions of the generalized KdV-mKdV equation.
When p ∈ Z + , α = 0, β = 0 and γ = 0, Eq. ( 1) becomes the generalized KdV-mKdV equation The generalized KdV-mKdV equation with time delay has more actual significance.It is natural to ask weather the generalized KdV-mKdV equation with the delay possesses traveling wave solution.Zhao and Xu [18,19] discussed generalized KdV equations with time delay and established existence results of solitary waves solutions.
Our purpose is to apply the geometric singular perturbation theorem to establish the existence of solitary wave solutions of Eq. ( 2) with time delay, such as where τ is time delay, U xxx represents the dispersion effect and U xx means the backward diffusion, when τ is small, which also is the perturbation, the convolution f * U is denoted by where the kernel f : [0, +∞) → [0, +∞) satisfies the following normalization assumption: The average delay for the distributed delay kernel f (t) is defined as Here we consider the average delay τ is small.Note that equations of various types can be derived from Eq. ( 3) by taking different delay kernels.For instance, when we take the kernel to be f (t) = δ(t), where δ denotes Dirac's delta function, Eq. ( 3) becomes the corresponding undelayed perturbed mKdV equation While taking f (t) = δ(t−τ ), Eq. ( 3) becomes the following equation with discrete delay: We usually use the gamma distribution delay kernel (see [20]) where α > 0 is a constant, n is a integer, with average delay τ = n/α > 0. Two special cases are called weak generic kernel and strong generic kernel, respectively.A solitary wave for Eq. ( 3) is a special traveling wave solution U (x, t) = ϕ(ξ) = ϕ(x − ct), where c > 0 is speed and ϕ(ξ) satisfies the following functional differential equation: with here = d/dξ and lim ξ→±∞ ϕ(ξ) = 0.
Taking the translation u = ϕ/ p √ c − α and (i) = d i /dz i , i = 1, 2, 3, z = √ c − αξ, Eq. ( 6) with Eq. ( 7) could be rewritten as with If we find a solitary solution (u, τ, c, z) to Eq. ( 8), then the corresponding (ϕ, τ, c, ξ) is our solitary wave solution to Eq. ( 6) and, therefore, to the original Eq. ( 3).Under the assumption that the distributed delay kernel f (t) is the weak generic kernel, our main idea is to change the existence problem for the functional differential Eq. ( 8) into the existence of a homoclinic in two-dimensional invariant manifold.
By using the linear chain trick, Eq. ( 6) with the weak generic delay can be transformed into a nondelay three-dimensional ordinary differential system.When the delay τ is sufficiently small, the three-dimensional ordinary differential system is a standard singularly perturbed system.The rest part of this paper is organized as follows.At first, we briefly introduce some basic lemmas.And we will use these lemmas to establish results on solitary wave for nondelay Eq. ( 6).At last, by employing the geometrical singular perturbation theory, we will prove there exist a solitary wave solution for Eq. ( 6) with the weak generic delay and τ is sufficiently small for a particular wave speed.

Preliminaries
For convenience, we present some results in the paper [9,21] and establish a lemma, which will be employed in the proof of our main theorem.
Lemma 1 (Geometric singular perturbation theorem).For the system where x ∈ R n , y ∈ R l and is a real parameter, f , g are C ∞ on the set V × I, where V ∈ R n+l and I is an open interval, containing 0. If when = 0, the system has a compact, normally hyperbolic manifold of critical points M 0 , which is contained in the set {f (x, y, 0) = 0}.Then for any 0 < r < +∞, if > 0, but sufficiently small, there exists a manifold M : (i) which is locally invariant under the flow of (9); (ii) which is C r in x, y and ; (iii) M = {(x, y): x = h (y)} for some C r function h (y) and y in some compact K; (iv) there exist locally invariant stable and unstable manifolds W s (M ) and W u (M ) that lie within O( ), and are diffeomorphic to W s (M 0 ) and W u (M 0 ).
Then it will be useful for referencing to know what the corresponding results are in the nondelay case.Because the questions, that we will address are the questions of persistence of solitary waves when the delay is small.
In traveling wave form, with U (x, t) = ϕ(ξ), ξ = x − ct and c > 0, the nondelay equation (that is, Eq. ( 4) with f (t) = δ(t), τ = 0, i.e., generalized KdV equation ( 6)) reads where = d/dξ.Using the boundary condition at ±∞ and integrating Eq. ( 10) once, we yield to the equation where • = d/dz.We get an equivalent form We are now in a position to yield the existence of a solitary wave solution of the nondelay equation.

Existence of solitary waves
In this section, we shall mainly analyze Eq. (8) for solitary waves, which in the particular case when the kernel f is the first case of the two in Eq. ( 5), the weak generic delay case.The corresponding calculations for the strong kernel are similar but a little complicated and will be omitted.Recall that the parameter τ measures the delay and u (2) means perturbation.The main result can be stated as follow.
Proof.According to (5), we take the function f (t) as and we define w by Differentiating with respect to z, we can obtain that Using the boundary condition at −∞, Eq. ( 8) can be integrated once to yield the equation where If we further denote u = v, v = w, then Eq. ( 8) with the kernel given above can be replaced by the system From system (12) and Eq. ( 13), one has Note that if τ = 0, then System (13) reduces to the following ordinary differential equation: For τ > 0, system (13) defines a system of ODEs whose solutions evolve in the three-dimensional (u, v, w) phase space.In this phase space, there are critical points at O(0, 0, 0), The linearized matrix of (13) at critical point O(0, 0, 0) is The eigenvalues λ of this matrix satisfy This equation has two real positive roots 1/(c A solitary wave solution of the original equation will exist if among the solutions of system ( 13), there exists a homoclinic orbit to the critical point O(0, 0, 0).
We want to show that for τ > 0 but very small, the positive branch of the onedimensional stable manifold of O(0, 0, 0) for system (13), W s τ (O), connect to the origin.When τ = 0, system (13) does not define a dynamical system in R 3 .By introducing a new independent variable η defined by z = τ η, the flow system (13) becomes the following fast system: since the time scale z is slow and η is fast, where • = d/dη.If τ > 0 and sufficiently small, flow system (13) and fast system ( 14) are equivalent.Consider the slow system (13), for τ = 0, then the flow of that system is confined to the set which is, therefore, a two-dimensional invariant manifold for system (13).If M 0 is normally hyperbolic, then for sufficiently small τ > 0, Lemma 1 provides us with a twodimensional invariant manifold M τ for system (13), studying the system (13) reduced to this manifold, the dimensionality is reduced back to 2 and the existence of the homoclinic orbit we are seeking can be established.
In order to check that the invariant manifold M 0 is a normally hyperbolic in the sense of Fenichel [9,21], we need to check that the linearization of the fast system (14), restricted to M 0 , has precisely dim M 0 eigenvalues on the imaginary axis, with the remainder of the spectrum being hyperbolic.The linearization of Eq. ( 14) restricted to M 0 is given by the following matrix: which has three eigenvalues 0, 0, 1/(c √ c − α).Thus M 0 is normally hyperbolic.According to geometric singular perturbation theory, there exist a locally invariant two-dimensional manifold M τ with τ > 0 but sufficiently small, under the flow of system (13), which can be written in the form Control, 2014, Vol. 19, No. 4, 551-564 where the function g is a smooth function defined on a compact domain, and it satisfies g(u, v, 0) = 0.
By substituting into slow system (13), g must satisfy Since τ is small, we attempt solutions of this partial differential equation in the form of regular perturbation series in τ .From Eq. ( 15), we can see when τ = 0, g is zero, then let g be g(u, v, τ Substituting Eq. ( 16) into Eq.( 15) and comparing powers of τ yields, which implies We would study the flow of system ( 13) restricted to M τ and show that it has a solitary solution.The slow system (13) restricted to M τ is, therefore, given by where G(z) = z −∞ u p−1 v 2 dξ, by the same way, we have G(z) = G(u, v, τ ).It is easily verified that for τ > 0, system (17) still has critical point (u, v) = (0, 0).
For convenience, we consider delay parameter τ and wave speed c as variables, then system ( 17) is equivalent to So we can study the flow in (u, v, τ, c) ∈ R 4 .We seek homoclinic orbits for system (18) with small τ .The critical point 0 can, in reference to system (18), be construed as a surface of critical point, say S, parameterized by c, τ , i.e., critical point (u, v) = (u(c, τ ), v(c, τ )) = (0, 0).This in turn spawns an unstable manifold W u τ (S) and stable manifold W s τ (S) which meet in the curve at τ = 0, namely, the homoclinic orbit found already in Lemma 2. Furthermore, by Lemma 1, W u τ (S) and W s τ (S) must still cross hyperplane v = 0.In the set v = 0, we parameterize W u and W s , respectively, near the intersection away from the critical point 0 as u = h − (c, τ ) and u = h + (c, τ ).
We next define and observe that zeroes of d render homoclinic orbits.From Lemma 2, there are homoclinic orbits independently of c when τ = 0, we have that d(c, 0) = 0, and thus let d(c, τ ) = τ d(c, τ ).Then we have .
If there exists a (unique) value of c = c(τ ) for τ small, near to c = c(0), such that hold, then it is a simple application of the Implication function theorem to see that there is a curve of homoclinic orbits exists.Now we consider the Eq. ( 17), for which an intersection of W u (S) and W s (S) is sought.We need to establish the existence of M (c) defined in (19).
In fact the variational equation for system (18), the differential form with τ = 0 can be calculated as For the tangent spaces Π ± (0) of the invariant manifold W u τ (S) and W s τ (S), there are three tangent vectors to W u and W s at z = 0 that are easily found (when τ = 0, v(0) = 0) where the u of η 2 satisfies u > p (p + 1)/β, i.e., δ < 0. It can be checked that where π is a permutation of (1, 2, 3).Then we can see that the equation for the form du ∧ dv ∧ dc can be calculated as (τ = 0) Similarly, the form du ∧ dv ∧ dc , when applied to the tangent space Π ± (z) at η i z, i = 1, 2, 3, can actually be calculated.Since So letting p ± (z) = du ∧ dv ∧ dc(Π ± (z), we obtain It is easy to check that p ± → 0 as z → ±∞.Eq. ( 20) can then be easily solved to render which shows that p ± (0) = δ(∂h ± /∂τ ).Then we get and From Eq. ( 21) and Eq. ( 22), we obtain that www.mii.lt/NA

Conclusion
In this work, we establish the existence result of solitary wave solutions for a generalized KdV-mKdV equation with time delay under assumption that the distributed delay kernel f (t) is the weak generic kernel.We could obtain the similar result for the strong generic kernel.Our methods are geometrical singular perturbation theory and the linear chain trick.Furthermore, we investigate the traveling wave solutions for the generalized KdV-mKdV equation by using (G /G)-expansion method.We will leave this for future work.