# Existence of Solitary Waves in a Perturbed KdV-mKdV Equation

Authors: Chengqun Li; Minzhi Wei; Yuanhua Lin
Journal: Journal of Mathematics
 (2021)

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## Abstract

In this paper, we establish the existence of a solitary wave in a KdV-mKdV equation with dissipative perturbation by applying the geometric singular perturbation technique and Melnikov function. The distance of the stable manifold and unstable manifold is computed to show the existence of the homoclinic loop for the related ordinary differential equation systems on the slow manifold, which implies the existence of a solitary wave for the KdV-mKdV equation with dissipative perturbation.


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## Body

## 1. Introduction

KdV-mKdV equation plays a most important role in soliton physics and appears in many practical scenarios such as thermal pulse and wave propagation of the bound particle [1]. Recently, many mathematicians are interested in studying traveling wave solutions for the KdV-mKdV equation, and there are many tools to find the traveling wave solutions, such as the inverse scattering method [2], the double subequation method [3], the sine-cosine method [4, 5], the Darboux transformation [6], the Exp-expansion method [7, 8], and the bifurcation method of dynamic systems [9]. KdV-mKdV equation is one of the most popular soliton equations; therefore, there were widely related investigations in [10–14]. On the contrary, based on the relationship between the solitary wave solution and the homoclinic orbit of the associated ordinary differential equations, Fan and Tian [15] proved that the solitary wave persists the singularly perturbed mKdV-KS equation from the geometric singular perturbation point of view, when the perturbation parameter is suitably small. Mansour proved the existence of traveling wave solutions of some singular perturbed equations [16] by using the geometric singular perturbation theory. Yan et al. [17] investigated a perturbed generalized KdV equation. They proved the existence of solitary waves and periodic waves by applying the geometric singular perturbation theory and the regular perturbation analysis for a Hamiltonian system.Solitary waves are a special form of ultrashort pulses, which maintain a constant shape, amplitude, and velocity during their propagation. From the point of view of physics, solitary wave is a special product of the nonlinear effects of matter. Mathematically, it is a class of stable, energy-finite, nondispersive solutions of some nonlinear partial differential equations. As the particular characteristics of the solitary wave, we shall use Fenichel persistence theorem [18, 19] to study the existence of the solution for the singularly perturbed KdV-mKdV equation in this paper.(1)Ut+UpUx+UqUx+Uxxx+εUxx+Uxxxx=0,where p,q∈Z+ and p<q, not studied by the same method before. We try to investigate the existence problem of solitary wave solutions of (1) by using the geometric singular perturbation technique.


## 2. Fenichel Theory

In this section, we review the necessary theory that we will use for our discussion. We take the exposition in Christopher [19]. For details, one can resort to Fenichel [18] or Jones [19]. Geometric singular perturbation theory was first given by Fenichel and is often referred to as Fenichel theory. A comprehensive overview of the theory, as well as new proofs of many theorems and detailed examples of applications, was given by Jones [19].Consider the standard fast-slow system:(2)f˙t=Fft,st,ε,g˙t=εGft,gt,ε,where ⋅ is the derivative to t, 0<ε≪1 is a real parameter, f=f1,f2,…,fnfT∈Rnf, g=g1,g2,…,gngT∈Rng, nf+ng=n, maxf˙=maxg˙, and f corresponds to fast directions and g corresponds to slow directions. f,g are C∞ on the set U×V, where U⊂Rn and V is an open interval containing 0. Assume that, for ε=0, the system has a compact normal hyperbolic manifold M0 which contained in the set fx,y,0=0. The manifold M0 is hyperbolic normally if the linearization of (2) at each point in M0 has exactly ng eigenvalues with zero real part, where ng is the dimension of the center point.Fenichel’s Persistence Theorem 1. Under the assumption described above, ifε>0 is small sufficiently, there exists a function Mε defined on D such that the manifold Mε is invariant locally under the flow of (2). Moreover, Mε is Cr-smooth for any r<+∞, and Mε is CrOε close to M0.With a change of time scaleτ=εt,′=d/dt, system (2) is equivalent to(3)εf′=Ff,g,ε,g′=εGf,g,ε,when ε≠0, systems (2) and (3) are equivalent, system (2) is called the fast system, and system (3) is called the slow system. Geometric singular perturbation theory exploits a differential equation’s geometric structures, such as its slow (center) manifolds and their fast stable and unstable fibers.


## 3. Singular Perturbation Analysis to the KdV-mKdV Equation

For a given constantc>0, substituting Ux,t=ux−ct=uξ into (1), it reduces to an ordinary differential equation(4)−cu′ξ+upu′ξ+uqu′ξ+u″′+εu″+u″″=0.Under the condition ofu±∞=0, u′±∞=0, and u″±∞=0, integrating this equation, and setting the integral constant to be zero, we have(5)−cu+1p+1up+1+1q+1uq+1+u″+εu′+u″′=0.Suppose thatuξ is a continuous solution of (5) for ξ∈−∞,+∞ and uξ satisfies limξ⟶∞uξ=m,limξ⟶−∞uξ=n. If m=n, uξ is called a solitary wave solution. If m≠n, uξ is called a kink (antikink) wave solution. A solitary wave solution corresponds to a homoclinic orbit of (5), a kink (antikink) wave solution corresponds to a heteroclinic orbit of (5), and a periodic orbit corresponds to a periodic traveling wave solution of (5).ODE (5) can be given as(6)dudξ=v,dvdξ=w,εdwdξ=cu−1p+1up+1−1q+1uq+1−εv−w.System (6) is obviously formulated on a slow time scale because of the location of the small parameter ε. The corresponding fast system is(7)dudσ=εv,dvdσ=εw,dwdσ=cu−1p+1up+1−1q+1uq+1−εv−w.Generally, system (6) is referred to as the slow system since the time scale ξ is slow. And system (7) is referred to as the fast system since the time scale σ is fast. The two systems are equivalent when ε>0.In this section, we will find the equilibrium points of system (7) and determine the local behavior in the neighborhood of these equilibria. Let z=u,v,wT and(8)GZ,c,ε=εvεwcu−1p+1up+1−1q+1uq+1−εv−w.System (7) can be formulated as(9)dZdσ=GZ,c,ε.There areq+1 equilibria satisfying GZ,c,ε=0, that is, Z0=0,0,0,Zi=ui,0,0,i=1−q, where c−1/puip−1/quiq=0.Let(10)A0=DFZ0,c,ε=0ε000εc−ε−1.Then, the characteristic equation forA0 is detA0−λI=−λ3−λ2−ε2λ+ε2c, and the discriminant of the above equation is Dc,ε=ε2−4ε4+1−18c−27c2ε2+4c; it means Dc,ε<0 for any positive c and ε, and matrix A0 has two conjugate complex and one real eigenvalues λ1,λ2,λ3. By the Viete theorem, λ1,λ2,andλ3 will satisfy(11)λ1+λ2+λ3=−1,λ1λ2λ3=ε2c.It is easy to know that ifλ3>0, both complex eigenvalues λ1 and λ2 have a negative real part. So, the stable subspace at Z0, named as ESZ0, is two-dimensional for all positive c and ε. And EUZ0, the unstable subspace at Z0, is one-dimensional. Let WSZ0 and WUZ0 be the local stable and unstable manifolds, respectively. By dynamical linearization theory, it holds that DimWSZ0=2 and DimWUZ0=1. As DimWSZ0+DimWUZ0=3 in R3, it is not enough to prove that a homoclinic orbit and periodic orbit exist. We shall solve this question in the next section.


## 4. Persistence of Solitary Waves

Settingε=0, (7) has Jacobian(12)000000cu−1p+1up+1−1q+1uq+10−1.SinceM0 can be characterized as the graph of a function, we have that(13)M0=u,v,w|w=gu,v,ε,where g depends on ε smoothly and satisfies gu,v,0=cu−1/p+1up+1−1/q+1uq+1. Therefore, (6) can be reduced to the following differential equation on Mε:(14)dudξ=v,dvdξ=gu,v,ε,which has the limiting form on M0of(15)dudξ=v,dvdξ=cu−1/p+1up+1−1/q+1uq+1.SinceM0 is smooth, the vector field in (6) is smooth, and Mε can be characterized as the graph of a function; we can expand gu,v,w in ε for ε>0 sufficiently small, that is,(16)w=cu−1p+1up+1−1q+1uq+1+εg1u,v+Oε2.Substituting (16) into the last equation of slow system (6) and comparing coefficients of ε yield(17)g1u,v=−c+1v+upv+uqv.In the following, we will check whether there exists a homoclinic orbit with smallε or not. From the original equations, we find the origin(18)Ou,v=uc,ε,vc,ε=uc,ε,0=cu−1p+1up+1−1q+1uq+1+Oε2=0,0,is a critical point for (6), and it is still a critical point and must belong to Mε; thus, we shall hunt for the orbits homoclinic to the origin O. The critical point O can be construed as a surface of critical points, noted as S, which is parameterized by c and ε; it is said that S=uc,ε,vc,ε, and it spans an unstable manifold WuS and a stable manifold WsS. When ε=0, WuS and WsS intersect in a curve, which is the curve (homoclinic orbits) we are hunting for, denoted by Γ. Therefore, under the condition v=0, for η1=h+c,ε and η2=h−c,ε, we parameterize WuS and WsS near the intersection away from the critical point, respectively.Definedc,ε=h+c,ε−h−c,ε and observe the zeros of d. The number of zeros of d corresponds to the number of homoclinic orbits. Since there are homoclinic orbits independent on c when ε=0, we have dc,0=0, and dc,ε=dc,0+εd˜c,0. The Melnikov function is given by(19)Mc,0=d˜c,0=∂h+∂ε−∂h−∂ε|ε=0.Forε=0, define the parameter equation of Γ as follows:(20)u0t=u1˜sechct,v0t=−u1˜csechcttanhct,where −c/2+1/p+1p+2u˜1p+1/q+1q+2u˜1q=0. For this, we use the argument of the Melnikov function [20]; the Melnikov function is expressed as(21)Mc,Z0=∫−∞+∞vcu−1p+1up+1−1q+1uq+1∧0−c−1v+upv+uqvdt=∫−∞+∞v−c−1v+upv+uqvdt,where ab=a1b2−a2b1,a=a1,a2∈R2,b=b1,b2∈R2.On different values ofp and q, different Melnikov function values are obtained. There are some values as follows:(22)1p,q=1,3,Mc,Z0=−16cu˜12124c+1+4+2πu˜1+πu˜13,(23)2p,q=1,4,Mc,Z0=cu˜12−23c+1+π8u˜1+16105u˜14,(24)3p,q=2,3,Mc,Z0=cu˜12−23c+1+445u˜12−π16u˜13,(25)4p,q=2,4,Mc,Z0=cu˜12−23c+1+445u˜12+16105u˜14.Formulas (22)–(25) show that the Melnikov function Mc,Z0 has unique zero in different p,q, respectively. It means that all the corresponding orbits have a unique intersection on the u-axis, which is a unique saddle. In other words, the homoclinic orbits exist when ε is sufficiently small; hence, the solitary wave solution of KdV-mKdV equation (3) persists when p,q=1,3, p,q=1,4, p,q=2,3, and p,q=2,4 (see Figure 1). Similarly, we can compute more values for p and q and then prove that Mc,Z0 has unique zero.Figure 1
The level set ofHu,v. (a) p,q=1,3. (b) p,q=1,4. (c) p,q=2,3. (d) p,q=2,4.
(a)(b)(c)(d)


## 5. Conclusion

The perturbed KdV-mKdV equation is considered by using the geometric singular perturbation theory, and the existence of solitary waves has been proved for sufficiently small perturbing parameters. We have proved the persistence of the slow manifold under perturbation, and then we have constructed the wave as homoclinic orbits in the transverse intersection of appropriate stable and unstable manifolds in this slow manifold. It is a pity that the geometric singular perturbation theory cannot give an explicit expression for the proved solitary wave solution. However, the introduced method should have wide applicability to persistence questions for the solitary wave solution of many other perturbed equations; it may be an interesting work in the future.



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