Grotunits=360

# Does solitary wave solution persist for the long wave equation with small
perturbations?

Hang Zheng1,2,3 Y-H. Xia1111Corresponding author: Y-H. Xia, Email:
<EMAIL_ADDRESS><EMAIL_ADDRESS>
1\. Department of Mathematics, Zhejiang Normal University, Jinhua, 321004,
China
<EMAIL_ADDRESS><EMAIL_ADDRESS>
2\. Department of Mathematics and Computer, Wuyi University, Wuyishan, 354300,
China
3\. Fujian key Laboratory of Big Data Application and Intellectualization for
Tea Industry
<EMAIL_ADDRESS><EMAIL_ADDRESS>

###### Abstract

Persistence of solitary wave solutions of the regularized long wave equation
with small perturbations are investigated by the geometric singular
perturbation theory. Two different kinds of the perturbations are considered
in this paper: one is the weak backward diffusion and dissipation, the other
is the Marangoni effects. Indeed, the solitary wave persists under small
perturbations. Furthermore, the different perturbations do affect the proper
wave speed $c$ ensuring the persistence of the solitary waves. Finally,
numerical simulations are utilized to confirm the theoretical results.

Key Words: Long wave equation, solitary wave solution, geometric singular
perturbation.

##  1 Introduction

###  1.1 History of the perturbed long wave equation

####  1.1.1 Unperturbed long wave equation

Shallow water models have been one of the research hotspots of nonlinear
partial differential equations. There are many shallow water models appearing
in different forms, such as KdV equation (Korteweg and de Vries [28]), CH
equation (Camassa and Holm [9]), DP equation (Degasperis and Procesi [11]),
BBM equation (Benjamin et al.[1]), GN equation (Green and Naghdi [21]), and so
on. The classical KdV equation is formulated as follows

$u_{t}+uu_{x}+u_{xxx}=0.$ (1)

In 1967, Peregrine [37, 36] put forward a regularized long-wave (RLW, for
short) equation which regarded as a alternative model to KdV equation of the
form

$u_{t}+u_{x}+uu_{x}-u_{xxt}=0.$ (2)

The equation is of many attractive features which can be concluded as follows
(see [2]): (i) more closely matching that of the full Euler equations to
describe the two-dimensional motion of the free-surface oscillations of an
ideal liquid under the force of gravity, (ii) it is better to integrate
numerically. Later, to describe surface water waves in a uniform channel, a
regularized form of the KdV equation was studied by Benjamin et al. (see [1]),
and the BBM equation is given by

$u_{t}+uu_{x}-u_{xxt}=0.$ (3)

####  1.1.2 KS perturbation

However, in a more realistic situation, it is unavoidable that the existence
of uncertainty or perturbations. Thus, small perturbation terms should be
considered in modelling the problems. For shallow water wave equation, weak
backward diffusion $u_{xx}$ and dissipation $u_{xxxx}$ are called Kuramoto-
Sivashinsky (KS, for short) which are one of common perturbations. To explain
the wave motions on a liquid layer over an inclined plane, Topper and Kawahara
[41] analyzed a PDE as follows:

$u_{t}+uu_{x}+\alpha u_{xx}+\beta u_{xxx}+\gamma u_{xxxx}=0,$ (4)

where $\alpha$, $\beta$ and $\gamma$ are positive parameters. If the inclined
plane and surface tension are relatively long and weak, it implies $u_{xx}$
and $u_{xxxx}$ are real small terms such that they are considered to be some
small perturbations. Then Eq. (4) is transformed to the following equation

$u_{t}+uu_{x}+u_{xxx}+\varepsilon(u_{xx}+u_{xxxx})=0,$ (5)

where $0<\varepsilon\ll 1$, Eq. (5) is usually called as a perturbed KdV
equation. There are many monographs studying the existence of the traveling
wave solution under KS perturbation. For instance, Derks and Gils [13], Ogawa
[35] investigated the existence of traveling wave solution of Eq. (5). Yan et
al. [44] considered a generalized KdV equation with KS perturbation and proved
the existence of solitary wave and periodic wave solution. Chen et al. [6, 4,
5] explained the existence of traveling wave solution of a generalized KdV
equation and BBM equation with KS perturbation.

####  1.1.3 Marangoni effect perturbation

In addition to KS perturbation, Marangoni effect (ME) is an alternative
perturbation in modelling, see [22]. Generally, the change of surface tension
gradient at the interface of two phases leads to the occurrence of ME. In
other words, ME presents an additional nonlinear term $(uu_{x})_{x}$ when it
is considered on the surface of a thin layer. Velarde et al. [42] proposed the
following equation

$u_{t}+2\alpha_{1}uu_{x}+\alpha_{2}u_{xx}+\alpha_{3}u_{xxx}+\alpha_{4}u_{xxxx}+\alpha_{5}(uu_{x})_{x}=0$
(6)

which involving ME into the one-way long-wave assumption. And nonlinear term
$(uu_{x})_{x}$ generates from ME, which describes the opposite to the Bénard
convection. Sun and Yu [40] proved the existence of periodic wave and solitary
wave solution of BBM equation under KS and generalized ME perturbations, the
equation is described by

$(u^{3})_{t}+(u^{4})_{x}+u_{xxx}+\epsilon E_{i}=0,\ i=1,2,$ (7)

where $E_{1}=u_{xx}+u_{xxxx}$ and
$E_{2}=((\alpha_{0}+\alpha_{1}u+\alpha_{2}u^{2})u_{x})_{x}$. It shows that
traveling wave are very sensitive to KS and weak ME perturbations.

####  1.1.4 Model formulation

In recent years, more and more papers concern the influence of distributed
delay in the evolution of equations. Delays are more closely to show realistic
phenomenon than the usual kind of models without delay. In 1989, Britton [3]
derived a model for a single biological population of the form

$u_{t}=u(1+\alpha u-(1+\alpha)(f\ast u))+\bigtriangleup u,$ (8)

where $f\ast u$ is a local convolution in the spatial-temporal and defined by

$(f\ast u)(x,t)=\int^{t}_{-\infty}f(t-s)u(x,s)ds.$ (9)

The kernel $f(t)$ satisfies $f:[0,\infty)\rightarrow[0,\infty)$ and

$\int^{\infty}_{0}f(t)dt=1,\ \ tf(t)\in L^{1}((0,\infty),\mathbb{R}).$

Here, we suppose that the kernel $f(t)$ is the strong generic delay kernel
given by

$f(t)=\frac{4t}{\tau^{2}}e^{-{\frac{2t}{\tau}}}$ (10)

with averaging delay $\tau=\displaystyle\int^{\infty}_{0}tf(t)dt$. In this
paper, we study a delayed RLW equation with two different kinds of
perturbations (KS and ME) as follows

$u_{t}+u_{x}+(f\ast u)u_{x}-u_{xxt}+\tau P_{i}=0,\ \ (i=1,2),$ (11)

where $0<\tau\ll 1$ (a sufficiently small parameter), $P_{1}=u_{xx}+u_{xxxx}$
and $P_{2}=u_{xx}+(uu_{x})_{x}+u_{xxxx}$. $P_{1}$ is the KS perturbation and
$P_{2}$ represents the ME perturbation. $f\ast u$ is a local convolution in
the spatial-temporal, which is defined in (9) and (10). It should be noted
that BBM equation can be derived from equation (11). Most of the articles
concern the traveling wave solution of unperturbed RLW equation (see e.g.,
Bona et al. [2], Dag et al. [14], Gardner et al. [23]). Sun and Yu [40]
considered the generalized ME perturbation in BBM equation by Chebyshev
criteria and Abelian integral. Note that it is actually a regular perturbation
when perturbed equation transformed into its travelling wave system. However,
there is no paper considered the perturbed RLW equation with two different
perturbations and delay by singular perturbation theory (GSP, for short). This
paper aims to study the persistence of solitary wave solution of (11) by GSP
theory.

###  1.2 Motivation and contributions

It is not difficult to show the existence of a solitary wave solution to an
unperturbed RLW equation. Does the solitary wave solution persist under small
perturbations? In this paper, we pay particular attention to answer the
question. When a perturbed equation is a singularly perturbed system, the
geometric singular perturbation theory (Fenichel [19] and Jones [27]) is an
effective tool to prove the persistence of traveling wave solution. There
exist a lot of articles on this issue including perturbed KdV equation (see
e.g., Chen et al. [6], Derks and Gils [13], Ogama [35]), perturbed CH equation
(see e.g., Du et al. [16, 15], Qiu et al. [38]), random systems (see e.g., Li
et al. [29, 30, 31]), perturbed BBM equation (see e.g., Chen et al. [4], Sun
and Yu [40]), Fitzhugh-Nagumo equation (see Liu et al. [33]), biological model
(see e.g., Chen and Zhang [10], Wang and Zhang [43]), perturbed sine-Gordon
equation (see Derks et al. [12]) and so on. Du and Qiao [18] utilized GSP
theory, Fredholm orthogonality and asymptotic theory to prove the existence of
traveling wave solutions in a Belousov-Zhabotinskii system with delay.
Combining Poincaré-Bendixson theorem, Du et al. [17] also confirmed the
persistence of solitary wave solution of a generalized Keller-Segel system
with small cell diffusion.

Usually, it is necessary to compute the Melnikov function by using the GSP
theory. For the perturbed Hamiltonian system, Melnikov function is associated
to the Abelian integral. It is an effective skill to prove existence of
traveling wave solutions by detecting monotonicity of the ratio of Abelian
integral (MRAI). For example, Derks and Gils [13], Ogawa [35], Chen et al. [4,
6, 5], Sun et al. [39, 40] computed the MRAI of corresponding perturbed
equations to prove the existence of traveling wave solution. Besides, another
effective way is to directly compute the exact representation of Melnikov
function. Qiu et al. [38], Zhu et al. [45], Cheng and Li [7] computed the
explicit expression of Melnikov function to discuss the existence of traveling
wave solution in a perturbed generalized BBM equation, generalized CH equation
delayed DP equation, respectively.

Motivated by aforementioned works, we use the singular perturbation theory to
study the delayed RLW equation (11) with two different kinds of perturbations
(KS and ME). We prove that the solitary wave solution does persist under small
perturbations. To this end, we proceed with three steps.

Step 1. The exact expression of solitary wave solution of unperturbed RLW
equation are obtained by using dynamical system method.

Step 2. The homoclinic orbits are constructing by tracking invariant manifolds
of corresponding ODEs such that the singular perturbed system reduces to a
regular perturbed system.

Step 3. The analytical expression of Melnikov integral is given for perturbed
RLW equation. Some elaborate analytical skills are employed to discuss the
simple zero of the Melnikov integral associated to the perturbed RLW equation,
due to its complexity. Finally, by Melnikov method and bifurcation theorem, we
conclude that the solitary wave solution persists at proper speed.

###  1.3 Outline of the paper

We present some preliminary concept and lemmas in next section. We obtain the
exact expression of solitary wave solution of unperturbed RLW equation in
section 3. In section 4, we construct the homoclinic orbits and compute the
Melnikov integral associated to the perturbed RLW equations. Then after
analysis of the Melnikov integrals, we present our main results in this
section. Finally, we present the numerical simulations to verify the
theoretical results.

##  2 Preliminaries

In this section, we introduce some known results which will be used in our
proof.

###### Lemma 1

(see [19]) Consider the system

$\begin{cases}x_{1}^{\prime}=f(x_{1},x_{2},\epsilon),\\\
x_{2}^{\prime}=\epsilon g(x_{1},x_{2},\epsilon),\end{cases}$ (12)

where ${}^{\prime}=\frac{d}{dt}$, $x_{1}\in\mathbb{R}^{n}$,
$x_{2}\in\mathbb{R}^{l}$ and $\epsilon$ is a real parameter and positive, the
function $f$ and $g$ are $C^{\infty}$ on a set $U\times I$ where
$U\subseteq\mathbb{R}^{n+l}$ is open, and $I$ is an open interval, containing
$0$. If $\epsilon=0$, the system has a compact, normally hyperbolic manifold
of points $M_{0}$ which is contained in the set
$\\{f(x_{1},x_{2},\epsilon)=0\\}$. Then for sufficiently small positive
$\epsilon$ and any $0<r<+\infty$. There exists a manifold $M_{\epsilon}$ such
that following hold:

(i) which is locally invariant under the flow of system (12);

(ii) which is $C^{r}$ in $x_{1}$, $x_{2}$ and $\epsilon$;

(iii) $M_{\epsilon}=\\{(x_{1},x_{2}):x_{1}=h^{\epsilon}(x_{2})\\}$ for some
$C^{r}$ function $h^{\epsilon}(x_{2})$ and $x_{2}$ in some compact $K$;

(iv) there exist locally invariant stable and unstable manifold
$W^{s}(M_{\epsilon})$ and $W^{u}(M_{\epsilon})$ lying within $O(\epsilon)$ and
being $C^{r}$ diffeomorphic to the stable and unstable manifold $W^{s}(M_{0})$
and $W^{u}(M_{0})$.

###### Lemma 2

If $\tau\rightarrow 0$, then $(f\ast u)(x,t)\rightarrow u(x,t)$.

_Proof._ According to expressions (9) and (10). The $(f\ast u)(x,t)$ is
defined by

$\begin{array}[]{rcl}(f\ast
u)(x,t)=\displaystyle\int^{t}_{-\infty}f(t-s)u(x,s)ds=\displaystyle\int^{t}_{-\infty}\frac{4(t-s)}{\tau^{2}}e^{-{\frac{2(t-s)}{\tau}}}u(x,s)ds.\end{array}$
(13)

Let $\lambda=-\frac{2(t-s)}{\tau}$, it yields

$\begin{array}[]{rcl}(f\ast u)(x,t)=-\displaystyle\int^{0}_{-\infty}\lambda
e^{\lambda}u(x,t+\frac{\lambda\tau}{2})d\lambda.\end{array}$ (14)

Hence, we have

$\begin{array}[]{rcl}\lim\limits_{\tau\rightarrow 0}(f\ast
u)(x,t)&=&-\lim\limits_{\tau\rightarrow
0}\displaystyle\int^{0}_{-\infty}\lambda
e^{\lambda}u(x,t+\frac{\lambda\tau}{2})d\lambda\\\
&=&-\displaystyle\int^{0}_{-\infty}\lambda e^{\lambda}u(x,t)d\lambda\\\
&=&-u(x,t)\displaystyle\int^{0}_{-\infty}\lambda e^{\lambda}d\lambda\\\
&=&u(x,t).\end{array}$ (15)

The proof is completed.

##  3 Solitary wave solution of the unperturbed RLW equation

In this section, the solitary wave solution of unperturbed equation
(11)$|_{\tau=0}$ is obtained by dynamical system method.

We first introduce the following transformations:

$u(x,t)=\phi(\xi),\ \ \ \xi=x-ct,$ (16)

then, equation (11)$|_{\tau=0}$ is converted to an ordinary differential
equation given by

$(1-c)\phi^{\prime}+\phi\phi^{\prime}+c\phi^{\prime\prime\prime}=0,$ (17)

where ${}^{\prime}=\frac{d}{d\xi}$. Integrating both sides of equation (17)
once with respect to $\xi$, we have

$(1-c)\phi+\frac{1}{2}\phi^{2}+c\phi^{\prime\prime}=g,$ (18)

where $g$ is the integration constant ($g\in\mathbb{R}$). According to actual
physical meanings, we are only interested in the case $c>0$.

Next, the bifurcation and phase portraits is studied by dynamical system
method.

###  3.1 Homoclinic orbit of the travelling wave system (18)

Eq. (18) is equivalent to the following planar system:

$\begin{cases}\phi^{\prime}=y,\\\
y^{\prime}=\displaystyle\frac{1}{c}\big{[}(c-1)\phi-\frac{1}{2}\phi^{2}+g\big{]}.\end{cases}$
(19)

It is easy to obtain the corresponding first integral

$\begin{array}[]{rcl}H(\phi,y)=\frac{1}{2}y^{2}-\frac{1}{c}[\frac{(c-1)}{2}\phi^{2}-\frac{1}{6}\phi^{3}+g\phi]=h.\end{array}$
(20)

Denote that

$\begin{array}[]{rcl}&&\Delta=(c-1)^{2}+2g,\ H(\phi_{1},0)=h_{1},\
H(\phi_{2},0)=h_{2},\\\ &&\phi_{1}=c-1-\sqrt{\Delta},\
\phi_{2}=c-1+\sqrt{\Delta}.\\\ \end{array}$ (21)

Obviously, system (19) has two equilibrium $(\phi_{1},0)$ and $(\phi_{2},0)$
when $\Delta>0$. Here, we suppose that $c>0$ in the sense of physics
significance.

###### Proposition 1

If $(c-1)^{2}>-2g$, system (19) has a homoclinic orbit $\Gamma$ surrounding
the center point $(\phi_{2},0)$ to one saddle point $(\phi_{1},0)$ when level
curves defined by $H(\phi,y)=h_{1}$. (see Fig. 1).

_Proof._ The linearized matrix of system (19) is

$A=\left(\begin{array}[]{ccccc}0&1\\\ \displaystyle\frac{c-1-\phi}{c}&0\\\
\end{array}\right).$ (22)

Thus, if $(c-1)^{2}>-2g$,

$J(\phi_{1},0)=\textmd{det}A_{(\phi_{1},0)}=\left|\begin{array}[]{ccccc}0&1\\\
\frac{\sqrt{\Delta}}{c}&0\\\ \end{array}\right|=-\frac{\sqrt{\Delta}}{c}<0,\ \
\textmd{Trace}(A)_{(\phi_{1},0)}=0,$ (23)

and

$J(\phi_{2},0)=\textmd{det}A_{(\phi_{2},0)}=\left|\begin{array}[]{ccccc}0&1\\\
-\frac{\sqrt{\Delta}}{c}&0\\\ \end{array}\right|=\frac{\sqrt{\Delta}}{c}>0,\ \
\textmd{Trace}(A)_{(\phi_{2},0)}=0.$ (24)

By the bifurcation theory of planar dynamical systems (see [32, 8]), we know
that $(\phi_{1},0)$ is a saddle point and $(\phi_{2},0)$ is a center point.
Then, if the level curves defined by $H(\phi,y)=h_{1}$, there exists a
homoclinic orbit $\Gamma$ (blue curve) surrounding the center point
$(\phi_{2},0)$ to one saddle point $(\phi_{1},0)$ (see Fig. 1).

This completes the proof.

(a) Phase portraits | (b) Homoclinic orbit
---|---

Fig. 1 Bifurcation of phase portraits of system (19) when $(c-1)^{2}>-2g$.

###  3.2 Solitary wave solution of the unperturbed RLW equation

According to Proposition 1 and (20), the expression of $y$ can be written as

$\begin{array}[]{rcl}y&=&\pm\sqrt{\frac{1}{3c}\phi^{3}+\frac{c-1}{c}\phi^{2}+\frac{2g}{c}\phi+\frac{2h_{1}}{c}}\\\
&=&\pm\sqrt{\frac{1}{3c}(\phi_{r}-\phi)(\phi-\phi_{1})^{2}},\end{array}$ (25)

where $\phi_{r}$ represents the right intersection of the homoclinic orbit
with the $\phi$ axis and $\phi_{r}=2\sqrt{\Delta}+c-1$, it leads to

$\frac{d\phi}{d\xi}=\pm\sqrt{\frac{1}{3c}(\phi_{r}-\phi)(\phi-\phi_{1})^{2}},$
(26)

which implies

$\xi=\int_{\phi_{r}}^{\phi}\frac{1}{\sqrt{\frac{1}{3c}(\phi_{r}-s)(s-\phi_{1})^{2}}}ds.$
(27)

Then, the parametric representation of homoclinic orbit (see Fig. 2(a)) can be
given by

$\begin{array}[]{rcl}\phi(\xi)=\phi_{r}-(\phi_{r}-\phi_{1})\textmd{tanh}^{2}(\frac{1}{2}\sqrt{\frac{\phi_{r}-\phi_{1}}{3c}}\xi).\end{array}$
(28)

Therefore, it corresponds to a bright solitary wave solution of Eq.
(11)$|_{\tau=0}$ (see Fig. 2(b)):

$\begin{array}[]{rcl}u(x,t)=\phi_{r}-(\phi_{r}-\phi_{1})\textmd{tanh}^{2}(\frac{1}{2}\sqrt{\frac{\phi_{r}-\phi_{1}}{3c}}(x-ct)).\end{array}$ (29) (a) Solitary wave | (b) Bright soliton
---|---

Fig. 2 Solitary wave solution and bright soliton when $(c-1)^{2}>-2g$.

Through the above analysis, we give the following theorem:

###### Theorem 1

For $(c-1)^{2}>-2g$, Eq. (11)$|_{\tau=0}$ has a solitary wave solution given
by (29).

##  4 Persistence of solitary wave solution with two kinds of small
perturbations

In this section, the GSP approach is employed to analyze the persistence of
solitary wave solution of Eq. (11) with KS and ME perturbations, respectively.

###  4.1 Solitary wave solution persists with KS perturbation

Substituting (16) into Eq. (11) with KS perturbation and integrating both
sides once with respect to $\xi$, we have

$(1-c)\phi+F+c\phi^{\prime\prime}+\tau(\phi^{\prime}+\phi^{\prime\prime\prime})=g,$
(30)

where $F=\displaystyle\int^{\xi}_{-\infty}\psi\phi^{\prime}ds$ and
$\psi(\xi)=\displaystyle\int^{\infty}_{0}\frac{4t}{\tau^{2}}e^{-{\frac{2t}{\tau}}}\phi(\xi+ct)dt$.

Next, we obtain

$\begin{array}[]{rcl}\displaystyle\frac{d\psi}{d\xi}&=&\displaystyle\int^{\infty}_{0}\frac{4t}{\tau^{2}}e^{-{\frac{2t}{\tau}}}\phi_{\xi}(\xi+ct)dt\\\
\vspace{.25cm}\hfil&=&\displaystyle\frac{1}{c}\displaystyle\int^{\infty}_{0}\frac{4t}{\tau^{2}}e^{-{\frac{2t}{\tau}}}\phi_{t}(\xi+ct)dt\\\
\vspace{.15cm}\hfil&=&\displaystyle\frac{1}{c}\displaystyle\int^{\infty}_{0}\frac{4t}{\tau^{2}}e^{-{\frac{2t}{\tau}}}d\phi\\\
\vspace{.15cm}\hfil&=&\displaystyle\frac{1}{c}\big{[}\displaystyle\frac{2}{\tau}\int^{\infty}_{0}\frac{4t}{\tau^{2}}e^{-{\frac{2t}{\tau}}}\phi(\xi+ct)dt-\frac{1}{\tau}\displaystyle\int^{\infty}_{0}\frac{4}{\tau}e^{-{\frac{2t}{\tau}}}\phi(\xi+ct)dt\big{]}\\\
\vspace{.15cm}\hfil&=&\displaystyle\frac{1}{c\tau}(2\psi-\zeta),\end{array}$
(31)

where
$\zeta=\displaystyle\int^{\infty}_{0}\frac{4}{\tau}e^{-{\frac{2t}{\tau}}}\phi(\xi+ct)dt$.

Similarly, differentiating both sides with respect to $\xi$, it gives

$\frac{d\zeta}{d\xi}=\frac{2}{c\tau}(\zeta-2\phi).$ (32)

Thus, combining Eq. (30), (31) and (32), we obtain a five-dimensional slow
system as follows

$\begin{cases}\phi^{\prime}=y,\\\ y^{\prime}=z,\\\ \tau z^{\prime}=(c-1)\phi-
F-cz+g-\tau y,\\\ c\tau\psi^{\prime}=2\psi-\zeta,\\\
c\tau\zeta^{\prime}=2(\zeta-2\phi).\end{cases}$ (33)

Setting $\tau=0$, the critical manifold is

$M_{0}=\\{(\phi,y,z,\psi,\zeta)\in\mathbb{R}^{5}\big{|}\psi=\frac{1}{2}\zeta=\phi,(c-1)\phi-\frac{1}{2}\phi^{2}-cz+g=0\\}.$

Taking $\chi=\tau\xi$, the corresponding fast system is

$\begin{cases}\dot{\phi}=\tau y,\\\ \dot{y}=\tau z,\\\ \dot{z}=(c-1)\phi-F-
cz+g-\tau y,\\\ \dot{\psi}=\displaystyle\frac{1}{c}(2\psi-\zeta),\\\
\dot{\zeta}=\displaystyle\frac{2}{c}(\zeta-2\phi),\end{cases}$ (34)

where $\cdot=\frac{d}{d\chi}$. Let $A$ be the linearized matrix of the fast
system (34) restricted to $M_{0}$. It is of the form

$A=\left(\begin{array}[]{ccccc}0&0&0&0&0\\\ 0&0&0&0&0\\\ \ast&0&-c&0&0\\\
0&0&0&\frac{2}{c}&-\frac{1}{c}\\\
-\frac{4}{c}&0&0&0&\frac{2}{c}\end{array}\right)$ (35)

that the eigenvalues of A are $0$, $0$, $-c$, $\frac{2}{c}$ and $\frac{2}{c}$.
Hence, $M_{0}$ is normally hyperbolic manifold (see [19]). And there exists a
submanifold $M_{\tau}$ which is a differentiable homeomorphism and locates
near $M_{0}$ at a distance $O(\tau)$ (see Lemma 1). The invariant submanifold
$M_{\tau}$ is represented by

$\begin{array}[]{rcl}M_{\tau}=\\{(\phi,y,z,\psi,\zeta)\in\mathbb{R}^{5}\big{|}\psi=\phi+p(\phi,y,\tau),\zeta=2\phi+q(\phi,y,\tau),\\\
(c-1)\phi-\frac{1}{2}\phi^{2}-cz+g=\omega(\phi,y,\tau)\\},\end{array}$ (36)

where $g$, $h$ and $\omega$ are smooth enough with respect to $\tau$ and
satisfy

$p(\phi,y,0)=q(\phi,y,0)=\omega(\phi,y,0)=0.$

Therefore, $g$, $h$ and $\omega$ can be Taylor’s expansion on $\tau$

$\begin{array}[]{rcl}p(\phi,y,\tau)&=&\tau p_{1}+\tau^{2}p_{2}+\cdots,\\\
q(\phi,y,\tau)&=&\tau q_{1}+\tau^{2}q_{2}+\cdots,\\\
\omega(\phi,y,\tau)&=&\tau\omega_{1}+\tau^{2}\omega_{2}+\cdots.\\\
\end{array}$ (37)

Substituting (37) into (33) yields

$\begin{array}[]{rcl}\tau\frac{1}{c}\big{[}(c-1)y-\phi
y-\tau(\frac{\partial\omega_{1}}{\partial\phi}y+\frac{\partial\omega_{1}}{\partial
y}z)+O(\tau^{2})\big{]}&=&-F+\frac{1}{2}\phi^{2}+\tau\omega_{1}-\tau
y+O(\tau^{2}),\\\ \tau\big{[}y+\tau(\frac{\partial
p_{1}}{\partial\phi}y+\frac{\partial p_{1}}{\partial
y}z)+O(\tau^{2})\big{]}&=&\tau\frac{2}{c}p_{1}+O(\tau^{2}),\\\
\tau\big{[}2y+\tau(\frac{\partial q_{1}}{\partial\phi}y+\frac{\partial
q_{1}}{\partial
y}z)+O(\tau^{2})\big{]}&=&\tau\frac{2}{c}q_{1}+O(\tau^{2}).\end{array}$ (38)

Comparing the coefficients of $\tau$, we have

$\begin{array}[]{rcl}p_{1}&=&\displaystyle\frac{c}{2}y,\\\ q_{1}&=&cy,\\\
\omega_{1}&=&\displaystyle\frac{c-1}{c}y-\frac{1}{c}\phi y+y.\end{array}$ (39)

Hence, slow system restricted to $M_{\tau}$ is

$\begin{cases}\phi^{\prime}=y,\\\
y^{\prime}=\displaystyle\frac{1}{c}\big{[}(c-1)\phi-\frac{1}{2}\phi^{2}+g\big{]}-\tau\frac{1}{c}\big{(}\frac{2c-1}{c}y-\frac{1}{c}\phi
y\big{)}+O(\tau^{2}).\end{cases}$ (40)

Secondly, let us consider the existence of solitary wave solutions. Here,
Melnikov method (see [34, 24]) is employed to detect the existence of solitary
wave solutions under KS perturbation.

The Melnikov function of system (40) is defined by

$\begin{array}[]{rcl}M_{\textmd{hom-
KS}}(c,g)=\displaystyle\frac{1}{c}\displaystyle\oint_{\Gamma}(\frac{2c-1}{c}y^{2}+\frac{c-1}{c}\phi
y^{2})d\xi=\frac{2c-1}{c^{2}}I_{1}-\frac{1}{c^{2}}I_{2},\end{array}$ (41)

where $I_{1}=\displaystyle\oint_{\Gamma}y^{2}d\xi$ and
$I_{2}=\displaystyle\oint_{\Gamma}\phi y^{2}d\xi$. Due to $I_{1}$ and $I_{2}$
are integrated over a closed curve, and the time variable $\xi$ is represented
by the state variable $\phi$ on homoclinic orbit $\Gamma$. Thus, by (25), we
have

$\begin{array}[]{rcl}I_{1}&=&\displaystyle\oint_{\Gamma}y^{2}d\xi=\displaystyle\oint_{\Gamma}yd\phi=2\sqrt{\frac{1}{3c}}\displaystyle\int_{\phi_{r}}^{\phi_{1}}\sqrt{(\phi_{r}-\phi)(\phi-\phi_{1})^{2}}d\phi\\\
&=&2\sqrt{\frac{1}{3c}}\displaystyle\int_{\phi_{r}}^{\phi_{1}}(\phi-\phi_{1})\sqrt{\phi_{r}-\phi}d\phi\\\
&=&-\frac{8}{45}\sqrt{\frac{3}{c}}(\phi_{r}-\phi_{1})^{\frac{5}{2}}\\\
&=&-\frac{24}{5}\sqrt{\frac{1}{c}}(c^{2}-2c+2g+1)^{\frac{5}{4}},\end{array}$
(42)

and

$\begin{array}[]{rcl}I_{2}&=&\displaystyle\oint_{\Gamma}\phi
y^{2}d\xi=\displaystyle\oint_{\Gamma}\phi yd\phi\\\
&=&2\sqrt{\frac{1}{3c}}\displaystyle\int_{\phi_{r}}^{\phi_{1}}\phi\sqrt{(\phi_{r}-\phi)(\phi-\phi_{1})^{2}}d\phi\\\
&=&2\sqrt{\frac{1}{3c}}\displaystyle\int_{\phi_{r}}^{\phi_{1}}\phi(\phi-\phi_{1})\sqrt{\phi_{r}-\phi}d\phi\\\
&=&2\sqrt{\frac{1}{3c}}(\displaystyle\int_{\phi_{r}}^{\phi_{1}}\phi^{2}\sqrt{\phi_{r}-\phi}d\phi-\phi_{1}\displaystyle\int_{\phi_{r}}^{\phi_{1}}\phi\sqrt{\phi_{r}-\phi}d\phi)\\\
&=&-\frac{8\sqrt{3}}{315}\sqrt{\frac{1}{c}}(\phi_{r}-\phi_{1})^{\frac{3}{2}}(4{\phi_{r}}^{2}-\phi_{r}\phi_{1}-3{\phi_{1}}^{2})\\\
&=&-\frac{8}{35}\sqrt{\frac{1}{c}}(15\sqrt{c^{2}-2c+2g+1}+21c-21)(c^{2}-2c+2g+1)^{\frac{5}{4}}.\end{array}$
(43)

Therefore, we substitute (42) and (43) into (41) to obtain the following
expression

$\begin{array}[]{rcl}M_{\textmd{hom-
KS}}(c,g)&=&\frac{8}{35c^{2}}\sqrt{\frac{1}{c}}(15\sqrt{c^{2}-2c+2g+1}+21c-21)(c^{2}-2c+2g+1)^{\frac{5}{4}}\\\
&&-\frac{24(2c-1)}{5c^{2}}\sqrt{\frac{1}{c}}(c^{2}-2c+2g+1)^{\frac{5}{4}}.\end{array}$
(44)

Since the complexity of expression of $M_{\textmd{hom-KS}}(c,g)$, it is hard
to solve $c$ from $M_{\textmd{hom-KS}}(c,g)$ directly. The expression of
$M_{\textmd{hom-KS}}(c,g)$ is simplified by

$\begin{array}[]{rcl}M_{\textmd{hom-
KS}}(c,g)=\frac{24}{5c^{2}}\sqrt{\frac{1}{c}}(c^{2}-2c+2g+1)^{\frac{5}{4}}M^{*}_{\textmd{hom-
KS}}(c,g),\end{array}$ (45)

where $M^{*}_{\textmd{hom-KS}}(c,g)=\frac{5}{7}\sqrt{c^{2}-2c+2g+1}-c$.
Because $c>0$ and $(c-1)^{2}>-2g$, it is easy to know that whether
$M_{\textmd{hom-KS}}(c,g)=0$ has a simple zero if only if $M^{*}_{\textmd{hom-
KS}}(c,g)=0$ has one.

Next, let us discuss the zero of $M^{*}_{\textmd{hom-KS}}(c,g)=0$. We have

$\begin{array}[]{rcl}\frac{\partial M^{*}_{\textmd{hom-KS}}(c,g)}{\partial
c}=\frac{5(c-1)}{7\sqrt{c^{2}-2c+2g+1}}-1.\end{array}$ (46)

To proceed, we divide into two cases.

Case (a) $g<0$. Since $(c-1)^{2}>-2g$, it implies that $c>1+\sqrt{-2g}$ or
$0<c<1-\sqrt{-2g}$. For $c>1+\sqrt{-2g}$, we see that $\frac{\partial
M^{*}_{\textmd{hom-KS}}(c,g)}{\partial c}>0$ when
$c\in(1+\sqrt{-2g},1+\frac{7}{6}\sqrt{-3g})$ and $\frac{\partial
M^{*}_{\textmd{hom-KS}}(c,g)}{\partial c}<0$ when
$c\in(1+\frac{7}{6}\sqrt{-3g},+\infty)$. Obviously, $M^{*}_{\textmd{hom-
KS}}(1+\frac{7}{6}\sqrt{-3g},g)$ is the maximum for $c>1+\sqrt{-2g}$.
Consequently, $M^{*}_{\textmd{hom-KS}}(c,g)\leq\max M^{*}_{\textmd{hom-
KS}}(c,g)=M^{*}_{\textmd{hom-
KS}}(1+\frac{7}{6}\sqrt{-3g},g)=-\frac{4\sqrt{3}}{7}\sqrt{-g}-1<0$. Therefore,
for any $c>1+\sqrt{-2g}$, the nonexistence of the simple zero of
$M^{*}_{\textmd{hom-KS}}(c,g)$ implies that $M_{\textmd{hom-KS}}(c,g)$ has no
simple zero.

For $0<c<1-\sqrt{-2g}$ (_i.e._ , $-\frac{1}{2}<g<0$), we obtain that
$\frac{\partial M^{*}_{\textmd{hom-KS}}(c,g)}{\partial c}<0$ when
$c\in(0,1-\sqrt{-2g})$. Clearly, $M^{*}_{\textmd{hom-KS}}(c,g)$ is
monotonically decreasing about $c\in(0,1-\sqrt{-2g})$ for any
$-\frac{1}{2}<g<0$. When $c\rightarrow 0$, then $M^{*}_{\textmd{hom-
KS}}(c,g)\rightarrow\frac{5}{7}\sqrt{2g+1}>0$. When $c\rightarrow
1-\sqrt{-2g}$, and $M^{*}_{\textmd{hom-KS}}(c,g)\rightarrow-1+\sqrt{-2g}<0$.
Further, $M^{*}_{\textmd{hom-KS}}(c,g)$ is continuous in
$c\in(0,1-\sqrt{-2g})$ for any $-\frac{1}{2}<g<0$. Therefore, the existence of
a simple zero of $M^{*}_{\textmd{hom-KS}}(c,g)$ implies that $M_{\textmd{hom-
KS}}(c,g)$ has one simple zero.

In summary, $M_{\textmd{hom-KS}}(c,g)$ has a simple zero for any
$0<c<1-\sqrt{-2g}$ when $-\frac{1}{2}<g<0$ (see Fig. 3(a)).

Case (b) $g\geq 0$. Obviously, we have $\frac{\partial M^{*}_{\textmd{hom-
KS}}(c,g)}{\partial c}<0$ for any $c>0$ when $g\geq 0$. Thus,
$M^{*}_{\textmd{hom-KS}}(c,g)$ is monotone about $c>0$ for any $g\geq 0$.
Then, the existence of a simple zero of $M^{*}_{\textmd{hom-KS}}(c,g)$ implies
that $M_{\textmd{hom-KS}}(c,g)$ has one simple zero (see Fig. 3(b) and Fig.
3(c)).

(a) $g=-0.2$ | (b) $g=0$ | (c) $g=2$
---|---|---

Fig. 3 The algebraic curve of $M^{*}_{\textmd{hom-KS}}(c,g)$ under different
value $g$.

Further, we plot the algebraic curve of $M_{\textmd{hom-KS}}(c,g)$ with
respect to $c$ by taking different values $g$ (see Fig. 4).

(a) $g=-0.2$ | (b) $g=0$ | (c) $g=2$
---|---|---

Fig. 4 The algebraic curve of $M_{\textmd{hom-KS}}(c)$ under different value
$g$.

With the help of Maple, we calculate the value of $c$. The Fig. 4(b) shows
that the curve intersects the $c$-axis at two points, which yields that there
exist two roots $c_{1}=0.4166666667$ and $c_{2}=1$. However, if $g=0$ and
$c_{2}=1$, it contradicts inequality $(c-1)^{2}>-2g$ such that we just take
unique $c_{1}=0.4166666667$.

Therefore, there exists an unique $c^{\ast}>0$ such that

$M_{\textmd{hom-KS}}(c^{\ast},g)=0\ \textmd{and}\ \frac{\partial
M_{\textmd{hom-KS}}(c^{\ast},g)}{\partial c}\neq 0.$ (47)

###### Proposition 2

For any $(c-1)^{2}>-2g$ and $0<\tau\ll 1$, there exists a function
$c(g,\tau)=c(g)+O(\tau)$ such that system (40) with $c=c(g,\tau)$ possesses a
homoclinic orbit near $\Gamma$.

_Proof._ According to [25, Chapter 4, Theorem 4.2.1] and (47) that system (40)
with $c(g,\tau)=c(g)+O(\tau)$ has a homoclinic orbit near $\Gamma$. This
completes the proof.

###### Theorem 2

For any $(c-1)^{2}>-2g$, there exists a function $c(g,\tau)=c(g)+O(\tau)$ such
that Eq. (11) with KS perturbation has a solitary wave solution
$u=u(x,t,g,\tau)$ with a wave speed $c=c(g,\tau)$. Moreover, if
$\tau\rightarrow 0$, the solitary wave solution $u=u(x,t,g,\tau)$ converge to
solitary wave solution (29) of the unperturbed Eq. $(\ref{PRLW})\mid_{\tau=0}$
with $c=c(g)$.

###  4.2 Solitary wave solution persists with ME perturbation

Similarly, substituting (16) into Eq. (11) with ME perturbation and
integrating both sides once with respect to $\xi$, we obtain

$(1-c)\phi+F+c\phi^{\prime\prime}+\tau(\phi^{\prime}+\phi\phi^{\prime}+\phi^{\prime\prime\prime})=g.$
(48)

Therefore, combining (31), (32) and Eq. (48), we can also obtain a five-
dimensional slow system as follows

$\begin{cases}\phi^{\prime}=y,\\\ y^{\prime}=z,\\\ \tau z^{\prime}=(c-1)\phi-
F-cz+g-\tau(y+\phi y),\\\ c\tau\psi^{\prime}=2\psi-\zeta,\\\
c\tau\zeta^{\prime}=2(\zeta-2\phi),\end{cases}$ (49)

and it corresponds fast system is

$\begin{cases}\dot{\phi}=\tau y,\\\ \dot{y}=\tau z,\\\ \dot{z}=(c-1)\phi-F-
cz+g-\tau(y+\phi y),\\\ \dot{\psi}=\displaystyle\frac{1}{c}(2\psi-\zeta),\\\
\dot{\zeta}=\displaystyle\frac{2}{c}(\zeta-2\phi).\end{cases}$ (50)

Obviously, the critical manifold $M_{0}$ of system (49) is the same as system
(33) such that it is also a normally hyperbolic manifold.

Same discussions as KS perturbation, we have

$\begin{array}[]{rcl}\widetilde{M}_{\tau}=\\{(\phi,y,z,\psi,\zeta)\in\mathbb{R}^{5}\big{|}\psi=\phi+p(\phi,y,\tau),\zeta=2\phi+q(\phi,y,\tau),\\\
(c-1)\phi-\frac{1}{2}\phi^{2}-cz=\widetilde{\omega}(\phi,y,\tau)\\},\end{array}$
(51)

where
$\widetilde{\omega}(\phi,y,\tau)=\tau(\displaystyle\frac{c-1}{c}y-\frac{1}{c}\phi
y+y+\phi y)+O(\tau^{2})$.

Hence, slow system restricted to $\widetilde{M}_{\tau}$ is presented by

$\begin{cases}\phi^{\prime}=y,\\\
y^{\prime}=\displaystyle\frac{1}{c}\big{[}(c-1)\phi-\frac{1}{2}\phi^{2}+g\big{]}-\tau\frac{1}{c}\big{(}\frac{2c-1}{c}y+\frac{c-1}{c}\phi
y\big{)}+O(\tau^{2}).\end{cases}$ (52)

The Melnikov function of system (52) is given by

$\begin{array}[]{rcl}M_{\textmd{hom-
ME}}(c,g)&=&\displaystyle\frac{1}{c}\displaystyle\oint_{\Gamma}(\frac{2c-1}{c}y^{2}+\frac{c-1}{c}\phi
y^{2})d\xi=\frac{2c-1}{c^{2}}I_{1}+\frac{c-1}{c^{2}}I_{2}\\\
&=&-\frac{8(c-1)}{35c^{2}}\sqrt{\frac{1}{c}}(15\sqrt{c^{2}-2c+2g+1}+21c-21)(c^{2}-2c+2g+1)^{\frac{5}{4}}\\\
&&-\frac{24(2c-1)}{5c^{2}}\sqrt{\frac{1}{c}}(c^{2}-2c+2g+1)^{\frac{5}{4}}.\end{array}$
(53)

The expression of $M_{\textmd{hom-ME}}(c,g)$ is simplified by

$\begin{array}[]{rcl}M_{\textmd{hom-
ME}}(c,g)=-\frac{24}{5c^{2}}\sqrt{\frac{1}{c}}(c^{2}-2c+2g+1)^{\frac{5}{4}}M^{*}_{\textmd{hom-
ME}}(c,g),\end{array}$ (54)

where $M^{*}_{\textmd{hom-
ME}}(c,g)=\frac{5(c-1)}{7}\sqrt{c^{2}-2c+2g+1}+c^{2}$. Similarly, we see that
whether $M_{\textmd{hom-ME}}(c,g)=0$ has a simple zero if only if
$M^{*}_{\textmd{hom-ME}}(c,g)=0$ has one.

Next, let us discuss the zero of $M^{*}_{\textmd{hom-ME}}(c,g)=0$. We have

$\begin{array}[]{rcl}\frac{\partial M^{*}_{\textmd{hom-ME}}(c,g)}{\partial
c}=\frac{2}{7}\frac{7c\sqrt{c^{2}-2c+2g+1}+5(c-1)^{2}+5g}{\sqrt{c^{2}-2c+2g+1}}.\end{array}$
(55)

To proceed, we divided into two cases.

Case ($\widetilde{a}$) $g<0$. Since $(c-1)^{2}>-2g$, it implies that
$c>1+\sqrt{-2g}$ or $0<c<1-\sqrt{-2g}$. For $c>1+\sqrt{-2g}$, we obtain that
$\frac{\partial M^{*}_{\textmd{hom-ME}}(c,g)}{\partial c}>0$ when
$c\in(1+\sqrt{-2g},+\infty)$. Obviously, $M^{*}_{\textmd{hom-
ME}}(1+\sqrt{-2g},g)$ is the minimum for $c>1+\sqrt{-2g}$. Consequently,
$M^{*}_{\textmd{hom-ME}}(c,g)\geq\min M^{*}_{\textmd{hom-
ME}}(c,g)=M^{*}_{\textmd{hom-ME}}(1+\sqrt{-2g},g)=(1+\sqrt{-2g})^{2}>0$.
Therefore, for any $c>1+\sqrt{-2g}$, the nonexistence of the simple zero of
$M^{*}_{\textmd{hom-ME}}(c,g)$ implies that $M_{\textmd{hom-ME}}(c,g)$ has no
simple zero.

For $0<c<1-\sqrt{-2g}$ (_i.e._ , $-\frac{1}{2}<g<0$), it obtains that
$\frac{\partial M^{*}_{\textmd{hom-ME}}(c,g)}{\partial c}>0$ when
$c\in(0,1-\sqrt{-2g})$. Clearly, $M^{*}_{\textmd{hom-ME}}(c,g)$ is
monotonically decreasing about $0<c<1-\sqrt{-2g}$ for any $-\frac{1}{2}<g<0$.
When $c\rightarrow 0$, then $M^{*}_{\textmd{hom-
ME}}(c,g)\rightarrow-\frac{5}{7}\sqrt{2g+1}<0$. When
$c\rightarrow+1-\sqrt{-2g}$, and $M^{*}_{\textmd{hom-
ME}}(c,g)\rightarrow(-1+\sqrt{-2g})^{2}>0$. Further, $M^{*}_{\textmd{hom-
ME}}(c,g)$ is continuous in $c\in(0,1-\sqrt{-2g})$ for any $-\frac{1}{2}<g<0$.
Then, the existence of a simple zero of $M^{*}_{\textmd{hom-ME}}(c,g)$ implies
that $M_{\textmd{hom-ME}}(c,g)$ has one simple zero.

In summary, $M_{\textmd{hom-ME}}(c,g)$ has a simple zero for any
$0<c<1-\sqrt{-2g}$ when $-\frac{1}{2}<g<0$ (see Fig. 5(a)).

Case ($\widetilde{b}$) $g\geq 0$. Obviously, we have that $\frac{\partial
M^{*}_{\textmd{hom-ME}}(c,g)}{\partial c}>0$ for any $c>0$ when $g\geq 0$.
Clearly, $M^{*}_{\textmd{hom-ME}}(c,g)$ is monotonically decreasing about
$c>0$ when $g\geq 0$. When $c\rightarrow 0$, then $M^{*}_{\textmd{hom-
ME}}(c,g)\rightarrow-\frac{5}{7}\sqrt{2g+1}<0$. When $c\rightarrow+\infty$,
and $M^{*}_{\textmd{hom-ME}}(c,g)\rightarrow+\infty$. Moreover,
$M^{*}_{\textmd{hom-ME}}(c,g)$ is continuous in $c\in(0,+\infty)$ for any
$g\geq 0$. Thus, the existence of a simple zero of $M^{*}_{\textmd{hom-
ME}}(c,g)$ implies that $M_{\textmd{hom-ME}}(c,g)$ has one simple zero.

We conclude that $M_{\textmd{hom-ME}}(c,g)$ has a simple zero for any $c>0$
when $g\geq 0$ (see Fig. 5(b) and Fig. 5(c)).

(a) $g=-0.2$ | (b) $g=0$ | (c) $g=2$
---|---|---

Fig. 5 The algebraic curve of $M^{*}_{\textmd{hom-ME}}(c,g)$ under different
value $g$.

Then, we plot the algebraic curve of $M_{\textmd{hom-ME}}(c,g)$ with respect
to $c$ by taking different values $g$ (see Fig. 6).

(a) $g=-0.2$ | (b) $g=0$ | (c) $g=2$
---|---|---

Fig. 6 The algebraic curve of $M_{\textmd{hom-ME}}(c)$ under different value
$g$.

The same discussion as before, there exists an unique $c^{\ast}$ such that

$M_{\textmd{hom-ME}}(c^{\ast},g)=0\ \textmd{and}\ \frac{\partial
M_{\textmd{hom-ME}}(c^{\ast},g)}{c}\neq 0.$ (56)

###### Proposition 3

For any $(c-1)^{2}>-2g$ and $0<\tau\ll 1$, there exists a function
$c(g,\tau)=c(g)+O(\tau)$ such that system (52) with $c=c(g,\tau)$ possesses a
homoclinic orbit near $\Gamma$.

_Proof._ Similarly, according to [25, Chapter 4, Theorem 4.2.1] and (56) that
system (52) with $c(g,\tau)=c(g)+O(\tau)$ has a homoclinic orbit near
$\Gamma$. This completes the proof.

###### Theorem 3

For any $(c-1)^{2}>-2g$, there exists a function $c(g,\tau)=c(g)+O(\tau)$ such
that Eq. (11) with ME perturbation has a solitary wave solution
$u=u(x,t,g,\tau)$ with a wave speed $c=c(g,\tau)$. Moreover, if
$\tau\rightarrow 0$, the solitary wave solution $u=u(x,t,g,\tau)$ converge to
solitary wave solution (29) of the unperturbed Eq. $(\ref{PRLW})\mid_{\tau=0}$
with $c=c(g)$.

##  5 Numerical simulations, comparison and conclusions

In this section, numerical simulations by maple are employed to verify the
theoretical results given in previous sections. Taking $\tau=0.01$, we have

Case I. KS perturbation

Let $g=-0.2$, $g=0$ and $g=2$. By (44), we obtain $c(-0.2)\approx
0.2661295689$, $c(0)\approx 0.4166666667$ and $c(2)\approx 1.466998871$,
respectively. According to Theorem 2, taking
$c=c(-0.2)+0.0001=0.2660295689+0.0001$ and
$c=c(0)+0.0001=0.4166666667+0.0001$, we set initial value to be
$(\phi(0),y(0))=(\phi_{r}-10^{-4},0)$ which the homoclinic orbit would pass
through. The phase portraits $(\phi,y)$ and time history curves $(\xi,\phi)$
of system (40) are plotted in Fig. (a) and Fig. (c) of table 1. Similarly,
taking $c=c(2)+0.0001=1.466998871+0.0001$, we set initial value to be
$(\phi(0),y(0))=(\phi_{r}-10^{-3},0)$, the phase portraits $(\phi,y)$ and time
history curves $(\xi,\phi)$ of system (40) are plotted in Fig. (e) of table 1.

The results show us that the homoclinic orbit of system (40) and solitary wave
of Eq. (11) with suitable speed $c$ still persist under the KS perturbation.
Thus, we verify the correctness of Proposition 2 and Theorem 2.

Case II. ME perturbation

Set $g=-0.2$, $g=0$ and $g=2$. Using (53), we have $c(-0.2)\approx
0.3286393802$, $c(0)\approx 0.4580398915$ and $c(2)\approx 0.6801960271$,
respectively. By Theorem 3, taking $c=c(-0.2)+0.0001=0.3286393802+0.0001$ and
$c=c(0)+0.0001=0.4580398915+0.0001$, we set initial value to be
$(\phi(0),y(0))=(\phi_{r}-10^{-4},0)$ which the homoclinic orbit would pass
through. The phase portraits $(\phi,y)$ and time history curves $(\xi,\phi)$
of system (52) are presented in in Fig. (b) and Fig. (d) of table 1.
Similarly, taking $c=c(2)+0.0001=0.6801960271+0.0001$, we set initial value to
be $(\phi(0),y(0))=(\phi_{r}-10^{-3},0)$, the phase portraits $(\phi,y)$ and
time history curves $(\xi,\phi)$ of system (52) are plotted in Fig. (f) of
table 1.

Similarly, the homoclinic orbit of travelling wave system (52) and solitary
wave of Eq. (11) with suitable speed $c$ still persist under the ME
perturbation. Thus, the correctness of Proposition 3 and Theorem 3 are
verified.

Table 1: Phase portraits of travelling system (40) and time history are presented for two kinds of perturbations under $\tau=0.01$. $g$ | The perturbation of KS | The perturbation of ME
---|---|---
$g=-0.2$ |  | (a1) $c=0.2660295689+0.0001$ (a2) $(\xi,\phi)$
---
(a) Initial value $(\phi(0),y(0))=(\phi_{r}-10^{-4},0)$
| (b1) $c=0.3286393802+0.0001$ (b2) $(\xi,\phi)$
---
(b) Initial value $(\phi(0),y(0))=(\phi_{r}-10^{-4},0)$
$g=0$ |  | (c1) $c=0.4166666667+0.0001$ (c2) $(\xi,\phi)$
---
(c) Initial value $(\phi(0),y(0))=(\phi_{r}-10^{-4},0)$
| (d1) $c=0.4580398915+0.0001$ (d2) $(\xi,\phi)$
---
(d) Initial value $(\phi(0),y(0))=(\phi_{r}-10^{-4},0)$
$g=2$ |  | (e1) $c=1.466998871+0.0001$ (e2) $(\xi,\phi)$
---
(e) Initial value $(\phi(0),y(0))=(\phi_{r}-10^{-3},0)$
| (f1) $c=0.6801960271+0.0001$ (f2) $(\xi,\phi)$
---
(f) Initial value $(\phi(0),y(0))=(\phi_{r}-10^{-3},0)$

###### Remark 1

The above simulations and the figures confirm the following facts:
1\. Indeed, the solitary wave persists under small perturbations, whether the
perturbations are KS or ME.
2\. The different perturbations do affect the proper wave speed $c$ ensuring
the persistence of the solitary waves. The essential reason is that the
nonlinear term $(uu_{x})_{x}$ induces different representations of Melnikov
functions. Consequently, solitary wave solutions of corresponding system exist
at different proper wave speed $c$ for different kinds of small perturbations.

##  6 Acknowledgement

The authors would like to thank Professor A.J. Roberts for his valuable
discussions when they are preparing the manuscript. This work was jointly
supported by the National Natural Science Foundation of China under Grant (No.
11671176, 11931016), Natural Science Foundation of Fujian Province under Grant
(No. 2021J011148), Fujian Province Young Middle-Aged teachers education
scientific research project (No. JAT210454, No. JAT200670) and Teacher and
Student Scientific Team Fund of Wuyi University (Grant No. 2020-SSTD-003).

##  7  Conflict of Interest

The authors declare that they have no conflict of interest.

##  8 Data Availability Statement

No data was used for the research described in the article.

##  9 Author contributions

All the authors have the same contributions to the paper.

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