2.1 Preliminaries

In this subsection, we recall the Fenichel invariant manifold theorem for slow–fast systems, or singularly perturbed systems. Consider the slow–fast systems

(2.1)\begin{equation} \begin{array}{c} x^{\prime}(t)=f(x,y,\varepsilon ),\\ y^{\prime}(t)=\varepsilon g(x,y,\varepsilon ), \end{array}\quad (x,y)\in\mathbb{R}^n\times\mathbb{R}^m \end{equation}

where $\varepsilon >0$ is a small real parameter, and $f$ and $g$ are $C^{\infty }$ functions in their variables. Here, the prime is the derivative in the time $t$. For $\varepsilon \ne 0$, after the time rescaling $\tau =\varepsilon t$, system (2.1) can be written as

(2.2)\begin{equation} \begin{array}{c} \varepsilon \dot x(\tau) =f(x,y,\varepsilon ), \\ \dot y (\tau) = g(x,y,\varepsilon ), \end{array}\quad (x,y)\in\mathbb{R}^n\times\mathbb{R}^m. \end{equation}

Systems (2.1) and (2.2) are called, respectively, the fast and slow systems, and their limits when $\varepsilon \rightarrow 0$

\[ \begin{array}{c} x^{\prime }(t)=f(x,y,0), \\ y^{\prime }(t)=0, \end{array}\quad (x,y)\in\mathbb{R}^n\times\mathbb{R}^m \]

and

\[ \begin{array}{c} 0=f(x,y,0), \\ \dot y (\tau)= g(x,y,0), \end{array}\quad (x,y)\in\mathbb{R}^n\times\mathbb{R}^m. \]

are called the layer and reduced systems, respectively.

The set $\Omega :=\{(x,y)\in \mathbb {R}^n\times \mathbb {R}^m|\ f(x,y,0)=0\}$ is called a critical set, and it plays a key role in the study of dynamics of system (2.1). Choose a connected branch $\Omega _0$ of $\Omega$. If the Jacobian matrix of $f(x,y,0)$ with respect to $x$ restricted to $\Omega _0$ has all eigenvalues with nonvanishing real parts, we call $\Omega _0$ normally hyperbolic.

A subset $M$ of $\mathbb {R}^n\times \mathbb {R}^m$ is locally invariant under the flow of system (2.1) if it has a neighbourhood $U$, as long as an orbit of the system leaves $M$, it will leave $U$.

We now state the Fenichel first invariant manifold theorem [Reference Du, Li and Li9, Reference Fenichel13, Reference Jones21], which will be one of our main tools in the next proofs of theorems 2.1 and 2.2.

Fenichel first invariant manifold theorem. Assume that

• • $M_0$ is a compact, normally hyperbolic critical manifold of system (2.1), and

• • it has a coordinate expression $M_0:=\{x=\psi (y)|\ y\in K_0\}$ with $K_0\subset \mathbb {R}^m$ compact and $\psi (y)\in C^\infty (K_0)$.

Then for $\varepsilon >0$ sufficiently small, system (2.1) has a locally invariant $C^r$ manifold $M_{\varepsilon }$, with any prescribed $r\in \mathbb {N}$, which has the expression $x=\psi _\varepsilon (y)$, $x\in K_0$, such that

\[ \lim\limits_{\varepsilon\rightarrow 0} \psi_{\varepsilon}(y)=\psi(y),\quad y\in K_0. \]

Restricted to $M_\varepsilon$, system (2.1) is reduced to

(2.3)\begin{equation} \dot y(\tau)=g(\psi_\varepsilon(y),y,\varepsilon), \end{equation}

which, as $\varepsilon \rightarrow 0$, has the limit $\dot y(\tau )=g(\psi (y),y,0)$.

The Fenichel first invariant manifold theorem indicates that in the normally hyperbolic case, system (2.3) is a regular perturbation of the reduced system $\dot y(\tau )=g(\psi (y),y,0)$. Consequently, all hyperbolic and structurally stable objectives of the reduced system will be preserved by system (2.3) for $0<\varepsilon \ll 1$.

2.2 Statement of the main results

In this subsection, we present our main results. Since we want to find the travelling wave solution of equation (1.6), we set $U(x,t) = \phi (\xi )=\phi (x-ct)$. Then equation (1.6) becomes the next ordinary differential equation

(2.4)\begin{equation} -c\phi'+a_{0}({\eta_{1}{\phi}^2})' +a_{1}\phi'''+\tau \phi''''=0, \end{equation}

where $'=\frac {d}{d\xi },$ and

\[ \displaystyle \eta_{1}=\int _{0}^{+\infty}\displaystyle\frac{1}{\tau}\,{\rm e}^{-\frac{s}{\tau}}\phi(\xi+cs)\,{\rm d}s. \]

Integrating equation (2.4) with respect to $\xi$ and omitting the integral constant, we obtain the following equation

(2.5)\begin{equation} -c\phi+a_{0}{\eta}_{1}{\phi}^2+a_{1}\phi''+\tau \phi'''=0. \end{equation}

After the rescalings $\phi (\xi )=\sqrt {c}u(z)$ and $\xi =\frac {z}{\sqrt {c}}$, equation (2.5) can be further written in

(2.6)\begin{equation} -u+a_{0}{\eta}{u}^2+a_{1}\ddot{u}+\sqrt{c}\tau\dddot{u}=0. \end{equation}

Here the dot denotes the derivative in $z$, and

(2.7)\begin{equation} \displaystyle \eta=\int _{0}^{+\infty}\frac{1}{\tau}\,{\rm e}^{-\frac{s}{\tau}}u(z+c^{\frac{3}{2}}s)\,{\rm d}s. \end{equation}

Direct computations show that $\eta \rightarrow u$ as $\tau \rightarrow 0.$ Thus, when $\tau \rightarrow 0,$ equation (2.6) has the limit

(2.8)\begin{equation} -u+a_{0}{u}^3+a_{1}\ddot{u}=0, \end{equation}

which is called unperturbed equation of equation (2.6). Equation (2.8) can be written in an equivalent way as the following Hamiltonian system

\[ \dot{u} =v, \quad \dot{v}=wu-b{u}^3, \]

with the Hamiltonian

\[ H(u,v)= \displaystyle\frac{{v}^2}{2}-\frac{w{u}^2}{2} +\frac{b{u}^{4}}{4}, \]

where $w= {a_{1}^{-1}}>0$ and $b= {a_{0}}{a_{1}^{-1}}>0.$ The level curve $H(u,v)=h$ is

• • a homoclinic orbit of the Hamiltonian system at the origin in the half plane $u>0$ when $h = 0$,

• • a periodic orbit for each $h\in (\displaystyle - {1}/({4a_{0}a_{1}}),0)$, which is in the interior region enclosed by the homoclinic orbit,

• • the centre when $h=-{1}/({4a_{0}a_{1}})$.

We remark that for $h<- {1}/({4a_{0}a_{1}})$ or $h>0$, the level curve $H(u,v)=h$ is out of our interest.

Now we can state our main results. The first one is on the existence of periodic waves and of a solitary wave, and their limiting properties.

The second one provides a relation between the wave speed and the wavelength of the periodic wave of equation (1.6).

Theorem 2.2 For $0<\tau \ll 1$, the wave speed $c_{0}$ and the wavelength $\lambda _{0}$ of the periodic wave solution to equation (1.6) satisfy

\[ c_{0}=\tilde{c}_{0}(\lambda_{0}),\quad \tilde{c}_{0}'(\lambda_{0})<0, \quad \lambda_{0} \in(a_{1}\pi,+\infty). \]

In the next section, we will prove theorem 2.1. The proof of theorem 2.2 will be given in § 4. In both sections, we will provide more information on the properties of the periods and the wave speeds of the periodic waves, respectively.

3.1 Existence of periodic waves

Write the third-order differential equation (2.6) in an equivalent way as a system of the first-order ordinary differential equations

(3.1)\begin{equation} \begin{aligned} \dot{u} & =v,\\ \dot{v} & =p,\\ \sqrt{c}\tau\dot{p} & =u-a_{0}\eta u^2-a_{1}p. \end{aligned} \end{equation}

Differentiating expression (2.7) with respect to $z$ yields $\displaystyle c^{\frac {3}{2}}\tau \dot {\eta }=\eta -u.$ It together with (3.1) forms the following system

(3.2)\begin{equation} \begin{aligned} \dot{u} & =v,\\ \dot{v} & =p,\\ \sqrt{c}\tau\dot{p} & =u-a_{0}\eta u^2-a_{1}p,\\ c^{\frac{3}{2}}\tau \dot{\eta} & =\eta-u. \end{aligned} \end{equation}

For $\tau >0$ sufficiently small, system (3.2) is a singularly perturbed system with two fast variables $p$ and $\eta$, and two slow variables $u$ and $v$.

According to the singular perturbation theory, we consider the critical set

\[ M_{0}=\left\{(u,v,p,\eta)|\ p=\ a_1^{{-}1}{u} - {a_{0}}{a_{1}^{{-}1}}\eta u^2,~\eta=u\right\}, \]

which is a two-dimensional smooth manifold. Restricted to this critical manifold, the reduced system of system (3.2) is

(3.3)\begin{equation} \begin{aligned} \dot{u} & =v, \\ \dot{v} & =wu-b{u}^3, \end{aligned} \end{equation}

where $w= {a_{1}^{-1}}>0, b= {a_{0}}{a_{1}^{-1}}>0.$ Obviously, the reduced system (3.3) has three equilibria: $(0,0)$ a saddle, and $(\pm \displaystyle \sqrt { {w}/{b}},0)$ both centres. Note that system (3.3) is a Hamiltonian one, with the Hamiltonian

\[ H(u,v) = \displaystyle\frac{{v}^2}{2}-\frac{w{u}^2}{2} +\frac{b{u}^{4}}{4}, \]

which has two symmetric homoclinic orbits to the origin contained in the level set $H(u,v)=0$, and a family of periodic orbits of the reduced system contained in the level set $H(u,v)=h$ for $h\in (-\displaystyle {{w}^2}/{(4b)},0).$

Now we are back to the full system (3.2). When $\tau \ne 0$, the slow system (3.2) can be written in an equivalent way, via the time rescaling $z=\tau s,$ as the next fast system

(3.4)\begin{equation} \begin{aligned} \dfrac{du}{ds} & =\tau v,\\ \dfrac{dv}{ds} & =\tau p,\\ \sqrt{c}\dfrac{dp}{ds} & =u-a_{0}\eta u^2-a_{1}p,\\ c^{\frac{3}{2}}\dfrac{d\eta}{ds} & =\eta-u. \end{aligned} \end{equation}

Some easy calculations show that the linearized matrix of (3.4) restricted to $M_{0}$ is

\[ \left(\begin{array}{cccc} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0\\ \displaystyle\dfrac{1}{\sqrt{c}}(1-2a_{0}\eta u) & 0 & -\displaystyle\dfrac{a_{1}}{\sqrt{c}} & -\displaystyle\dfrac{a_{0}}{\sqrt{c}}u^2\\ - \displaystyle c^{-\dfrac{3}{2}} & 0 & 0 & \displaystyle c^{-\dfrac{3}{2}} \end{array}\right), \]

which has four eigenvalues: $0,\ 0, - \displaystyle a_{1} c^{-\frac 12},\ \displaystyle c^{-\frac {3}{2}}$. This means that the critical manifold $M_{0}$ is normally hyperbolic. By the Fenichel first invariant manifold theorem, for $0<\tau \ll 1$, system (3.2) has an invariant slow manifold $M_\tau$, which can be expressed as

\[ M_{\tau}=\left\{(u,v,p,\eta)\in \mathbb{R}^{4}|\ p= {a_{1}^{{-}1}}{u}- {a_{0}}a_{1}^{{-}1}\eta u^2+g(u,v,\tau),\ \eta=u+h(u,v,\tau)\right\}, \]

where $g$ and $h$ depend smoothly on their variables, and $g(u,v,0)=h(u,v,0)=0.$ To compute the asymptotic expression of the functions $g$ and $h$, we expand them in the parameter $\tau$ as

(3.5)\begin{equation} \begin{aligned} & g(u,v,\tau)=\tau g_{1}(u,v)+\tau^{2} g_{2}(u,v)+\cdots,\\ & h(u,v,\tau)=\tau h_{1}(u,v)+\tau^{2} h_{2}(u,v)+\cdots. \end{aligned} \end{equation}

Plugging the expression of $M_\tau$ with (3.5) into the slow system (3.2), we obtain

\begin{align*} \sqrt{c}\tau \left(\displaystyle\frac{v}{a_{1}}-\displaystyle\frac{a_{0}}{a_{1}}\dot{\eta} u^2- \displaystyle\frac{2a_{0}}{a_{1}}\eta uv +O(\tau) \right)& ={-}a_{1}g_{1}\tau-a_{1}g_{1}\tau^2+o(\tau^2),\\ \displaystyle c^{\frac{3}{2}}\tau\left(v+\tau(\frac{\partial h_{1}}{\partial u}\dot{u}+\frac{\partial h_{1}}{\partial v}\dot{v})+o(\tau)\right)& =\tau h_{1}(u,v)+\tau^{2} h_{2}(u,v)+o(\tau^2). \end{align*}

Equating the coefficients of $\tau ^1$ in the both sides of the above equations yields

\begin{align*} g_{1}(u,v)& ={-}\displaystyle\frac{\sqrt{c}}{a_{1}}\left(\frac{v}{a_{1}}-\frac{3a_{0}}{a_{1}}u^2v\right),\\ h_{1}(u,v)& =c^{\frac{3}{2}}v. \end{align*}

Then the slow manifold $M_{\tau }$ has a more precise expression

\[ \begin{array}{c} p=\dfrac{u}{a_{1}}-\dfrac{a_{0}}{a_{1}} u^3-\dfrac{\sqrt{c}}{a_{1}}\left(a_{0}cu^2v+\dfrac{v}{a_{1}}-\dfrac{3a_{0}}{a_{1}}u^2v\right)\tau+o(\tau) ,\\ \eta=u+c^{\dfrac{3}{2}}\tau v +o(\tau), \end{array}\quad (u,v)\in\mathbb{R}^2. \]

Restricted to the invariant slow manifold $M_{\tau }$, the slow system (3.2) is reduced to the following two-dimensional system

(3.6)\begin{equation} \begin{aligned} \dot{u} & =v, \\ \dot{v} & =wu-b{u}^3-\displaystyle\frac{\sqrt{c}}{a_{1}}\left(a_{0}cu^2v+\frac{v}{a_{1}} -\frac{3a_{0}}{a_{1}}u^2v\right)\tau +o(\tau), \end{aligned} \end{equation}

with $w=\displaystyle {a_{1}^{-1}}>0$ and $b=\displaystyle {a_{0}}{a_{1}^{-1}}>0.$ By the Fenichel invariant manifold theorem, it follows that system (3.6) is a regular perturbation of system (3.3).

Now we investigate the existence of the periodic orbits of system (3.6), which are the perturbation of the periodic orbits inside the period annulus of the Hamiltonian system (3.3). We should mention that the periodic orbits of system (3.6) is also the periodic orbits of system (3.2), and so they provide the periodic waves of equation (1.6).

To prove the existence of the periodic orbits of system (3.6), we will use the Melnikov method. Fix an initial data $(\alpha, 0)$ with $0 < \alpha <\sqrt {\displaystyle {w}/{b}}$, and let $(u_{\tau }(z), v_{\tau }(z))$ be the solution of system (3.6) satisfying $(u_{\tau }, v_{\tau })(0) =(\alpha, 0)$. Then there exist $z_{1}>0$ and $z_{2}<0$, so that

\[ v_{\tau}(z)> 0 ~\mathrm{for} \ 0 < z < z_{1},\quad v_{\tau}(z_{1}) = 0 \]

and

\[ v_{\tau}(z)< 0\ \mathrm{for} \ z_{2} < z < 0,\quad v_{\tau}(z_{2}) = 0. \]

Let $(\alpha,0)$ be the point on the level curve $H(u_{\tau },v_{\tau })=h$ for $h\in (- {w^2}/{(4b)},0)$. Then $\alpha$ is a monotonic function in $h$. For exhibiting dependence of the solution $(u_{\tau }(z),v_{\tau }(z))$ on all the variables and parameters that are involved, we also write

\[ u_{\tau}(z)=u(\tau,h,c,z),\quad v_{\tau}(z)=v(\tau,h,c,z). \]

Direct calculation by (3.6) gives

\[ \frac{\partial}{\partial z}u(\tau,h,c,0)=0,\quad \frac{\partial^{2}}{\partial z^{2}}u(\tau,h,c,0)>0. \]

According to Carr [Reference Carr4, Chapter 4], we define the function

(3.7)\begin{equation} \Phi(h,c,\tau) :=\displaystyle\int^{ z_{1}}_{ z_{2}}\dot{H}(u_{\tau},v_{\tau})\,{\rm d}z, \end{equation}

where $\dot H$ is the derivative of the Hamiltonian $H$ along the orbit of system (3.6). Some calculations show that

(3.8)\begin{equation} \dot{H}(u_{\tau},v_{\tau})=\displaystyle\frac{\tau \sqrt{c}}{a_{1}}\left(a_{0}c{u}_{\tau}^2{v}_{\tau}^2+\frac{{v}_{\tau}^2}{a_{1}} -\frac{3a_{0}}{a_{1}}{u}_{\tau}^2{v}_{\tau}^2+O(\tau)\right). \end{equation}

Note that $\Phi (h,c,\tau )$ is, in fact, the difference of the values of $H$ at the two first intersecting points of the positive and negative orbits passing $(\alpha,0)$ with the positive $u$-axis, i.e.

\[\Phi(h,c,\tau) = H(u_{\tau}(z_{1}), v_{\tau}(z_{1})) - H(u_{\tau}(z_{2}), v_{\tau}(z_{2})). \]

Hence, $\Phi (h,c,\tau ) =0$ if and only if $(u_{\tau }(z),v_{\tau }(z))$ is a periodic solution of system (3.6).

Our next goal is to find solutions $c=c(h,\tau )$ of $\Phi (h,c,\tau ) =0$. By (3.7) and (3.8), we have the expression

\[ \Phi(h,c,\tau) =\tau \tilde{\Phi}(h,c,\tau), \]

and the limit

\[ \lim_{\tau \rightarrow 0}\tilde{\Phi}(h,c,\tau)=\displaystyle\int^{ z_{10}}_{ z_{20}} \displaystyle\displaystyle\frac{\sqrt{c}}{a_{1}}(a_{0}c{u_{0}}^2{v_{0}}^2+\frac{{v_{0}}^2}{a_{1}} -\frac{3a_{0}}{a_{1}}{u_{0}}^2{v_{0}}^2)\,{\rm d}z=:\sqrt{c}a_1^{{-}2}M(h,c), \]

which follows from the classical qualitative theory, where $(u_{0}, v_{0})$ is the solution of the unperturbed system (3.3) starting from the initial point $(\alpha,0)$, and $z_{10}$ and $z_{20}$ are respectively the positive and negative times of this orbit arriving again at the $u$-axis. After these manipulations, we can write $\Phi (h,c,\tau )$ in (3.7) as

\[ \Phi(h,c,\tau)=\sqrt{c}a_1^{{-}2}M(h,c)\tau+O(\tau^2). \]

The function $M(h,c)$ is called the Melnikov function, or the Pontryagin–Melnikov function.

Note that

\[ \displaystyle\frac{\partial M}{\partial c}(h,c)=a_{0}a_1\displaystyle\int^{ z_{10}}_{ z_{20}} \displaystyle {u_{0}}^2{v_{0}}^2\,{\rm d}z>0. \]

So for each $h=H(\alpha, 0) \in (-\displaystyle {{w}^2}/{(4b)},0)$, the functional equation $M(h,c)=0$ has a unique simple solution $c=c_0(h)$, whose precise expression will be obtained from (3.10) below. Hence, it follows from the implicit function theorem that for each $h=H(\alpha, 0) \in (-\displaystyle {{w}^2}/{(4b)},0)$, the functional equation $\Phi (h,c,\tau )=0$ has a unique simple solution $c=c(h,\tau )$ defined in a small neighbourhood of $(h,0)$. This proves that for any $h=H(\alpha, 0) \in (-\displaystyle {{w}^2}/{(4b)},0)$, and $0<\tau \ll 1$, if $c=c(h,\tau )$ system (3.6) has an isolated periodic orbit passing $(\alpha,0)$ in the $(u,v)$ plane. Consequently, system (3.2) has an isolated periodic orbit when $c=c(h,\tau )$. It provides a periodic wave of equation (1.6).

Up to now, we have proved statements $(a)$, $(b)$ and $(c)$ of theorem 2.1 on the periodic waves.

Next we will study the properties of the wave speed $c(h,\tau )$ associated to the periodic wave of equation (1.6).

3.2 Properties of the wave speed of the periodic wave

This subsection is to investigate the properties of the limit wave speed $c_{0}(h)$ of $c(\tau,h)$ as ${\tau \rightarrow 0}$, where $c(\tau,h)$ is the wave speed of the periodic wave to equation (1.6), which is determined by the solutions $(u_{\tau }(z),v_{\tau }(z))=(u(\tau,h,c(h,\tau ),z),v(\tau,h,c(h,\tau ), z))$ of system (3.2) with $(u_{\tau }(0),v_{\tau }(0))=(\alpha,0)$. Recall that the relation between $\alpha$ and $h$ is given by $H(\alpha,0)=h\in (-\displaystyle {{w}^2}/{(4b)},0)$,

First, for $h=H(\alpha, 0) \in (-\displaystyle {{w}^2}/{(4b)},0)$, one has

\begin{align*} \int^{ z_{10}}_{ z_{20}}{u_{0}}^2{v_{0}}^2\,{\rm d}z& = \int^{ z_{10}}_{ z_{20}}({u_{0}}^2-\displaystyle\frac{w}{b})\dot{u}_{0}\,{\rm d}u_{0}+\frac{w}{b}\int^{ z_{10}}_{ z_{20}}{v_{0}}^2\,{\rm d}z \\ & ={-}2\int^{ z_{10}}_{ z_{20}}{u_{0}}^2{v_{0}}^2\,{\rm d}z+\displaystyle\frac{1}{b}\int^{ z_{10}}_{ z_{20}}{\ddot{u}_{0}}^2\,{\rm d}z+\displaystyle\frac{w}{b}\int^{ z_{10}}_{ z_{20}}{v_{0}}^2\,{\rm d}z. \end{align*}

It follows that

\[ \displaystyle\int^{ z_{10}}_{ z_{20}}{u_{0}}^2{v_{0}}^2\,{\rm d}z=\displaystyle\displaystyle\frac{1}{3b}\int^{ z_{10}}_{ z_{20}}{\ddot{u}_{0}}^2\,{\rm d}z+\displaystyle\frac{w}{3b}\int^{ z_{10}}_{ z_{20}}{v_{0}}^2\,{\rm d}z. \]

Then the equation $M(h,c)=0$ is equivalent to

(3.9)\begin{equation} (a_{0}a_{1}c -3a_{0})\int^{ z_{10}}_{ z_{20}}{u_{0}}^2{{v}_{0}}^2\,{\rm d}z+\int^{ z_{10}}_{ z_{20}}{{v}_{0}}^2\,{\rm d}z=0, \end{equation}

and is further equivalent to

(3.10)\begin{equation} \frac{1}{a_{0}a_{1}c-3a_{0}}={-}\displaystyle\frac{1}{3b}\frac{\int^{ z_{10}}_{ z_{20}}{\ddot{u}_{0}}^2\,{\rm d}z}{\int^{ z_{10}}_{ z_{20}}{\dot{u}_{0}}^2\,{\rm d}z}- \frac{w}{3b}, \end{equation}

with $0< c < {3}{a_{1}^{-1}}=3w.$

Next, for simplification to notations, we replace $(u_{0}(z),v_{0}(z))$ by $(u(z),v(z))$. Set $k=4h \in (\displaystyle -{1}/{(a_{0}a_{1})},0)$, and let $\alpha (k)$ and $\beta (k)$ be the two positive real roots of the equation

(3.11)\begin{equation} 2wu^2-bu^4+k=0, \end{equation}

with $0< \alpha (k) < \beta (k)$. Recall that the functional equation (3.11) is, in fact, $H(u,0)=k/4$. Let

\[ v:=E(u)=\sqrt{2wu^2-bu^4+k}, \quad u\in[\alpha(k), \beta(k)], \]

be the closed segment $\gamma _0^+$ of the periodic orbit $\Gamma _0(k)$ over the $u$-axis, which is contained in $H(u,v)=k/4$. Let

\[ P(k):=\displaystyle\frac{1}{2}\displaystyle\int^{ z_{10}}_{ z_{20}}{\ddot{u}}^2\,{\rm d}z,\quad Q(k):=\frac{1}{2}\int^{ z_{10}}_{ z_{20}}{\dot{u}}^2\,{\rm d}z. \]

Then by (3.3) one has

(3.12)\begin{equation} \begin{aligned} P(k) & = \displaystyle\frac{1}{2}\int^{ z_{10}}_{ z_{20}}(wu-bu^3)^2\,{\rm d}z= \sqrt{2}\int^{\beta}_{\alpha} \displaystyle\frac{w^2u^2+b^2u^6-2wbu^4}{E(u)}\,{\rm d}u,\\ Q(k) & = \displaystyle\frac{1}{2}\int^{ z_{10}}_{ z_{20}}{\dot{u}}^2\,{\rm d}z = \displaystyle\frac{\sqrt{2}}{2}\int^{\beta}_{\alpha} E(u)\,{\rm d}u. \end{aligned} \end{equation}

For $k\in (\displaystyle -{1}/{(a_{0}a_{1})},0)$, we denote by $c(k)$ the solution of equation (3.9), instead of the previous $c_0(h)$. The next result characterizes the properties of the limit $c(k)$ of the wave speed of the periodic wave.

Proposition 3.1 The function $c(k)$ has the next properties$:$

\[ c'(k)<0, \quad k \in \left(\displaystyle\frac{-1}{a_0a_{1}},\ 0\right), \]

\[ \lim_{k\rightarrow -{1}/{(a_{0}a_{1})}}c =\displaystyle\frac{2}{a_{1}} , \quad \lim_{k\rightarrow 0}c=\frac{7}{4a_{1}}. \]

To prove this proposition, according to (3.9), (3.10) and (3.12), we define the functions

\[ J_{n}(k)=\displaystyle\int^{\beta}_{\alpha}u^nE(u)\,{\rm d}u,\quad n=0, 1, 2,\cdots. \]

Then

\[ J_{n}'(k)=\displaystyle\frac{1}{2}\displaystyle\int^{\beta}_{\alpha}\frac{u^n}{E(u)}\,{\rm d}u, \]

which follows from $E(u)=\sqrt {2wu^2-bu^4+k}$ and $\frac {{\rm d}E}{{\rm d}k}=\frac {1}{2E}$. Under these notations, $P$ and $Q$ can be written as

(3.13)\begin{equation} \begin{aligned} P & =2\sqrt{2}w^2J_{2}'(k)+2\sqrt{2}b^2J_{6}'(k)-4\sqrt{2}wbJ_{4}'(k),\\ Q & = \frac{\sqrt{2}}{2}J_{0}(k). \end{aligned} \end{equation}

Since $E^2=2wu^2-bu^4+k$, we have

\[ E\displaystyle\frac{{\rm d}E}{{\rm d}u}=2wu-2bu^3. \]

Lemma 3.2

\[ \left(\begin{array}{c} J_{0} \\ J_{1} \\ J_{2} \\ \end{array}\right)=\left(\begin{array}{ccc} \displaystyle\dfrac{4k}{3} & 0 & \dfrac{4w}{3} \\ 0 & \dfrac{w^2}{b}+k & 0 \\ \dfrac{4wk}{15b} & 0 & \dfrac{16w^2+12kb}{15b} \\ \end{array}\right)\left(\begin{array}{c} J_{0}' \\ J_{1}' \\ J_{2}' \\ \end{array}\right) \]

Proof. Hereafter, for simplicity, we omit the upper and lower limits of the integrals. Direct calculations show that

\begin{align*} J_{0}& =\int(2wu^2-bu^4+k)\frac{{\rm d}u}{E} =\int\left(2wu^2-u(wu-\frac{E{\rm d}E}{2{\rm d}u})+k\right)\frac{{\rm d}u}{E}\\ & =w\int\frac{u^2}{E}\,{\rm d}u+\frac{1}{2}\int u\,{\rm d}E+k\int\frac{{\rm d}u}{E} =2wJ_{2}'- \frac{1}{2}J_{0}+2kJ_{0}'. \end{align*}

It induces

\[ J_{0}= \frac{4k}{3}J_{0}'+\frac{4w}{3}J_{2}'. \]

Similar manipulation gives the expressions of $J_1$ and $J_2$ in $J_0'$, $J_1'$ and $J_2'$.

Next result exhibits the expressions of $J_3, J_4, J_5$ and $J_6$ in $J_0, J_1$ and $J_2$.

Lemma 3.3

\begin{align*} J_{3}& =\frac{w}{b}J_{1},\\ J_{4}& =\frac{8w}{7b}J_{2}+\frac{k}{7b}J_{0},\\ J_{5}& =\frac{5w^2+kb}{4b^2}J_{1},\\ J_{6}& =\frac{4wk}{21b^2}J_{0}+\frac{2}{9} \frac{96w^2+21kb}{14b^2}J_{2}. \end{align*}

Furthermore, we need to compute the expressions of the second-order derivatives of $J_0$ and $J_2$.

Lemma 3.4

\[ \left(\begin{array}{c} J_{0}'' \\ J_{2}'' \\ \end{array} \right)=\Delta(k) \left(\begin{array}{ccc} -k & -w - \dfrac{wk}{b} & k \\ \end{array} \right)\left(\begin{array}{c} J_{0}' \\ J_{2}' \\ \end{array}\right),\quad \Delta(k)=\dfrac{b}{4k(w^2+kb)} . \]

Proof. According to lemma 3.2, we have

(3.14)\begin{equation} J=A(k)J', \end{equation}

where $J=(J_{0},J_{2})^ \mathrm {T}$ and

\[ A(k)=\left( \begin{array}{cc} \dfrac{4k}{3} & \dfrac{4w}{3} \\ \dfrac{4wk}{15b} & \dfrac{16w^2+12kb}{15b} \\ \end{array}\right). \]

Then the lemma follows from

\[ J'=A'J'+AJ'', \]

and

\[ A^{{-}1}(E-A')=\Delta(k) \left( \begin{array}{ccc} -k & -w - {wk}{b^{{-}1}} & k \\ \end{array} \right). \]

Here

(3.15)\begin{equation} A^{{-}1}=\dfrac{1}{4k(w^2+kb)}\left( \begin{array}{ccc} 4w^2+3kb & -5bw \ - wk & 5bk \\ \end{array}\right). \end{equation}

Combining (3.13) and lemmas 3.2 and 3.3 together with some direct calculations, one gets

\[ P(k)={-}\frac{\sqrt{2}w}{2}J_{0}+\frac{3\sqrt{2}b}{2}J_{2}. \]

Let

\[ X(k)=\displaystyle\frac{P(k)}{Q(k)} ={-}w+3b\frac{J_{2}}{J_{0}}. \]

By (3.10) one has

(3.16)\begin{equation} c(k)=\frac{3}{a_{1}}\left(1-\frac{1}{a_{1}X(k)+1}\right). \end{equation}

Observe that the monotonicity of $c(k)$ and $X(k)$ is coincident. We turn to study monotonicity of $X(k)$. It is determined clearly by the properties of $J_0$ and $J_2$. Set

\[ B(p, q)=\int ^{1}_{0}x^{p-1}(1-x)^{q-1}\,{\rm d}x,\quad p>0,\ q>0, \]

which is the beta function.

Lemma 3.5

\[ J_{0}(0)=\displaystyle\frac{w\sqrt{2w}}{b}B\left(1,\frac{3}{2}\right), \quad J_{2}(0)=\frac{2w^2\sqrt{2w}}{b^2}B\left(2,\frac{3}{2}\right), \quad \frac{J_{2}(0)}{J_{0}(0)}=\frac{4w}{5b}. \]

Proof. By definition, one has clearly $\alpha (0)=0, \beta (0)=\sqrt { {2w}{b^{-1}}},$ and

\[ J_{0}(0)=\displaystyle\int ^{\sqrt{\frac{2w}{b}}} _{0} \sqrt{2wu^2-bu^4}\,{\rm d}u. \]

Simple calculations show that

\[ J_{0}(0)=\frac{1}{2}\int^{\frac{2w}{b}}_{0}\sqrt{2w-bt}\,{\rm d}t =\frac{\sqrt{2w}}{2}\int^{\frac{2w}{b}}_{0}\sqrt{1-\frac{bt}{2w}}\,{\rm d}t =\frac{w\sqrt{2w}}{b}B\left(1,\frac{3}{2}\right). \]

The expression $J_2(0)$ follows from the similar calculations. Applying the relation between the beta and gamma functions

\[ B(p,q)=\displaystyle\frac{\Gamma(p) \Gamma(q)}{\Gamma(p+q)}, \quad \Gamma(s+1)=s\Gamma(s), \]

yields

\[ \displaystyle\frac{J_{2}(0)}{J_{0}(0)}=\frac{4w}{5b}. \]

Recall that $\Gamma (s)=\int ^{+\infty }_{0}x^{s-1}\,{\rm e}^{-x}\,{\rm d}x$.

Lemma 3.6

\[ \lim_{k \rightarrow -\frac{w^2}{b}}\frac{J_{2}(k)}{J_{0}(k)}=\frac{w}{b}. \]

Proof. Since $0< \alpha (k) \leq u\leq \beta (k)$ along the periodic orbit, one has

\[ \int{\alpha}^2E(u)\,{\rm d}u \leq \int u^2 E(u)\,{\rm d}u \leq \int{\beta}^2E(u)\,{\rm d}u. \]

It follows

\[ {\alpha}^2 \leq \displaystyle\frac{J_{2}(k)}{J_{0}(k)} \leq {\beta}^2. \]

Since ${\alpha }^2 \rightarrow \dfrac {w}{b}\ \mathrm {and}\ {\beta }^2 \rightarrow \dfrac {w}{b}$ as $k \rightarrow -\dfrac {w^2}{b},$ we arrive the conclusion.

Proof. The assumption $X'(k_{0})=0$ is equivalent to $J'_{2}(k_{0})J_{0}(k_{0})=J_{2}(k_{0})J'_{0}(k_{0}),$ i.e.

(3.17)\begin{equation} \frac{J_{2}(k_{0})}{J_{0}(k_{0})}=\frac{J'_{2}(k_{0})}{J'_{0}(k_{0})}, \end{equation}

because ${J_{0}(k_{0})},{J'_{0}(k_{0})}> 0.$ Let

\[ \tilde{X}(k)=\displaystyle\frac{J_{2}(k)}{J_{0}(k)},\quad \hat{X}(k)=\frac{J'_{2}(k)}{J'_{0}(k)}. \]

Then

\[ J_{2} =J_{0}\tilde{X},\quad J_{2}' =J_{0}'\tilde{X}+J_{0}\tilde{X}',\quad J_{2}'' =J_{0}''\tilde{X}+2J_{0}'\tilde{X}'+J_{0}\tilde{X}'', \]

and

\[ J_{2}' =J_{0}'\hat{X},\quad J_{2}'' =J_{0}''\hat{X}+J_{0}'\hat{X}'. \]

Note that the sign of $X''(k_{0})$ coincides with that of $\tilde {X}''(k_{0})$, and that $\tilde {X}'(k_{0})=0$ and $\tilde {X}(k_{0})=\hat {X}(k_{0})$ by (3.17). Then

\begin{align*} \tilde{X}''(k_{0})& = \left. \frac{J_{2}''-J_{0}''\tilde{X}-2J_{0}'\tilde{X}'}{J_{0}} \right|_{k=k_{0}} = \left.\frac{J_{0}''(\hat{X}-\tilde{X})+J_{0}'\hat{X}'}{J_{0}}\right|_{k=k_{0}} =\frac{J_{0}'(k_{0})}{J_{0}(k_{0})}\hat{X}'(k_{0}), \end{align*}

and

\begin{align*} \hat{X}'(k_{0})& = \frac{J_{2}''(k_{0})J_{0}'(k_{0})-J_{2}'(k_{0})J_{0}''(k_{0})}{(J_{0}'(k_{0}))^2}\\ & \mathop{=}\limits^{{\rm lemma~3.4}} \Delta(k_{0})\left(2k_{0}\hat{X}(k_{0})+w\hat{X}^2(k_{0})-\frac{k_{0}w}{b}\right). \end{align*}

Obviously, $2k_{0}\hat {X}(k_{0})+w\hat {X}^2(k_{0})-\displaystyle {k_{0}w}/{b}$ is always positive, and $\Delta (k_{0})$ is negative. It follows that $\hat {X}'(k_{0})<0.$ Consequently

\[ \tilde{X}''(k_{0}) =\displaystyle\frac{J_{0}'(k_{0})}{J_{0}(k_{0})}\hat{X}'(k_{0})<0. \]

Proof. According to lemma 3.2, we have

(3.18)\begin{equation} 3J_{0}=4kJ_{0}'+4wJ_{2}', \end{equation}

and

(3.19)\begin{equation} 15bJ_{2}=4wkJ_{0}'+(16w^2+12kb)J_{2}'. \end{equation}

Subtracting the multiplication of (3.18) by $4w$ with (3.19) gives

\[ 12wJ_{0}-15bJ_{2}=12wkJ_{0}'-12kbJ_{2}'. \]

It can be written, via lemma 3.2, as

\[ 4w-5b\frac{J_{2}}{J_{0}}=4k\left(w\frac{J_{0}'}{J_{0}}-b\frac{J_{2}'}{J_{0}}\right) =4k\left(w-b\frac{J_{2}'}{J_{0}'}\right)\frac{J_{0}'}{J_{0}}. \]

Since $\displaystyle \frac {J_{0}'(k_{0})}{J_{0}(k_{0})}>0$ and $\tilde {X}(k_{0})=\hat {X}(k_{0}),$ we get

\[ 4k_{0}(4w-5b\tilde{X}(k_{0}))(w-b\tilde{X}(k_{0}))>0. \]

This shows

\[ \displaystyle\frac{4w}{5b}<\tilde{X}(k_{0})<\frac{w}{b}. \]

Moreover, by lemmas 3.5 and 3.6, we get

\begin{align*} \lim_{k\rightarrow -\frac{w^2}{b}}X(k)& =\lim_{k\rightarrow -\frac{w^2}{b}}\left({-}w+3b \frac{J_{2}(k)}{J_{0}(k)}\right) =2w,\\ \lim_{k\rightarrow 0}X(k)& =\lim_{k\rightarrow 0}\left({-}w+3b\frac{J_{2}(k)}{J_{0}(k)}\right) =\frac{7w}{5}. \end{align*}

This proves the lemma.

Combining lemmas 3.7 and 3.8 we achieve the next result.

Lemmas 3.7, 3.8 and 3.9 together with equation (3.16) verify the two limits of $c(k)$ in proposition 3.1. This completes the proof of the proposition.

By proposition 3.1 and its proof, we have proved statement $(d)$ of theorem 2.1.

In the next subsection, we will prove statements $(a)$, $(b)$ and $(c)$ of theorem 2.1 on the solitary wave.

3.3 Existence of the solitary wave

Subsection 3.1 provides a proof on the existence of the periodic waves of equation (1.6), which is a result from the perturbation of a period annulus of a Hamiltonian system with the total energy belonging to $(-\displaystyle {{w}^2}/{(4b)},0).$ This subsection is to prove the existence of the solitary wave, which bifurcates from the homoclinic orbit of system (3.3) in the $u>0$ half plane. To prove the existence of the solitary wave of equation (1.6), we only need to prove the existence of the homoclinic orbit of system (3.6) at the origin. Let $\gamma _1:=(u_{1}(z),v_{1}(z))$ be the $1$-dimensional unstable manifold of system (3.6) at the origin with $v_1(z)>0$, and let $\gamma _2:=(u_{2}(z),v_{2}(z))$ be the $1$-dimensional stable manifold of system (3.6) at the origin with $v_2(z)<0$. Then for $0<\tau \ll 1$, the two invariant manifolds will intersect the positive $u$-axis. We denote these two intersecting points as their initial points of the unstable and stable manifolds. Then we have $v_1(0)=0$ and $v_1(z)>0$ for $-\infty < z<0$, and $v_2(0)=0$ and $v(z)<0$ for $0< z<+\infty$.

To prove the existence of a homoclinic orbit at the origin, we need to prove that the stable and unstable manifolds at the origin coincide on the positive $u$-axis. This is the same as $u_1(0)=u_2(0)$, which is equivalent to $H(u_1(0),0)=H(u_2(0),0)$. Recall that $H(u,v)=0$ contains the homoclinic orbit $\gamma _0$ at the origin of the unperturbed system (3.3), and that when $\tau \rightarrow 0$, $\gamma _1\cup \gamma _2$ has $\gamma _0$ as their limit.

To find the conditions ensuring $H(u_1(0),0)=H(u_2(0),0)$, we define

\[ \Psi(c,\tau):=\displaystyle\int^{0}_{-\infty}\dot{H}(u_1,v_1)\,{\rm d}z-\int^{+\infty}_{0}\dot{H}(u_2,v_2)\,{\rm d}z. \]

By the standard Melnikov method together with some calculations, one gets from (3.6) that

\[ \Psi(c,\tau) =\tau M(c)+O(\tau^2), \]

where

\begin{align*} M(c )& =\int^{+\infty}_{-\infty}\left(\begin{array}{c} v \\ wu-bu^3 \\ \end{array}\right) \wedge \left(\begin{array}{c} 0 \ - \displaystyle\dfrac{\sqrt{c}}{a_{1}}(a_{0}cu^2v+\dfrac{v}{a_{1}} -\dfrac{3a_{0}}{a_{1}}u^2v) \\ \end{array}\right) {\rm d}z\\ & ={-}\int^{+\infty}_{-\infty}\displaystyle\frac{\sqrt{c}}{a_{1}}(a_{0}cu^2v^2+\frac{v^2}{a_{1}} -\frac{3a_{0}}{a_{1}}u^2v^2)\,{\rm d}z\\ & ={-}\frac{\sqrt{c}}{a_{1}^2}\left((a_{0}a_{1}c-3a_{0})\int^{+\infty}_{-\infty}u^2v^2\,{\rm d}z+\int^{+\infty}_{-\infty}v^2\,{\rm d}z\right), \end{align*}

where $(u(z),v(z))$ is the homoclinic orbit of system (3.3), the unperturbed system of system (3.6). Recall that the wedge operator $\wedge$ is defined as $f\wedge g=f_{1}g_{2}-f_{2}g_{1}$ for $f=(f_1,f_2)$ and $g=(g_1,g_2)$.

Obviously, the Melnikov function $M(c)$ has a unique positive zero, which is simple, and is denoted by $c=c_{0}$. The implicit function theorem shows that for $0<\tau \ll 1$, $\Psi (c,\tau )=0$ has a unique positive solution $c=c(\tau )$, which satisfies $\lim _{\tau \rightarrow 0}c(\tau )=c_0$. This proves the existence of the homoclinic orbit at the origin for system (3.6) when $c=c(\tau )$ for $0<\tau \ll 1$. Consequently, equation (1.6) has a solitary wave solution with the wave speed $c=c(\tau )$, whose leading term in the expansion in $\tau$ is

\[ c_{0}=\lim_{k\rightarrow 0}\displaystyle\frac{3}{a_{1}}\left(1-\frac{1}{a_{1}X(k)+1}\right)=\frac{7}{4a_{1}}. \]

Here $k$ and $X(k)$ have the same definitions as those in subsection 3.2, with difference only in replacing the limits $\alpha$ and $\beta$ of the integrals by $0$ and $u(0)$, respectively.

Up to now, we have completed the proof of statements $(a)$, $(b)$ and $(c)$ of theorem 2.1 on the solitary wave, and consequently of the theorem.